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*NEW* Backup & Recovery Software Buyer's Guide: GET IT HERE
BUDR Solutions Directory
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Data Protection Solutions Directory
BUDR Buyer's Guide
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Data Protection Buyer's Guide
Enterprise Backup and Disaster Recovery Solutions Directory
Below is a directory of Backup and Disaster Recovery vendors, tools, appliances and software solutions including company overviews, links to social media and contact information for the Top 28 providers. If you would like a printed version of this page including complete solutions profiles and a list of the top questions to ask in an RFP – Request for Proposal, click here for a Free PDF.
Acronis offers backup, disaster recovery, and secure file sync and share solutions. The company also provides data protection in any environment, including virtual, physical, cloud, and mobile. Acronis True Image is personal backup software that enables users to make a duplicate of their system, effectively capturing all of their data for system recovery or disk migration.
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Acronis serves small and medium-sized businesses with less than 25 servers. However, it has at least one client with more than 200 servers under management. In 2018, the provider was named a Gartner Peer Insights Customers' Choice for Data Center Backup and Recovery Software. According to verified user reviews, complex deployments with Acronis are a simple and swift process.
300 Trade Center
Woburn, MA
www.acronis.com
Actifio enables a "data time machine" for the enterprise, eliminating backup and restore windows and creating virtual, point-in-time copies of data on demand, for use by any business application. Instant data recovery means no delays in restoring data for any use, and it's enabled through a simple consumer grade interface that is SLA-driven and application centric in applying data protection. Actifio's differentiated features in backup and recovery deliver functionality that can speed up backup modernization, improve recovery point and recovery time service levels, and apply granular protection at the application level and at the VM level.
Actifio delivers a single platform that integrates disaster recovery, backup and long-term data retention. It protects a range of workloads, including large-scale applications and virtual environments. Actifio pioneered copy data management and offers scalable instant recovery for apps, files and virtual machines.
www.actifio.com
Altaro provides users with Virtual Machine Backup and recovery solutions. The provider's flagship product, Altaro VM Backup, has a straightforward UI that facilitates its use, while being full-featured as to create a strong backup strategy. The solution comes equipped with an outstanding support team determined to help clients succeed in protecting their environment. Everything can be managed and monitored from a single dashboard or by using the vendor's online management console. Altaro was founded in 2009.
Altaro VM Backup is an easy-to-use backup solution for Hyper-V and VMware. The solution boasts VM backup in less than 15 minutes. Though user reviews from G2 Crowd show that large, offsite backups to USB drives can take time to update, users also state that the solution's Continuous Data Protection (CDP) enables them to protect virtual machines as often as every five minutes. Users report quick deployment and implementation times as well.
3737 Glenwood Ave
www.altaro.com
Arcserve Unified Data Protection – Delivers comprehensive Assured Recovery for virtual and physical environments with a next generation unified architecture and an easy to use console. This solution enables organizations to scale their IT environments easily while delivering against their recovery point and recovery time objectives, on premise or in cloud. Arcserve UDP provides a spectrum of enterprise-level functionality such as imaging, replication and high availability. It also enables the automated disaster recovery testing of business-critical systems, applications and data, without business downtime or impact on production systems.
Arcserve Unified Data Protection is most popular in Europe and Asia and appeals to mid-market companies. The solution provides continuous data protection, comprehensive backup, high availability, and disaster recovery for virtual, physical, and cloud environments.
8911 Columbine Rd
www.arcserve.com
Asigra is built for cloud computing environments and designed to offer backup efficiencies by allowing enterprises to capture, ingest and store less data. Designed for compatibility with public, private and hybrid cloud architectures, the Asigra Cloud Backup solution is equipped with agentless software architecture, global deduplication and data compression technology along with NIST FIPS 140-2 certified security.
Asigra Cloud Backup is an enterprise-class backup and recovery solution that provides a single data protection interface for physical servers, virtual machines, endpoint devices, and cloud apps. The company was recently named as one of the 20 Coolest Cloud Storage Vendors of the 2018 Cloud 100 by CRN.
79 Brisbane Rd
www.asigra.com
Creates a fully-deduplicated mirror the business locally and in the cloud so that it can easily be accessed and recover files, folders, applications, servers and even the entire office with one click. Axcient's backup and disaster recovery solution offers granular recovery options, system snapshots, bare metal restores and full virtualization. Axcient protects physical and virtual environments with one backup and disaster recovery solution; Windows, Mac, Linux, VMWare, Hyper-V and XenServer. All recovery points can quickly be accessed, searched or restored.
Axcient typically serves small and mid-size businesses. The vendor's Business Recovery Cloud tool is recommended for smaller physical and virtual x86 environments, while Axcient Fusion is a better fit for larger VMWare environments.
www.axcient.com
A cloud-integrated solution for protecting physical and virtual environments that includes software, appliance, and offsite replication. Barracuda Backup is simple to deploy, easy to manage and offers unlimited cloud storage. With an extensive range of supported environments and integration with Copy file sync and share services, organizations can replace multi-vendor piecemeal backup solutions with an all-in-one Barracuda Backup appliance. Barracuda Backup supports replication to another Barracuda appliance for private-cloud data protection or secure transfer to the Barracuda Cloud.
3175 Winchester Blvd
www.barracuda.com
EVault by Carbonite provides cloud-based backup, recovery and storage solutions to enterprises. EVault centralizes the backup and recovery of data on all computers distributed throughout an organization's locations. Agents automatically back up the data over the internet to a highly secure data center.
Carbonite's Data Protection Platform offers local and cloud backup for physical, virtual and legacy servers. The solution provides high availability for tier-one systems, workload migration to lessen the impact of planned and unplanned downtime, and endpoint protection for highly mobile workforces. The installation and implementation of the solution is simple, according to Gartner Peer Insights user reviewers.
2 Avenue de Lafayette
www.carbonite.com
Cobalt Iron's flagship SaaS-based backup solution, Compass reduces complexity and the amount of time spent on backup. Additionally, the tool improves overall data protection performance. The enterprise data protection platform offers four main product components: Commander, Analytics Engine, Accelerators, and Accelerator Operating System. With this solution, users have access to a range of analytics-driven data management capabilities through a unified web user experience. Additionally, the solution offers ransomware detection, alerting and notification, and remediation capabilities.
Cobalt Iron serves medium to large enterprises with multiple petabytes of data. Cobalt Iron's Compass platform is recommended for enterprises looking to cut CapEx and OpEx expenses, move off of legacy solutions, incorporate cloud workloads, or reduce backup times while increasing backup success rates. Compass was recently named one of DCIG's Top Five Anti-Ransomware Enterprise Backup Solutions, as well as one of CRN's Top 20 Coolest Data Protection Companies.
1421 Research Park Dr
www.cobaltiron.com
Code42 offers backup, disaster recovery, and data loss protection solutions. The provider's data loss protection solution, (Next-Gen DLP), detects insider threats, satisfies regulatory compliance requirements, automatically monitors file activity across computers and cloud, and facilitates incident response. The vendor's security, IT, and compliance professionals have the ability to protect endpoint and cloud data from loss, leak, and theft while maintaining a collaborative culture for employees. Code42is backed by Accel Partners, JMI Equity, NEA, and Split Rock Partners and was founded in 2001.
Code42 offers an intuitive backup solution that does not require training for end-users. This solution is recommended for backing up user data in cloud or on-prem for compliance and simple recovery. The provider is praised for its platform's easy implementation process, as well as the capability to run the tool automatically in the background of business operations. While the platform does not detect bugs and notify the user, incremental backups are smooth and do not require any interaction.
100 Washington Ave S
www.code42.com
Cohesity is a data management company that manages, protects, and extracts value from enterprise data. The provider's flagship tool, Cohesity DataProtect, safeguards a wide range of data sources on a single web-scale platform. The solution can be deployed on-premises on qualified platforms either in the data center or at the edge, or in the public cloud. Additionally, the platform utilizes a scale-out clusterlike design that starts with a minimum of three nodes and scales by adding nodes to the cluster. Through this solution, users have the ability to use backup data directly from the backup appliance without needing to restore it, which allows for the consolidation of secondary storage use cases. Cohesity is based out of San Jose, California with offices and partners around the globe.
Cohesity's DataProtect is recommended primarily for the backup of standard enterprise applications and VM infrastructures. However, the vendor also exports the backup data in native formats to allow for additional use cases, such as test and development, disaster recovery, and analytics. Though Cohesity does not support backup for databases hosted in VMware raw device mapping logical unit numbers, the vendor offers an integrated appliance that scales both backup performance and storage capacity linearly, along with cloud tiering, replication, and recovery capabilities. Additionally, Cohesity provides 24/7 global support with one-hour response times.
www.cohesity.com
Commvault's disaster recovery for cloud data offers cost-effective disaster recovery on-premises, in the cloud and virtual environments. The platform features built-in automation, orchestration, replication, alerting, and reporting. Commvault enables customer's to streamline and automate disaster recovery operations, orchestrate complex or uncommon disaster recovery scenarios, optimize costs by tailoring recovery objectives to multiple business and application requirements, and gain more security across workloads.
Commvault is a good fit for mid-sized organizations and the enterprise. Gartner's Magic Quadrant ranks it as having leading support for the public cloud, as well as protection of O365 and Salesforce. In addition, the IntelliSnap feature offers support for integrating with and exploiting storage hardware platform snapshots.
1 Commvault Way
Tinton Falls, NJ
www.commvault.com
A family of enterprise business continuity solutions, available in both physical and virtual platforms. The new Backup Insights tool gives you a complete picture of how files and folders have changed over time, adding versioning to the file restore capabilities. Datto utilizes Hybrid Cloud technology to improve redundancy and give users greater protection of their systems and data. Hybrid Cloud technology leverages the advantages of local backup and the security of the cloud. Should a business experience a server failure, the system can be virtualized instantly on the Datto device.
Datto is recommended for small to mid-sized businesses and is cost-effective. The vendor does not limit the number of vCPU and vRAM in the cloud. However, there is a limit of 8 vCPU and 16 GB vRAM per machine on the portal without customization.
101 Merritt 7
www.datto.com
Dell EMC data protection solutions are scalable, reliable, and cloud-enabled. Dell EMC Data Domain provides cost-effective backup, archive, and disaster recovery. The Dell EMC Integrated Data Protection Appliance is a comprehensive data protection tool for a broad application ecosystem. It combines protection storage, protection software, search, and analytics in an integrated appliance. And Dell EMC Data Domain Virtual Edition is software-defined protection storage that's easy to deploy and configure.
Dell EMC's solutions are a good fit for enterprise organizations. Though the data protection portfolio often results in lock-in with Data Domain and other proprietary hardware, it's important to note that the vendor offers low-cost cloud object storage for backup-based disaster recovery.
www.dellemc.com
Druva Phoenix delivers data availability and governance for virtual machines and physical servers with a unique cloud-first approach — combining high- performance, scalable backup, disaster recovery, archival and analytics to simplify data protection, improve visibility and dramatically reduce the risk, cost and effort of managing today's complex data sprawl.
Druva Phoenix makes it easier and more cost-effective for users to protect all of their data across complex infrastructures with cloud backup, disaster recovery, and archival. This solution is a good fit for small and medium organizations and offers a pay-as-you-go pricing model. Reviews from Gartner Peer Insights show that users find Druva Phoenix is easy to use, implement, and install on an end-user's device.
150 Mathilda Pl
A disk-based backup solution that efficiently protects business-critical data such as databases, email and files. CDP enabled fast recovery from disasters that range from as little as five minutes for files and under four hours for critical data. Centralized backup can be performed off of secondary CDP snapshot volumes. This out-of-band configuration isolates all backup related loads from physical or virtual production servers. Archive requirements can be addressed with the FalconStor HyperTrac Backup Accelerator option, which automates the selection and mounting of CDP snapshot volumes to backup media servers.
FalconStor supports application-specific data protection for most business applications including Microsoft, Oracle, and SAP. The vendor works with a variety of clients in different verticals but is recommended for small to mid-sized businesses.
701 Brazos St
www.falconstor.com
Hitachi Vantara is a wholly owned subsidiary of Hitachi, Ltd., and offers an expansive portfolio of products for integrating, managing and analyzing data. Hitachi's portfolio of data management solutions are best suited for modern environments, and can help organizations to quickly improve their key performance metrics, including business continuity, windows backup and operational recovery and disaster recovery. The tool also provides data protection and recovery for complex enterprise architectures.
For those looking to move beyond traditional backup and recovery products, Hitachi Vantara is an interesting option. Hitachi's backup and recovery capabilities feature an In-System Replication Bundle with volume-cloning and space-efficient snapshots. It also includes a Virtual Infrastructure Integrator that ensures quick recovery for VMware virtual environments. Hitachi provides AI-driven data center management as well, and offers machine learning and advanced analytics to optimize your data center. Solutions Review recently named Hitachi Vantara a Data Integration Tools Vendor to Watch in 2019.
www.hitachivantara.com
HYCU specializes in multi-cloud data backup, management, migration, protection and recovery for on-premises and hyper-converged (HCI), Google Cloud, Azure Cloud and multi-cloud infrastructures. Headquartered in Boston, Mass., HYCU harnesses 25 years of sophisticated IT experience, insights from over one million users, and works with more than 25,000 customers worldwide. The result is alignment with industry leaders and a competitive advantage in the multi-cloud space. HYCU's flagship products, a purpose-built Data Protection solution for Nutanix, a managed Data Protection as a service for Google Cloud Platform and Azure Cloud, and HYCU Protégé a Multi-Cloud Data Protection Solution offer one-click cross-cloud migration, disaster recovery, and consolidated management.
HYCU provides purpose-built data protection solutions for virtualized (VMware and Nutanix) and cloud infrastructures. HYCU solutions are deployed typically as virtualized, physical or hyperconverged infrastructure, or as SaaS directly from Google Cloud Marketplace or Azure Marketplace. The tight integration provides users of companies of all size to operate in a friendly and familiar environment that looks like an extension of their virtualization or cloud platform of choice. HYCU also offers a SaaS-based management platform, HYCU Protégé, that delivers cross-platform data analytics, management, migration and disaster recovery services.
www.hycu.com
Provides backup, snapshot, archive, recovery, space management, bare machine recovery and disaster recovery capabilities. Can help protect data on systems of all sizes, including virtual machines, file servers, email, databases, enterprise resource planning (EPR) systems, mainframes and desktops- all from a single environment that expands as data grows. Tivoli Storage Manager is available in a range of configuration options, including solution bundles, appliances and cloud services.
IBM Spectrum Protect can simplify data protection where data is hosted in physical, virtual, software-defined or cloud environments; making it ideal for mid-to-large businesses. Deduplication is not global across all Spectrum Protect server instances and is constrained to a single storage pool.
1 New Orchard Rd
Armonk, NY
www.ibm.com
Infrascale is an enterprise-grade disaster recovery solution that provides rapid failover to a second site with the flexibility to boot from the appliance or cloud. Founded in 2006, the company aims to give every company the ability to recover from a disaster – quickly, easily and affordably.
Infrascale is a good fit for organizations with a hybrid infrastructure that does not include a mainframe or that has needs outside of backup for Unix or IBM system servers. The vendor offers a storage-based monthly pricing fee that includes unlimited testing as well.
999 N. Sepulveda Blvd
www.infrascale.com
The vendor merged with HPE in the fall of 2017 and now offers Data Protector, a comprehensive, centrally-managed backup and recovery software for physical and virtual environments designed for data-driven data centers.
Micro Focus Data protector is a good fit for mid-to-large organizations. Data Protector is a unified, meaning-based data protection solution that utilizes an intelligent data management approach to protect data. Users are able to immediately install and configure the infrastructure to meet the needs of their organization, as well as set up a support IT community within the company's support model.
Founded in 2012, NAKIVO develops a fast, reliable, and affordable data protection solution for VMware, Hyper-V, and AWS EC2 environments. With 20 consecutive quarters of double-digit growth, 5-star online community reviews, 97.3% of customers happy with support, and more than 10,000 deployments worldwide including Coca-Cola, Honda, China Airlines, VeriFone, and many others NAKIVO is one of the fastest-growing vendors in the industry. Over 150 hosting, managed, and cloud services providers are using NAKIVO's software to deliver VM BaaS and DRaaS to their customers.
NAKIVO Backup & Replication is a reliable VM backup solution for protecting virtualized and cloud environments. Users describe it as easy to implement and use, but the vendor does not offer support on weekends or federal holidays.
4894 Sparks Blvd
+ 1(408) 335-7367
www.nakivo.com
sales@nakivo.com
Rubrik primarily provides data protection and data management in hybrid IT environments with its flagship product. Rubrik Cloud Data Management (RCDM) is a scale-out-architecture-based data protection tool with cloud integration, live mount for Oracle Databases, and support for Office 365 backup, and SAP HANA backup. In, 2018 Rubrik acquired Datos IO, a company specializing in data protection of NoSQL databases. The vendor also offers Polaris Radar, a solution which provides ransomware detection and remediation. Rubrik is headquartered in California and was founded in 2014.
Rubrik's solution is recommended for those looking to protect highly virtualized on-prem environments and hybrid environments that leverage Microsoft Azure and Amazon Web Services (AWS). Many users reportedly utilize RCDM for the protection of physical and virtualized databases. Rubrik also provides a granular set of features for backing up data to the cloud and protecting public-cloud-Infrastructure as a Service- based instances in AWS and Microsoft Azure. The solution's Live Mount capability also provides automated recovery and test/dev for Oracle and SQL databases, as well as for Hyper-V and VMware virtual machines.
1001 Page Mill Rd
Palo Alto CA,
www.rubrik.com
SolarWinds is an IT infrastructure management software solution provider. The vendor's backup solution, SolarWinds Backup, is designed to provide a cloud-first data protection service for physical and virtual servers, workstations, business documents, and Microsoft Office 365 data in a single web-based dashboard. SolarWinds Backup is compatible with on-prem, cloud, and hybrid environments. Additionally, the provider's solution can assist with reducing the cost and complexity of data protection, without sacrificing speed or reliability. The vendor also includes storage in its global private cloud. SolarWinds was founded in 1999 and is headquartered in Texas.
SolarWinds Backup is recommended to small- to mid-sized businesses. The provider offers technical support for installation, configuration, setup, operation, and other product-related issues 24 hours a day, 7 days a week. SolarWinds also provides a customer portal for additional support purposes. According to G2 Crowd user reviews, the solution is easy to use and installation is simple. While some users reported difficulty when using the solution with multiple backup sets, each backup is encrypted from the beginning and remains encrypted when in transit, or at rest.
3030 Slater Rd
www.solarwindsmsp.com
StorageCraft File Backup and Recovery with Backup Analyzer is able to identify, protect and recover critical data on laptops and remote computers that are not regularly backed up to corporate servers. Users are able to install, configure and monitor backups from a central location via a web-based interface.
StorageCraft typically serves small to medium-sized businesses. The vendor also provides support in various languages including German, French, Italian, Spanish, Japanese, and Portuguese. Buyers should keep in mind that though StorageCraft will accept calls from end-users, partners are responsible for testing and disaster recovery events.
380 Data Dr
www.storagecraft.com
A virtual backup appliance that can be deployed on Microsoft Hyper-V or VMware vSphere for enterprise-class virtual deep virtual, physical protection and unified compute protection. It offers backup archiving; disaster recovery and business continuity delivered via flexible deployment options with Unitrends Cloud storage and DRaas (disaster recovery-as-a-service) offering featuring application-level recovery assurance via ReliableDR. Flexible licensing with protected capacity or per-socket/server licensing is available as well.
Unitrends is a veteran in the backup and recovery software marketplace, and it appeals to mid-market organizations. Its all-in-one licensing and centralized management are also popular. While the solution only offers NAS backup over NFS or CIFS, Unitrends provides users with a predictive analytics algorithm that can detect a ransomware infection.
200 Summit Dr
www.unitrends.com
A data protection and disaster recovery solution for virtual environments of any size. It provides fast, flexible and reliable recovery of virtualized applications and data. It unifies backup and replication in a single solution, increases value of backup and reinvents data protection for VMware VSphere and Microsoft Hyper-V virtual environments. Veeam Backup & Replication supports entire virtual infrastructures with industry leading features such as instant file-level recovery and streamlined VM recovery, scalability, 2-in-1 backup and replication, built-in deduplication and centralized management.
Veeam works well for companies of all sizes and offers a rich capabilities portfolio along with recovery options for the VMWare and Hyper-V environment. While Veeam users report that pricing options and license management could use some improvement, the provider continues to be one of the fastest-growing companies in the backup industry.
Linden Park, Lidenstr, 16,
www.veeam.com
Veritas Technologies is headquartered in California and was founded in 2015. It offers NetBackup, which is a single and scalable solution that can protect physical, virtual and cloud workloads. Multitenant support is optional.
Veritas offers a broad support matrix and boasts positive reference customer user reviews. Continued compliance audits reportedly cause some clients to look for alternative solutions, but the quality and strength of the product have been rated highly for the last two years. Users should be aware that the vendor's field support team, regarding sales people and support engineers in particular, has seen some turnover in recent years.
500 East Middlefield Rd
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May 26, 2022 in Southeast
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Q: How to host an ASP.NET Core Web Application with enabled enforced encryption with Nginx? I am a beginner and I am trying to host a .NET Core web application with enforced HTTPS enabled on Ubuntu Server. When I try to access my website from another computer I receive an error. However, when I create a project with enforced HTTPS off it works perfectly.
I need to have the enforced HTTPS turned on in the web application as I want to use the individual user account option, and for that, enforced HTTPS is needed.
I am using an Nginx server to make a reverse proxy for the web app.
After I run the dotnet command on Linux server and access the website, I receive this error in the Ubuntu server:

fail: Microsoft.AspNetCore.Server.Kestrel[13]
      Connection id "0HLNOTQ3C92Q3", Request id "0HLNOTQ3C92Q3:00000001": An unhandled exception was thrown by the application.
System.ArgumentNullException: Value cannot be null.
Parameter name: connectionString
   at Microsoft.EntityFrameworkCore.Utilities.Check.NotEmpty(String value, String parameterName)
   at Microsoft.EntityFrameworkCore.SqlServerDbContextOptionsExtensions.UseSqlServer(DbContextOptionsBuilder optionsBuilder, String connectionString, Action`1 sqlServerOptionsAction)
   at TurismRomaniaWebsite.Startup.<ConfigureServices>b__4_1(DbContextOptionsBuilder options) in D:\Proiecte C#\Turism Romania\TurismRomaniaWebsite\TurismRomaniaWebsite\Startup.cs:line 41
   at Microsoft.Extensions.DependencyInjection.EntityFrameworkServiceCollectionExtensions.<>c__DisplayClass1_0`2.<AddDbContext>b__0(IServiceProvider p, DbContextOptionsBuilder b)
   at Microsoft.Extensions.DependencyInjection.EntityFrameworkServiceCollectionExtensions.DbContextOptionsFactory[TContext](IServiceProvider applicationServiceProvider, Action`2 optionsAction)
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   at Microsoft.Extensions.DependencyInjection.ServiceProviderServiceExtensions.GetRequiredService(IServiceProvider provider, Type serviceType)
   at Microsoft.AspNetCore.Mvc.Razor.Internal.RazorPagePropertyActivator.<>c__DisplayClass8_0.<CreateActivateInfo>b__1(ViewContext context)
   at Microsoft.Extensions.Internal.PropertyActivator`1.Activate(Object instance, TContext context)
   at Microsoft.AspNetCore.Mvc.Razor.Internal.RazorPagePropertyActivator.Activate(Object page, ViewContext context)
   at Microsoft.AspNetCore.Mvc.Razor.RazorPageActivator.Activate(IRazorPage page, ViewContext context)
   at Microsoft.AspNetCore.Mvc.Razor.RazorView.RenderPageCoreAsync(IRazorPage page, ViewContext context)
   at Microsoft.AspNetCore.Mvc.Razor.RazorView.RenderPageAsync(IRazorPage page, ViewContext context, Boolean invokeViewStarts)
   at Microsoft.AspNetCore.Mvc.Razor.RazorView.RenderAsync(ViewContext context)
   at Microsoft.AspNetCore.Mvc.TagHelpers.PartialTagHelper.RenderPartialViewAsync(TextWriter writer, Object model, IView view)
   at Microsoft.AspNetCore.Mvc.TagHelpers.PartialTagHelper.ProcessAsync(TagHelperContext context, TagHelperOutput output)
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   at Microsoft.AspNetCore.StaticFiles.StaticFileMiddleware.Invoke(HttpContext context)
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   at Microsoft.AspNetCore.Diagnostics.ExceptionHandlerMiddleware.Invoke(HttpContext context)
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   at Microsoft.AspNetCore.Server.Kestrel.Core.Internal.Http.HttpProtocol.ProcessRequests[TContext](IHttpApplication`1 application)
info: Microsoft.AspNetCore.Hosting.Internal.WebHost[2]
      Request finished in 19.4666ms 500

Could you please recommend the right approach for hosting my web app with enabled enforced encryption with Nginx?

A: You need to specify connectionString inside your appsettings.json when you have deployed your application on any host.
Usually connection strings are like below:
{
 'connectionStrings':{
   'connectionString': 'your-connection-string'
 }
}

redpajama_set_name: RedPajamaStackExchange


The Church of St. Anne is located near the Lion's Gate in the Muslim Quarter of the Old City. Built between 1131 and 1168 AD, the church is best known for its typical medieval architecture and near-perfect acoustics that make it a destination for soloists and choirs.
Amazing experience inside, an American choir was sining, the acoustics are exceptional so our choir performed Ubi caritas...and we had to do an encore... my best experience in Jerusalem that is ovecrowded with tourists and cars. Sadly, I found two kittens in a poor state in the gardens of Saint Anne, I cleaned their eyes a ittle as they could not see and wrote to the French Consulate in Jerusalem to do something about this. But no reply. My heart bleeds.
This is a simple and beautiful church that has the most amazing acoustics for singing! It is located next to the Bethesda pools and the gardens surrounding it are well tended.
Highly recommended visit at the start of the Via Dolorosa, coming from Lion's Gate. And if you can, visit at a time when there is a choir singing or for a service.


Under-fire presidential candidate François Fillon cancelled a key campaign event on Wednesday and will speak to the press at midday. Is he about to throw the towel in or come out fighting once again?
fair in Paris and would hold a press conference at midday from his campaign HQ.
Fillon's campaign offered no explanation for the last-minute change, saying only that a new date for a visit would be published later. Even those politicians close to the centre-right candidate admitted they had no idea why the visit was postponed.
France Journal du Dimanche newspaper claimed Fillon cancelled the visit because he had been summoned to appear before judges investigating the "fake jobs" scandal known as Penelopegate that has engulfed him since the Le Canard Enchainé's revelations on January 24th.
Last week a formal judicial investigation was opened into suspected "misuse of public money" surrounding the allegations his wife was paid hundreds of thousands for a "fake job" as her husband's parliamentary assistant.
Magistrates have the power to place Fillon under formal investigation (mise en examen) which is the nearest equivalent to being charged, as long as he is present.
The latest media reports in France suggest both he and his British wife Penelope have received summons to appear before the investigating magistrates with a view to being placed under formal investigation in the next two weeks.
Reports by Mediapart website that Penelope was already in police custody on Wednesday were dismissed by those close to Fillon.
"She's at home, she's doing well," a Fillon aide told Le Figaro newspaper.
According to Le Figaro newspaper Fillon was holding a hastily organised meeting with senior MPs from his Republicans party on Wednesday morning.
There are also unconfirmed reports that Alain Juppé, who lost out in the Republicains party primary in November will be alongside Fillon when he makes his announcement. Juppé has repeatedly stated he would not stand in if Fillon pulls out.
His visit to the massive farm show, seen as a must for anyone hoping to win the election, was supposed to begin early Wednesday morning and several reporters and members of his campaign staff were already on the scene to welcome him.
The organisation Agriculture et Liberté which was carrying out a poll of farmers to find out which presidential election candidate was the most popular told The Local that Fillon came out on top ahead of Marine Le Pen.
READ ALSO: What happens if François Fillon falls?
Fillon's campaign appearances are frequently attended by protesters supporting Communist-backed candidate Jean-Luc Melenchon.
Despite difficulties on the campaign trail and attempts by a small number of rebel Republicans MPs to pressure Fillon to stand aside, the candidate has so far stubbornly refused to throw the towel in.


Now, we may be seeing the rise of games that ditch single player entirely. It's not a great PR message. Many are going to be reluctant to actually pull the trigger. But that may be a disservice to everyone involved. Players go into the game thinking they can get something they can't, and developers are forced to compromise a gameplay experience, knowing it's not what they're truly building. That's a lose-lose.


In this article we are going to cover How to Display Memory Configuration.
How to Display Memory Configuration?
This script will display Memory Configuration .
Hope you will like How to Display Memory Configuration.


text:
\section{Introduction}
The $j$-multiplicity was defined in 1993 by Achilles and Manaresi in \cite{AM} as a generalization of the Hilbert-Samuel multiplicity for arbitrary ideals in a Noetherian local ring. Several results on the Hilbert-Samuel multiplicity have been successfully extended to more general classes of ideals using the $j$-multiplicity, for example \cite{FM1}, \cite{PX}, and \cite{Ciu}. The main result of this paper may be viewed as one of these extensions.

Let $(R,\mathfrak{m},k)$ be a Noetherian local ring of dimension $d$, and $I\subset R$ an ideal. The $j$-multiplicity of $I$ is defined as the limit
\[j(I)=\lim_{n\rightarrow \infty}\frac{(d-1)!}{n^{d-1}}\lambda_R\big(\HH{0}{\mathfrak{m}}{I^n/I^{n+1}}\big)\,.\]

There have been previous approaches for computing the $j$-multiplicity. For example, in \cite{AM} and \cite{Xie} it is proven that if $k$ is infinite, then for general elements $a_1,\ldots,a_{d}$ in $I$, and $\alpha=(a_1,\ldots,a_{d-1})$, we have \[j(I)=\lambda_R \big(R/((\alpha:_R\,I^{\infty})+a_d R)\big)\,.\]
This formula is applied to compute specific examples in \cite{NU}.

Let $R$ denote now the polynomial ring $k[x_1,\ldots,x_d]$ over the field $k$, $\mathfrak{m}$ the homogeneous maximal ideal $(x_1,\ldots,x_d)$, and $I$ a monomial ideal of $R$. The {\it Newton polyhedron} of $I$ is the convex hull of the points in $\mathbb R^d$ that correspond to monomials in $I$, which we will denote by $\operatorname{conv}(I)$. In this paper we generalize the classical result that describes the Hilbert-Samuel multiplicity of an $\mathfrak{m}$-primary ideal as the normalized volume of the complement of its Newton polyhedron in $\mathbb R^d_{\geqslant 0}$, see \cite{Tes}. If $I$ is not $\mathfrak{m}$-primary, the complement of $\operatorname{conv}(I)$ is infinite, but we can define the analogue of this region in the general case by considering the truncated cone from the origin to the union of the bounded faces of $\operatorname{conv}(I)$. This truncated cone will be denoted by $\operatorname{pyr}(I)$. With this notation, we can state our main result:

\

{\bf Theorem \ref{main}.} {\it Let $I\subset R$ be a monomial ideal. Then $j(I)=d!\operatorname{vol}(\operatorname{pyr}(I))$.}

\

Earlier unpublished work of J. Validashti obtains this formula in dimension two. The rest of the paper is organized as follows: In the second section we set up the notation and also present some results that will be used in the proof of the main theorem. The third section will include the proof of Theorem~\ref{main}. In the fourth section we provide an extension of this result to pointed normal affine toric varieties. In the fifth section we will apply our characterization of the saturation of a monomial ideal in $R$ in Lemma~\ref{satur} to give a geometric description of the $\varepsilon$-multiplicity. The paper ends with some examples in a sixth section.

\section{Preliminaries}
Let $R=k[x_1,\ldots,x_d]$ be a polynomial ring over a field $k$ and $\mathfrak{m}=(x_1,\ldots,x_d)$ its homogeneous maximal ideal. Let $I$ be a monomial ideal of $R$ minimally generated by $x^{v_1},x^{v_2},\ldots, x^{v_n}$ where $v_i=(v_{i,1},\ldots,v_{i,d})$ and $x^{v_i}=x_1^{v_{i,1}}\cdots x_d^{v_{i,d}}$. For a monomial ideal $L$ in $R$ we denote by $\Gamma(L)$ the set of lattice points in $\mathbb R^d$ corresponding to the exponents. Additionally, if $L_1\supseteq L_2$ are monomial ideals, we will write $\Gamma(L_1/L_2)$ for $\Gamma(L_1)\setminus\Gamma(L_2)$.

We denote by $\operatorname{conv}(I)$ the {\it Newton polyhedron} of $I$, that is:
\[\operatorname{conv}(I):= \operatorname{conv}(v_1,\ldots, v_n)+\mathbb{R}^d_{\geqslant 0}\, , \]
where $+$ denotes the Minkowski sum. It is worth noting that the collection of bounded facets of the Newton polyhedron is not convex, and thus is not a polytope, but rather has the structure of a polytopal complex. Notice also that $\operatorname{conv}(I)=\operatorname{conv}(\Gamma(I))$. Since every polyhedron is defined by the intersection of finitely many closed half spaces, we can define $H_i=\{x\in \mathbb{R}^n\mid \langle x,b_i\rangle = c_i\}$, with $b_i\in \mathbb{Q}^d$, $c_i\in \mathbb{Q}$ for $i=1,\ldots, w$ to be the {\it supporting hyperplanes} of $\operatorname{conv}(I)$ such that
\[\operatorname{conv}(I)= H_{1}^+\cap H_{2}^+\cap\cdots \cap H_{w}^+ \, ,\]
where $H_i^+=\{x\in \mathbb{R}^n\mid \langle x,b_i\rangle \geqslant c_i\}$. Let $\mathcal{F}_i=H_i\cap \operatorname{conv}(I)$ for $i=1,\ldots, w$ be the facets of $\operatorname{conv}(I)$. We will assume that $H_1,\ldots,H_u$, are the hyperplanes corresponding to unbounded facets.

It can be shown that all the vectors $b_i$ have nonnegative components, and that $b_i\in \mathbb{R}^d_{>0}$ if and only if $\mathcal{F}_i$ is a bounded facet, as in \cite[Lemma~1.1]{Sin}. This forces the $c_i$ to be nonnegative, and in fact positive in the case of a bounded facet.

Recall that the {\it analytic spread} of an arbitrary ideal $I$, denoted by $l(I)$, is defined to be the dimension of its {\it special fiber ring} $\operatorname{gr}_I (R)\otimes_R R/\mathfrak{m}=\bigoplus_{n=0}^{\infty} I^n/\mathfrak{m} I^{n},$ where $\operatorname{gr}_I(R)$ is the associated graded algebra of $I$, i.e., $\operatorname{gr}_I(R)=\bigoplus_{n=0}^{\infty} I^n/I^{n+1}$. We will say that $I$ has \emph{maximal analytic spread} if $l(I)=\dim(R)$. If $I$ is monomial, $l(I)$ can be computed as $c+1$ where $c$ is the highest dimension of a bounded facet of $\operatorname{conv}(I)$, see \cite[Theorem~2.3]{Biv} or \cite[Corollary~4.10]{Sin}.

We denote by $\operatorname{vert}(I)$ the set of vertices of $\operatorname{conv}(I)$, and set $\operatorname{bd}(I)=\bigcup_{i=u+1}^{w}\mathcal{F}_i$ for the union of the bounded facets of $\operatorname{conv}(I)$.  If $\mathcal{P}$ is a polytope, we will write $\operatorname{pyr}(\mathcal{P})$ for $\operatorname{conv}(\mathcal{P},\boldsymbol{0})$, the truncated cone, or pyramid, over $\mathcal{P}$. By abuse of notation, we will write $\operatorname{pyr}(I)$ for $\bigcup_{i=u+1}^{w}\operatorname{pyr}(\mathcal{F}_i)$. Note that the monomials corresponding to the points in $\operatorname{vert}(I)$ are part of the set of minimal generators of $I$, so we will assume $\operatorname{vert}(I)=\{v_1,\ldots,v_s\}$ for some $1\leqslant s\leqslant n$.
We will also find it convenient to define the \emph{$n^{\text{th}}$ cone section} of a polytope $\mathcal{P}$ as
\[\conen{n}(\mathcal{P}):=\big((n+1)\operatorname{pyr}(\mathcal{P})\setminus (n+1)\mathcal{P}\big) \, \setminus \, \big(n\operatorname{pyr}(\mathcal{P})\setminus n\mathcal{P}\big) \, ,
\]
which we may alternatively write as $\bigcup_{n\leqslant s < n+1} s\mathcal{P}$. We again abuse notation by writing $\conen{n}(I)$ for  $\bigcup_{i=u+1}^{w}\conen{n}(\mathcal{F}_i)$.

\begin{figure}
\centering
\mbox{\subfigure{\includegraphics[width=2in]{fig1acol.pdf}}\quad
\subfigure{\includegraphics[width=2in]{fig1bcol.pdf} }}
\caption{Various regions for the monomial ideal $I=(xy^5,x^2y^3,x^3y^2)$} \label{fig1}
\end{figure}

For the monomial ideal $I=(xy^5,x^2y^3,x^3y^2)$, in Figure~\ref{fig1}, on the left we mark $\operatorname{vert}(I)$ with black dots, $\operatorname{bd}(I)$ with dark red lines, and the unbounded facets with pink lines. In this example, the green region with its boundary forms $\operatorname{pyr}(I)$, and the yellow region with its boundary forms $\operatorname{conv}(I)$. In the graph on the right, we shade $\conen{2}(I)$ in purple, where the top dotted segments of the boundary are not included.

The following description of the faces of the Newton polyhedron will be useful.
\begin{lem}\cite[Lemma~3.1]{Sin} \label{faces}
Let $\mathcal{F}$ be a face of $\operatorname{conv}(I)$ with supporting hyperplane
$H=\{x\in \mathbb{R}^n\mid \langle x,b\rangle=c\}$. Then $\mathcal{F}\cap \operatorname{vert}(I)=\{v_{i_1},\ldots, v_{i_r}\}$ is non-empty, and
\[\mathcal{F}=\operatorname{conv}(v_{i_1},\ldots, v_{i_r})+\sum_{j : b_j=0}\mathbb{R}_{\geqslant 0}\, e_j\] where $e_j$ is the unit vector with nonzero $j^{\text{th}}$ component.

\end{lem}

\

Recall that for any submodule $N$ of an $R$-module $M$, the {\it saturation} of $N$, denoted $(N:_M \mathfrak{m}^{\infty})$, is the set of elements $a$ in $M$ for which there exists $n\in \mathbb N$ such that $a\mathfrak{m}^n\in N$. The \emph{zeroth local cohomology module} of $M$ is defined to be $(0:_M\,\mathfrak{m}^{\infty})$ and is denoted by $\HH{0}{\mathfrak{m}}{M}$. Notice that in general, $\HH{0}{\mathfrak{m}}{M/N}=(N :_M \mathfrak{m}^{\infty})/N$. If $I$ is a monomial ideal, then $(I:_R\,\mathfrak{m}^{\infty})$ is also monomial.
\

The {\it integral closure} of an arbitrary ideal $J$ is the set of elements $x$ in $R$ that satisfy an integral relation $x^n+a_1 x^{n-1}+\cdots+a_{n-1}x+a_n$ where $a_i\in J^i$ for $i=1,\ldots, n$. It is denoted by $\overline{J}$ and it is an ideal.
For monomial ideals, it is possible to give a geometric description of the integral closure, namely $\Gamma(\overline{I})=\mathbb Z^d\cap \operatorname{conv}(I)$, i.e., $\operatorname{conv}(\overline{I})=\operatorname{conv}(I)$; see \cite[Proposition~7.25]{Vas}.

\

\begin{prop}\label{satur}

$\displaystyle{\Gamma(\overline{I} :_R \,m^\infty)=H_1^+\cap\cdots \cap H_u^+\cap \mathbb Z^d_{\geqslant 0}}$\,.
\end{prop}
\begin{proof}

Let $v\in H_1^+\cap\cdots \cap H_u^+\cap \mathbb Z^d_{\geqslant 0}$; then $\langle v, b_i\rangle\geqslant c_i$ for $i=1,\ldots, u$. For $t\in \mathbb R_{\geqslant 0}$, one has \[\langle v+te_j, b_i\rangle\geqslant c_i+tb_{i,j}\geqslant c_i\] for $j=1,\ldots, d$ and $i=1,\ldots, u$. Also, if $u+1\leqslant i\leqslant w$ then all the entries of $b_i$ are positive, so we also have that $\langle v+te_j, b_i\rangle \geqslant c_i$ for $t\gg 0$. We conclude $x_j^tx^v\in \overline{I}$
for $j=1,\ldots, d$ and $t\gg 0$, that is $v\in \Gamma(\overline{I} :_R\, m^\infty)$.

Conversely, if $v\in \Gamma(\overline{I} :_R \,m^{\infty})$, then $v+te_j\in \operatorname{conv}(I)\subset H_1^+\cap\cdots \cap H_u^+$ for
$j=1,\ldots, d$ and $t\gg 0$. Now, suppose $v\not\in H_i^+$ for some $1\leqslant i\leqslant u$, then $\langle v, b_i\rangle<c_i$. By Lemma~\ref{faces}, since $\mathcal{F}_i$ is an
unbounded facet, we can pick $j$ such that $b_{i,j}=0$, and hence $\langle v+te_j, b_i\rangle=\langle v, b_i\rangle < c_i$ for every $t\in \mathbb R$, which is a contradiction.
\end{proof}

\begin{lem}\label{satincone}
 $\displaystyle{\Gamma(\overline{I} :_R \,\mathfrak{m}^{\infty})\subset \operatorname{conv}(I)\cup \operatorname{pyr}(I)\,.}$
\end{lem}

\begin{proof}

The result follows immediately by Proposition \ref{satur} if $\operatorname{conv}(I)$ does not have bounded facets. We will assume that $u<w$.

Let $v$ be a nonzero vector in $\mathbb Z^d_{\geqslant 0} \setminus (\operatorname{pyr}(I)\cup
\operatorname{conv}(I))$. We will proceed by contradiction. Suppose $v\in
\Gamma(\overline{I} :_R\, m^\infty)$, then $v\in H_{1}^+\cap\cdots
\cap H_u^+$ by Proposition~\ref{satur}.

Note that since $b_i$ has positive entries for each $i=u+1,\ldots, {w}$,
we have $\langle v,b_i\rangle\neq 0$. For each $u+1\leqslant
i\leqslant {w}$ we can find a real number $t_i$ such that $t_i v\in H_i$, each of which is positive because $c_i>0$. Suppose, without
loss of generality, that $t_{w}$ is largest among $t_{u+1},\ldots, t_{w}$. Since $v\not\in
H_i^+$ for some $u+1 \leqslant i\leqslant {w}$ we have $\langle v,
b_i\rangle<c_i$, then $t_i>1$, and so $t_{w}>1$.

Now, $\langle t_{w} v , b_i\rangle\geqslant\langle t_i v , b_i\rangle= c_i $ for $i=u+1,\ldots, {w}$, so
\[
t_{w} v\in H_{1}^+\cap H_2^+\cap\cdots\cap H_{{w}-1}^+\cap H_{w}=\mathcal{F}_{w}\,.\]
Then we have $v\in\operatorname{pyr}(I)$ which is a contradiction.
\end{proof}

In the following lemma, we will use the notion of the \emph{Hausdorff distance} between compact sets $A$ and $B$ in $\mathbb R^d$, which is defined as
\[\rho(A,B):=\inf\{\lambda\geqslant 0 \,|\, A\subseteq B+\lambda U \,,\, B\subseteq A+\lambda U\}\,,\]
where $U$ is the unit ball. We will use a related notion for polytopes: for a convex polytope $\mathcal{P}=\operatorname{conv}(v_1,\dots,v_t)$ in $\mathbb R^d$, we will say that another convex polytope with $t$ vertices $\mathcal{P}'=\operatorname{conv}(v'_1,\dots,v'_t)$ is an \emph{$\varepsilon$-shaking} of $\mathcal{P}$ if $|v_j-v'_j|<\varepsilon$ for all $j$.

\begin{lem}\label{close} Fix $\mathcal{P}=\operatorname{conv}(v_1,\dots,v_t)$ in $\mathbb R^d$. Let $(\mathcal{P}^{(n)}_1,\dots,\mathcal{P}^{(n)}_s)_{n\in \mathbb N}$ be a sequence of $s$-tuples of polytopes such that each $\mathcal{P}^{(n)}_j$ is a $(1/n)$-shaking of $\mathcal{P}$. Then,
\[\lim_{n\rightarrow\infty} \operatorname{vol} \left( \mathcal{P}\cap\bigcap_{i=1}^s \mathcal{P}^{(n)}_i \right) = \operatorname{vol}(\mathcal{P})\,.\]
\end{lem}
\begin{proof}

Let $\mathcal{P}'=\operatorname{conv}(v'_1,\dots,v'_t)$ be an $\varepsilon$-shaking of $\mathcal{P}$, and write
\[
\mathcal{P}'\amalg\mathcal{P}=\operatorname{conv}(v'_1,\dots,v'_t,v_1,\dots,v_t)\,.
\]
Note that $\mathcal{P}'\cup\mathcal{P}\subseteq \mathcal{P}'\amalg\mathcal{P}$. Also, $\rho(\mathcal{P},\mathcal{P}\amalg\mathcal{P}')< \varepsilon$, since for $q\in \mathcal{P}\amalg\mathcal{P}'$, we may write $q=\sum{\lambda_j} v_j+\sum{\lambda'_j} v'_j$ with $\sum\lambda_i+\sum\lambda'_i=1$, and
\[|q- \sum{\lambda_j} v_j-\sum{\lambda'_j} v_j| \,=\, |\sum{\lambda'_j} v'_j- \sum{\lambda'_j} v_j| \,\leqslant\, \sum{\lambda'_j} |v'_j- v_j| \,<\, \varepsilon\,,\]
where $\sum{\lambda_j} v_j+\sum{\lambda'_j} v_j\in \mathcal{P}$.
Similarly $\rho(\mathcal{P},\mathcal{P}')< \varepsilon$.

We have
\begin{align*}
0\leqslant \operatorname{vol}(\mathcal{P})-\operatorname{vol}(\mathcal{P}\cap\bigcap_{i=1}^s \mathcal{P}^{(n)}_i)
&\leqslant \sum_{i=1}^s {\big(\operatorname{vol}(\mathcal{P})-\operatorname{vol}(\mathcal{P}\cap\mathcal{P}^{(n)}_i)\big)}  \\
&= \sum_{i=1}^s {\big(\operatorname{vol}(\mathcal{P}\cup\mathcal{P}^{(n)}_i)-\operatorname{vol}(\mathcal{P}^{(n)}_i)\big)}   \\
 &\leqslant \sum_{i=1}^s {\big(\operatorname{vol}(\mathcal{P}\amalg\mathcal{P}^{(n)}_i)-\operatorname{vol}(\mathcal{P}^{(n)}_i)\big)}\,.
\end{align*}

Then, by continuity of volume with respect to Hausdorff distance, see \cite[Theorem~6.2.17]{Web}, we have that $\operatorname{vol}(\mathcal{P}^{(n)}_i)$ and $\operatorname{vol}(\mathcal{P}\amalg\mathcal{P}^{(n)}_i)$ both converge to $\operatorname{vol}(\mathcal{P})$ as $n\rightarrow \infty$. Thus, $\operatorname{vol}(\mathcal{P}\cap\bigcap_{i=1}^s \mathcal{P}^{(n)}_i)\rightarrow \operatorname{vol}(\mathcal{P})$ as $n\rightarrow \infty$, as required.
\end{proof}

Recall that the \emph{Ehrhart function} of a polytope $\mathcal{P}\subset \mathbb R^d$ is defined as
\[ E_{\mathcal{P}}(n):=\#(\mathbb Z^d \cap n\mathcal{P})\,. \]
Ehrhart \cite{Ehr} showed that if the vertices of $\mathcal{P}$ have integer coordinates, $E_{\mathcal{P}}(n)$ is a polynomial of degree $\dim(\mathcal{P})$ with leading coefficient equal to the relative volume of $\mathcal{P}$ (cf., \cite[Chapter~12]{MS}). We will employ a strengthening of this fact. Recall that a function $f\colon\mathbb N\rightarrow \mathbb Z$ is called a \emph{quasi-polynomial} if there is an $m\in \mathbb N$ and polynomials $f_0,\dots,f_{m-1}$ such that $f(n)=f_{(n\, \text{mod}\, m)}(n)$ for all $n\in \mathbb N$. The \emph{grade} of $f$ is the least $\delta$ such that the $i^{\text{th}}$ coefficient of each of the $f_j$ is the same for all $i>\delta$.
The following was conjectured by Ehrhart \cite{Ehr}, and proved by McMullen \cite{McM} and Stanley \cite{Sta} separately (see also \cite{BI} for a proof based on monomial ideal techniques).

\begin{thm}[McMullen, Stanley]\label{MS} Let $\mathcal{P} \subset \mathbb R^d$ be a polytope with vertices in $\mathbb Q^d$. Suppose that the affine span of each $t$-dimensional face of $\mathcal{P}$ contains a lattice point. Then $\# (\mathbb Z^d \cap n\mathcal{P})$ as a function of $n$ is given by a quasi-polynomial of degree equal to the dimension of $\mathcal{P}$ and grade less than $t$.
\end{thm}

\begin{prop}\label{poly} $\phantom{M}$

\begin{enumerate}
\item[a)] Let $\mathcal{P} \subset \mathbb R^d$ be a polytope with vertices in $\mathbb Q^d$, and $\dim \mathcal{P}<d$. Suppose that the affine span of $\mathcal{P}$ contains a point in the integer lattice $\mathbb Z^d$, or that the dimension of $\mathcal{P}$ is less than $d-1$.
Then $\# (\mathbb Z^d \cap \conen{n}(\mathcal{P}))$ as a function of $n$ is given by a quasi-polynomial of the form
\[
p(n)=d\operatorname{vol}(\operatorname{pyr}(\mathcal{P}))\, n^{d-1} + O(n^{d-2})\,.
\]
\item[b)] If $I$ is a monomial ideal, then $\# (\mathbb Z^d \cap \conen{n}(I))$ is given by a quasi-polynomial of the form
 \[p(n)=d\operatorname{vol}(\operatorname{pyr}(I))\, n^{d-1} + O(n^{d-2})\,.\]
\end{enumerate}
\end{prop}

\begin{proof} Write
\begin{align*} \# (\mathbb Z^d \cap \conen{n}(\mathcal{P})) &= \# \bigg(\mathbb Z^d \cap  \Big(\big((n+1)\operatorname{pyr}(\mathcal{P})\setminus (n+1)\mathcal{P}\big)\setminus \big(n\operatorname{pyr}(\mathcal{P})\setminus n\mathcal{P}\big)\Big) \bigg)  \\
&= \big(E_{\operatorname{pyr}(\mathcal{P})}(n+1)-E_{\operatorname{pyr}(\mathcal{P})}(n)\big)-\big(E_{\mathcal{P}}(n+1)-E_{\mathcal{P}}(n)\big)\,.
\end{align*}
Notice that the hypothesis in part a) ensures the affine span of each $(d-1)$-dimensional face of $\operatorname{pyr}(\mathcal{P})$ contains a lattice point: if $\dim \mathcal{P}=d-1$, because the affine span of $\mathcal{P}$ has a lattice point and every other $(d-1)$-dimensional face contains $\boldsymbol{0}$. If $\dim\mathcal{P}=d-2$, the only $(d-1)$-dimensional face contains $\boldsymbol{0}$, and the condition is vacuous otherwise. By Theorem~\ref{MS}, in this situation $E_{\operatorname{pyr}(\mathcal{P})}$ and $E_{\mathcal{P}}$ are quasi-polynomials of the form
\begin{align*}
E_{\operatorname{pyr}(\mathcal{P})}(n) &=  a_{d} n^d + a_{d-1} n^{d-1} + O(n^{d-2}) \\
E_{\mathcal{P}}(n) &= b_{d-1} n^{d-1} + O(n^{d-2})\,,
\end{align*}
where $a_d$, $a_{d-1}$, and $b_{d-1}$ are constants; specifically, they do not depend on $n$. Further, from the definition of the Riemann integral we compute that
\[a_d=\lim_{n\rightarrow\infty} n^{-d}\cdot\#\big( (\tfrac{1}{n}\mathbb Z)^d \cap \operatorname{pyr}(\mathcal{P})\big) = \operatorname{vol}(\operatorname{pyr}(\mathcal{P}))\,.\]
Part a) now follows from the formula above.

For part b), we first show that that for two different bounded faces $\mathcal{F}$, $\mathcal{F}'$ of  $\operatorname{conv}(I)$ we have $\conen{n}(\mathcal{F})\cap \conen{n}(\mathcal{F}')=\conen{n}(\mathcal{F}\cap \mathcal{F}')$. Indeed, let $v$ be a nonzero element of $\cone(\mathcal{F})\cap \cone(\mathcal{F})$  and $t, r\geqslant 1$ such that $tv\in \mathcal{F}$ and $rv\in \mathcal{F}'$. If $t\geqslant r $, then $\frac{t}{r}(rv)=tv\in \mathcal{F}$ implies that $tv\in rv+\mathbb R^d_{\geqslant 0}$, but $tv$ is on the boundary of $\operatorname{conv}(I)$ so $t=r$ and $v\in \cone(\mathcal{F}\cap \mathcal{F}')$. Now, the claim follows from the definition of $\conen{n}$ and the fact that $n\mathcal{P}\cap n\mathcal{P}'=n(\mathcal{P}\cap \mathcal{P}')$ for any pair of polytopes $\mathcal{P}$, $\mathcal{P}'$.
Then by inclusion-exclusion we have
 \begin{align*}
\# (\mathbb Z^d \cap \conen{n}(I))=& \sum_{i=u+1}^{w}  \# (\mathbb Z^d \cap \conen{n}(\mathcal{F}_i))-\!\!\!\!\!\!\sum_{u+1\leq i<j\leq w}\!\!\!\!\!\!\# (\mathbb Z^d \cap \conen{n}(\mathcal{F}_i\cap \mathcal{F}_j))\\ &+\,\cdots\, \pm \# (\mathbb Z^d \cap \conen{n}(\mathcal{F}_{u+1}\cap \cdots \cap \mathcal{F}_w))\,.
\end{align*}
The conclusion now follows from part a).
\end{proof}

\section{The $j$-multiplicity of monomial ideals}

In order to be consistent with the definition of $j$-multiplicity, which is defined for ideals in a local ring, in this chapter we will consider $R=k[x_1,\ldots,x_d]_{\mathfrak{m}}$ and $I$ an ideal generated by monomials. All the results of the second section still hold in this setting, because all the ideals involved are monomial ideals. Moreover, the analytic spread does not change.

For an $R$-module $M$, we can define the {\it $j$-multiplicity} of $I$ with respect to $M$ as
\[j(I,M)=\lim_{n\rightarrow \infty}\frac{(d-1)!}{n^{d-1}}\lambda_R\big(\HH{0}{\mathfrak{m}}{I^nM/I^{n+1}M}\big)\,.\]
In the case $M=R$, $j(I,R)$ will be denoted $j(I)$ as in the introduction.

\

The following proposition shows that we can compute $j(I)$ using the filtration $\{\overline{I^n}\}_{n\in\mathbb N}$. The proof is similar to \cite[Proposition~2.10]{FM1}.
\begin{prop}\label{jbar} Let $I$ be a monomial ideal, then
\[\displaystyle{j(I)=\lim_{n\rightarrow\infty}\frac{(d-1)!}{n^{d-1}}\lambda_R\big(\HH{0}{\mathfrak{m}}{\overline{I^n}/\overline{I^{n+1}}}\big)}\,.\]
\end{prop}

\begin{proof}
By \cite[Theorem~7.29]{Vas}, $\overline{I^{n+1}}=I\overline{I^n}$ for
$n\geqslant d$. From \cite[Theorem~3.11]{NU}, and the following exact sequence of $R$-modules

\[0\rightarrow \overline{I^d}\rightarrow R \rightarrow
R/\overline{I^d}\rightarrow 0 \,,\]
we obtain  $j(I,R)=j(I,\overline{I^d})+j(I,R/\overline{I^d})$.
But $I^d(R/\overline{I^d})=0$, so
$j(I,R/\overline{I^d})=0$ and then $j(I)=j(I,R)=j(I,\overline{I^d}).$

Now,
\begin{align*}
j(I)=j(I,\overline{I^d})&=\lim_{t\rightarrow\infty}\frac{(d-1)!}{t^{d-1}}\lambda_R\big(\HH{0}{\mathfrak{m}}{I^{t}\overline{I^{d}}/I^{t+1}\overline{I^{d}}}\big)\\
&= \lim_{t\rightarrow\infty}\frac{(d-1)!}{t^{d-1}}\lambda_R\big(\HH{0}{\mathfrak{m}}{\overline{I^{t+d}}/\overline{I^{t+d+1}}}\big)\\
&= \lim_{n\rightarrow\infty}\frac{n^{d-1}}{(n-d)^{d-1}} \frac{(d-1)!}{n^{d-1}}\lambda_R\big(\HH{0}{\mathfrak{m}}{\overline{I^{n}}/\overline{I^{n+1}}}\big)\,,
\end{align*}
and the result follows from the equality $\displaystyle{\lim_{n\rightarrow\infty}
\frac{n^{d-1}}{(n-d)^{d-1}}=1}$.
\end{proof}

\begin{thm}\label{main} Let $I\subset R$ be a monomial ideal. Then $j(I)=d!\operatorname{vol}(\operatorname{pyr}(I))$.
\end{thm}
\begin{proof}
By Lemma~\ref{jbar}, we compute
\[
j(I)=\lim_{n\rightarrow\infty}\frac{(d-1)!}{n^{d-1}}\lambda_R\big(\HH{0}{\mathfrak{m}}{\overline{I^{n}}/{\overline{I^{n+1}}}}\big)\,.
\]

\subsection*{Step 1:} The proof that $j(I)\leqslant d!\operatorname{vol}(\operatorname{pyr}(I))$:

Recall that $\HH{0}{\mathfrak{m}}{\overline{I^{n}}/\overline{I^{n+1}}}=\big((\overline{I^{n+1}}: \mathfrak{m}^{\infty})\cap \overline{I^{n}}\big) /\overline{I^{n+1}}$, so that

\begin{align*}
\lambda_R\big(\HH{0}{\mathfrak{m}}{\overline{I^{n}}/\overline{I^{n+1}}}\big)&=\#\Big(\mathbb Z^d\cap \big(\Gamma(\overline{I^{n+1}}: \mathfrak{m}^{\infty})\cap \operatorname{conv}(I^{n})\setminus \operatorname{conv}(I^{n+1})\big)\Big)\\
&\leqslant \#\Big(\mathbb Z^d\cap \big(\operatorname{conv}(I^{n+1})\cup \operatorname{pyr}(I^{n+1})\big)\cap \operatorname{conv}(I^{n})\setminus \operatorname{conv}(I^{n+1})\Big)
\end{align*}
where the last inequality holds by Lemma~\ref{satincone}.
By \cite[Corollary~3.4]{Sin}, we have $n\operatorname{conv}(I)=\operatorname{conv}(I^n)$, $n\operatorname{pyr}(I)=\operatorname{pyr}(I^n)$, and $n\operatorname{bd}(I)=\operatorname{bd}(I^n)$ for every $n\geqslant 1$. Note that $\operatorname{bd}(I)=\operatorname{conv}(I)\cap\operatorname{pyr}(I)$; then,
\begin{equation*}\label{eq:convconvconen}
\begin{aligned}
&(\operatorname{pyr}(I^{n+1})\cap \operatorname{conv}(I^{n}))\setminus \operatorname{conv}(I^{n+1})\\
&=
 \big((n+1)\operatorname{pyr}(I)\cap n\operatorname{conv}(I)\big)\setminus (n+1)\operatorname{conv}(I)\\
 &=\big((n+1)\operatorname{pyr}(I)\cap n\operatorname{conv}(I)\big) \setminus (n+1)\operatorname{bd}(I)\\
 &= \big((n+1)\operatorname{pyr}(I)\setminus (n+1)\operatorname{bd}(I)\big)\setminus \big(n\operatorname{pyr}(I)\setminus n\operatorname{bd}(I)\big)=\conen{n}(I)\,.
\end{aligned}
\end{equation*}
It follows that $\lambda_R\big(\HH{0}{\mathfrak{m}}{\overline{I^{n}}/\overline{I^{n+1}}}\big)\leqslant \#(\mathbb Z^d\cap\conen{n}(I)).$
Therefore, by Proposition~\ref{poly}, part b),
\[j(I)=\lim_{n\rightarrow\infty}\frac{(d-1)!}{n^{d-1}}\lambda_R\big(\HH{0}{\mathfrak{m}}{\overline{I^{n}}/\overline{I^{n+1}}}\big)\leqslant d!\operatorname{vol}(\operatorname{pyr}(I))\,.\]

\subsection*{Step 2:} The proof that $j(I)\geqslant d!\operatorname{vol}(\operatorname{pyr}(I))$:
\subsection*{Step 2a:} Reduction to the case of an ideal corresponding to a single facet:\\
\indent First we claim that it suffices to verify the inequality for a monomial ideal whose Newton polyhedron has a single bounded facet. Indeed, if the inequality holds for such ideals, write $J_1,\dots, J_t \subset I$ for the monomial ideals corresponding to the bounded facets of $I$ and $\mathcal{F}_1,\dots,\mathcal{F}_t$ for the corresponding facets, so that we have $\operatorname{bd}(I)=\bigcup_i \mathcal{F}_i$ and $\operatorname{bd}(J_i)=\mathcal{F}_i$. Then
since we have

\[\bigcup_{i=1}^t\Gamma\big((\overline{J_i^{n+1}}:_{\,\overline{J_i^n}}\mathfrak{m}^{\infty})/\overline{J_i^{n+1}}\big)\ \subseteq \  \Gamma\big((\overline{I^{n+1}}:_{\,\overline{I^n}}\mathfrak{m}^{\infty})/\overline{I^{n+1}}\big)\,,\]
and
\[\bigcup_{i=1}^t \conen{n}(\mathcal{F}_i)=\conen{n}(I)\,,\]
we have a containment
\begin{align*}
\conen{n}(I)\setminus\Gamma\big((\overline{I^{n+1}}:_{\,\overline{I^n}}\mathfrak{m}^{\infty})/\overline{I^{n+1}}\big) &\subseteq \bigcup_{i=1}^t \conen{n}(\mathcal{F}_i)\setminus \bigcup_{i=1}^t\Gamma\big((\overline{J_i^{n+1}}:_{\,\overline{J_i^n}}\mathfrak{m}^{\infty})/\overline{J_i^{n+1}}\big)\\
&\subseteq \bigcup_{i=1}^t (\conen{n}(\mathcal{F}_i)\setminus\Gamma\big((\overline{J_i^{n+1}}:_{\,\overline{J_i^n}}\mathfrak{m}^{\infty})/\overline{J_i^{n+1}})\big)\,.
\end{align*}
Notice $\#\Gamma\big((\overline{J^{n+1}}:_{\,\overline{J^n}}\mathfrak{m}^{\infty})/\overline{J^{n+1}}\big)=\lambda_R\big(\HH{0}{\mathfrak{m}}{\overline{J^n}/\overline{J^{n+1}}}\big)$ for any ideal $J$, so
\begin{align*}
\# (\conen{n}(I)\cap \mathbb Z^d)& - \lambda_R\big(\HH{0}{\mathfrak{m}}{\overline{I^n}/\overline{I^{n+1}}}\big)\\
&\leqslant \sum_{i=1}^t \Big(\# (\conen{n}(\mathcal{F}_i)\cap \mathbb Z^d) - \lambda_R\big(\HH{0}{\mathfrak{m}}{\overline{J_i^n}/\overline{J_i^{n+1}}}\big)\Big)\,.
\end{align*}
Thus, if the claimed inequality holds for each $J_i$, we have that
\begin{align*}
& \lim_{n\rightarrow\infty}\frac{(d-1)!}{n^{d-1}}\Big(\# \big(\conen{n}(I)\cap \mathbb Z^d\big) - \lambda_R\big(\HH{0}{\mathfrak{m}}{\overline{I^n}/\overline{I^{n+1}}}\big)\Big)\\
&\leqslant \sum_i \left(\lim_{n\rightarrow\infty}\frac{(d-1)!}{n^{d-1}} \# \big(\conen{n}(\mathcal{F}_i)\cap \mathbb Z^d \big) - \lim_{n\rightarrow\infty}\frac{(d-1)!}{n^{d-1}} \lambda_R\big(\HH{0}{\mathfrak{m}}{\overline{J_i^n}/\overline{J_i^{n+1}}}\big)\right)\\
&=\sum_i\big(d!\operatorname{vol}(\operatorname{pyr}(J_i))-j(J_i)\big)\leqslant 0\,.
\end{align*}
It follows that
\begin{align*}
 j(I)&=\lim_{n\rightarrow\infty}\frac{(d-1)!}{n^{d-1}}\lambda_R\big(\HH{0}{\mathfrak{m}}{\overline{I^n}/\overline{I^{n+1}}}\big)\\
&\geqslant\lim_{n\rightarrow\infty}\frac{(d-1)!}{n^{d-1}}\# \big(\conen{n}(I)\cap \mathbb Z^d\big)=d!\operatorname{vol}(\operatorname{pyr}(I))\,,
\end{align*}
where the last equality follows from Proposition~\ref{poly}, part b).

We subsequently assume that the Newton polyhedron of $I$ has a single bounded facet $\mathcal{F}$.

\subsection*{Step 2b:} Description of a rational polytope containing points contributing to $j(I)$:

Let $\mathcal{F}$ be $\operatorname{bd}(I)$. If $\dim(\mathcal{F})<d-1$, then $\dim(\cone(\mathcal{F}))<d$, so $\operatorname{vol}(\cone(\mathcal{F}))=0$, and there is nothing to show. Assume $\dim(\mathcal{F})=d-1$, and let $H$ be the affine $(d-1)$-plane spanned by $\mathcal{F}$. Let $\langle x,b \rangle = c$ be a defining equation for $H$. Recall that each entry of $b$ is positive for a bounded face, so that after rescaling $b$, we may assume that $c=1$, with each $b_j>0$.\\

\indent We now describe a region $\mathcal{R}_n\subset\conen{n}(\mathcal{F})$ such that for any $\alpha\in\mathbb Z^d\cap\mathcal{R}_n$, one has $x^{\alpha}\in
(\overline{I^{n+1}}:_{\,\overline{I^n}}\mathfrak{m}^{\infty})$.
Note that $x^{\alpha}\in
(\overline{I^{n+1}}:_{R} x_i^{\infty})$ if and only if $\alpha$ is in the image of $\pi_i$, the projection in the $e_i$ direction onto $\langle \alpha, b \rangle H$. That is,
\[
\Gamma(\overline{I^{n+1}}:_{R} x_i^{\infty})\cap \langle \alpha, b \rangle H=\mathbb Z^d\cap \pi_i\big((n+1)\mathcal{F}\big)\,.\]
Let $\mathcal{F}=\operatorname{conv}(v_1,\dots,v_t)$. Then
\[
\pi_i\big((n+1) v_j\big)=(n+1) v_j-\frac{(n+1)-\langle \alpha, b \rangle}{\langle e_i, b \rangle}e_i\,,
\]
so that
\[
\pi_i\big((n+1)\mathcal{F}\big)=\operatorname{conv}\!\Big((n+1) v_1-\frac{(n+1)-\langle \alpha, b \rangle}{\langle e_i, b \rangle}e_i,\dots,(n+1) v_t-\frac{(n+1)-\langle \alpha, b \rangle}{\langle e_i, b \rangle}e_i\Big)\,.
\]
Since each $b_j>0$, this is well-defined. Now,
\[
(\overline{I^{n+1}}:_{\,\overline{I^n}}\mathfrak{m}^{\infty})=\overline{I^n}\cap\bigcap_{i=1}^d(\overline{I^{n+1}}:_{R} x_i^{\infty})\,.
\]
We define a region
\[\mathcal{R}_n:=\conen{n}(\mathcal{F})\cap\bigcap_{i=1}^d \big((n+1)\mathcal{F}+\mathbb R_{\leqslant 0}\, e_i \big)\]
so that, by the above, $\mathcal{R}_n\cap \mathbb Z^d$ is contained in $\Gamma\big((\overline{I^{n+1}}:_{\,\overline{I^n}}\mathfrak{m}^{\infty})/\overline{I^{n+1}}\big)$. (In fact, these two sets are quickly verified to be equal, but we will only use the stated containment.)
We remark that for $n\leqslant s< (n+1)$,
\begin{equation*}\label{rncontain}
\mathcal{R}_n\cap sH \ \supseteq \  sH \cap (\mathcal{R}_n\cap nH + \mathbb R^d_{\geqslant 0}) \ \supseteq \
 \frac{s}{n}(\mathcal{R}_n \cap nH) \ = \ \frac{s}{n}(\mathcal{R}_n \cap n\mathcal{F})
\end{equation*}
where the second containment holds because the vectors in $\mathcal{R}_n\cap nH$ have nonnegative components. If we set $\tau_{n}=\frac{1}{n}(\mathcal{R}_n \cap n\mathcal{F})$, then $\conen{n}(\tau_{n})\subset \mathcal{R}_n$.
Note that $\tau_{n}$ has vertices in $\mathbb Q^d$.

We also claim that $\tau_{n}\subset \tau_{n+1}$. We have $\alpha\in\tau_{n}$ if and only if
\[
\frac{n}{n+1}\alpha+\frac{1}{n+1}\frac{1}{\langle e_i, b \rangle}e_i\in \mathcal{F}
\]
 for all $i$. By convexity of $\mathcal{F}$ since $\alpha\in\mathcal{F}$, if $\lambda \alpha + (1-\lambda) \frac{1}{\langle e_i, b \rangle}e_i \in \mathcal{F}$ then also $\lambda' \alpha + (1-\lambda') \frac{1}{\langle e_i, b \rangle}e_i \in \mathcal{F}$ for $0\leqslant\lambda\leqslant \lambda' \leqslant 1$. In particular,
\[
\frac{n+1}{n+2}\alpha+\frac{1}{n+2}\frac{1}{\langle e_i, b \rangle}e_i\in \mathcal{F}
\]
for all $i$, so that $\alpha\in\tau_{n+1}$. It follows by induction that $\tau_{n}\subset \tau_{n'}$ for $n\leqslant n'$.

\subsection*{Step 2c:} Using $\operatorname{pyr}(\tau_{n})$ to give a lower bound:\\
\indent Consider the distance between a vertex of $n\mathcal{F}$ and the corresponding vertex of $\pi_i\big((n+1)\mathcal{F}\big)$. We compute this distance as
\[
|nv_j-((n+1) v_j - \frac{1}{\langle e_i, b \rangle}e_i)|=|v_j-\frac{1}{\langle e_i, b \rangle}e_i|\,,
\]
which is bounded above uniformly in $n$ by $L:=\max_{i,j}\{|v_j-\frac{1}{\langle e_i, b \rangle}e_i|\}$. Then the region
${n\tau_{n}=\mathcal{R}_n \cap n\mathcal{F}}$ is the intersection of $(d+1)$ many polytopes, each of which is the convex hull of $t$ points $v'_1,\dots,v'_t$ such that $|v_j-v'_j|< L$ for all $j$. That is, each such polytope is an $L$-shaking of $n\mathcal{F}$ in the sense of Lemma~\ref{close}. Dividing through by $n$ we see that $\tau_{n}$ is the intersection of $(d+1)$ many polytopes that are all $\frac{L}{n}$-shakings of the polytope $\mathcal{F}$.

Given $0<c<1$, we may now apply Lemma~\ref{close} in the affine subspace $H$. We obtain, for a sufficiently large $M$, a $\tau_M$ such that $\operatorname{vol} (\tau_M) \geqslant c \operatorname{vol} (\mathcal{F})$ in $H$, and hence $\operatorname{vol} (\operatorname{pyr}(\tau_M)) \geqslant c \operatorname{vol} (\operatorname{pyr}(\mathcal{F}))$.

For $n>M$, we have from the previous step that $\tau_n\supseteq\tau_M$, so
\[
\conen{n}(\tau_M)\subseteq\conen{n}(\tau_n)\subseteq \mathcal{R}_n\,.
\]
Thus, if $\alpha\in \mathbb Z^d \cap \conen{n}(\tau_{M})$, then
\[
x^{\alpha} \in (\overline{I^{n+1}}:_{\,\overline{I^n}}\mathfrak{m}^{\infty})/\overline{I^{n+1}}=\HH{0}{\mathfrak{m}}{\overline{I^n}/\overline{I^{n+1}}}\,.
\]
Thus,
\begin{align*}
j(I) &=\lim_{n\rightarrow\infty}\frac{(d-1)!}{n^{d-1}}\lambda_R\big(\HH{0}{\mathfrak{m}}{\overline{I^n}/{\overline{I^{n+1}}}}\big)\\
&\geqslant \lim_{n\rightarrow\infty}\frac{(d-1)!}{n^{d-1}} \#\big(\mathbb Z^d \cap \conen{n}(\tau_{M})\big)=d!\operatorname{vol} (\operatorname{pyr}(\tau_M))\,,
\end{align*}
where the last equality follows from Proposition~\ref{poly}. Therefore, for all $c<1$, we have the inequality $j(I)\geqslant c (d!\operatorname{vol} (\operatorname{pyr}(\mathcal{F})))$, so
\[j(I)\geqslant d!\operatorname{vol} (\operatorname{pyr}(\mathcal{F}))=d!\operatorname{vol} (\operatorname{pyr}(I))\,,\]
as required.
\end{proof}

\begin{rmk}
If $I$ is an $\mathfrak{m}$-primary monomial ideal, the $j$-multiplicity is equal to the Hilbert-Samuel multiplicity, and $\operatorname{pyr}(I)$ is the complement of the Newton polyhedron in $\mathbb R_{\geqslant 0}^d$. In this way, Theorem~\ref{main} agrees with Teissier's result for $\mathfrak{m}$-primary monomial ideals.
\end{rmk}

\section{Normal Affine Semigroup Rings}

We next record that, with slight modifications, we can use the same proof to establish a similar result for a wider class of rings. In this section, we state this generalization, and describe necessary modifications to the argument.

By an \emph{affine semigroup ring}, we mean a ring $A=k[Q]$ that has as a $k$ vector space basis $\{x^q\,|\,q\in Q\}$, where $Q$ is a subsemigroup of $\mathbb Z^d$ (with the operation~$+$), and multiplication given by $x^{q_1} x^{q_2}=x^{q_1+q_2}$. We denote by $\mathfrak{m}_A$ its maximal homogeneous ideal. If $A$ is a normal ring, then there is a cone $\sigma\subseteq \mathbb R^d$ with finitely many extremal rays, each of which contains a lattice point ($\sigma$ is a \emph{rational cone}) and such that $A\cong k[\mathbb Z^d \cap \sigma]$, see \cite[Chapters~7~and~10]{MS}. We assume henceforth that $A$ is presented in this form. Additionally, suppose that $\sigma$ is \emph{pointed}, i.e., that it contains no nontrivial linear subspace of $\mathbb R^d$, and that $\dim(\sigma)=d$.

Set $r_1,\dots,r_s$ to be \emph{ray generators} for $\sigma$, i.e., minimal lattice points along the extremal rays of $\sigma$. Let $I$ be a monomial ideal of $R$ minimally generated by monomials $x^{v_1},x^{v_2},\ldots, x^{v_n}$. In this context, we define
\[\operatorname{conv}(I):= \operatorname{conv}(v_1,\ldots, v_n)+\sigma\,,\]
and, as in section two, $H_i=\{x\in \mathbb{R}^n\mid \langle x,b_i\rangle = c_i\}$, with $b_i\in \mathbb{Q}^d$, $c_i\in \mathbb{Q}$ for $i=1,\ldots, w$ to be the supporting hyperplanes of $\operatorname{conv}(I)$ so that
\[\operatorname{conv}(I)= H_{1}^+\cap H_{2}^+\cap\cdots \cap H_{w}^+ \, ,\]
where $H_i^+=\{x\in \mathbb{R}^n\mid \langle x,b_i\rangle \geqslant c_i\}$. We again assume that $H_1,\ldots,H_u$, are the hyperplanes corresponding to unbounded facets. We retain the other definitions, e.g., $\operatorname{pyr}$ and $\conen{n}$, from the preliminary section. Note that since we have chosen an embedding of our semigroup in $\mathbb Z^d\subset \mathbb R^d$, it makes sense to talk about volume. Our main result from the previous section holds in this context:

\begin{thm}\label{toric} Let $A=k[\mathbb Z^d \cap \sigma]_{\mathfrak{m}_A}$, where $\sigma$ is a $d$-dimensional pointed rational cone. Let $I\subset A$ be a monomial ideal. Then $j(I)=d!\operatorname{vol}(\operatorname{pyr}(I))$.
\end{thm}
\begin{proof}
The proof of Theorem~\ref{main} applies, after some slight changes:

It is easy to see that the inequalities $\langle r_j, b_i \rangle \geqslant 0$ hold for all $i,j$, and $\langle r_j, b_i \rangle > 0$ for $u+1 \leqslant i \leqslant w$ and all $j$, as in \cite[Lemma~1.1]{Sin}. Apply these inequalities in the proof of Lemma~\ref{satincone} mutatis mutandis to obtain the same conclusion. Step 1 in the proof of Theorem~\ref{main} follows.

To prove the other inequality in this setting, first note that for any monomial ideal $J$, we have $\alpha\in\Gamma(J :_A \mathfrak{m}_A^{\infty})$ if and only if there exist $a_j\in \mathbb R_{\geqslant 0}$ such that $\alpha+a_j r_j\in \Gamma(J)$ for all $j$. Thus, for a monomial ideal $I$ with a single bounded face $\mathcal{F}$, we may define the region
\[ \mathcal{R}_n:=\conen{n}(\mathcal{F})\cap\bigcap_{j=1}^m \big((n+1)\mathcal{F}+\mathbb R_{\leqslant 0}\, r_j \big)\]
so that $\Gamma\big((\overline{I^{n+1}}:_{\,\overline{I^n}}\mathfrak{m}^{\infty})/\overline{I^{n+1}}\big)=\mathbb Z^d\cap \mathcal{R}_n$. This region has all of the salient properties from the case of the polynomial ring $R$  (i.e., $\mathcal{R}_n$ is a rational polytope such that $\mathcal{R}_n \cap s\mathcal{F} \supseteq \frac{s}{n}(\mathcal{R}_n\cap n\mathcal{F})$ for  $n\leqslant s < (n+1)$), and we employ this to complete the proof of Step 2 as before.
\end{proof}

\begin{rmk}
The $j$-multiplicity of an ideal $I$ is greater than zero if and only if $I$ has maximal analytic spread (\cite[Lemma~3.1]{NU}). Then it follows from Theorems~\ref{main} and \ref{toric} that a monomial ideal of a normal affine semigroup ring has maximal analytic spread if and only if $\operatorname{conv}(I)$ has some bounded facet of dimension $d-1$. In the polynomial case, this fact is easily deduced from the characterization of analytic spread of monomial ideals stated in section~2.
\end{rmk}

\section{The $\varepsilon$-multiplicity as a volume}

In this section we follow the notation from the third section, i.e., \linebreak $R=k[x_1,\ldots,x_d]_{\mathfrak{m}}$. The $\varepsilon$-multiplicity was defined by Ulrich and Validashti \cite{UV} in 2011 as a generalization of the Buchsbaum-Rim multiplicity for submodules of free modules with arbitrary colength. In its simpler form for ideals, the $\varepsilon$-multiplicity is defined by
\[
\varepsilon(I)=\limsup_{n\rightarrow \infty}\frac{d!}{n^d}\lambda_R(\HH{0}{\mathfrak{m}}{R/I^n})\,,
\]
where the limit of the sequence has been shown to exist in wide generality, see \cite{Cut}.
For monomial ideals, the limit is known to exist and is a rational number as shown in \cite[Corollary~2.5]{HPV}. Nevertheless, unlike the $j$-multiplicity, there are examples of ideals for which the $\varepsilon$-multiplicity is not an integer; see \cite[Example~2.4]{CHS}. In this section, we will give a combinatorial proof of the existence and rationality of the limit in the monomial case, identifying $\varepsilon(I)$ with the normalized volume of a region with rational vertices.

\

Let $\operatorname{out}(I)$ be the region $(H_1^+\cap\cdots\cap H_u^+)\cap (H_{u+1}^-\cup\cdots\cup H_{w}^-)$ if the ideal $I$ has maximal analytic spread, and empty otherwise. Notice that by the proof of Lemma~\ref{satincone} we have $H_1^+\cap\cdots\cap H_u^+\subset \operatorname{conv}(I)\cup \cone(I)$ and then $\operatorname{out}(I)$ is contained in $\cone(I)\cup \operatorname{bd}(I)$ which is bounded.
\begin{thm}
Let $I\subset R$ be a monomial ideal. Then the limit in the definition of $\varepsilon(I)$ exists, and $\varepsilon(I)=d!\operatorname{vol}(\operatorname{out}(I))$, which is a rational number.
\end{thm}

\begin{proof}
Since the functor $\HH{0}{\mathfrak{m}}{-}$ is sub-additive on short exact sequences, we have
\[
\lambda_R\big(\HH{0}{\mathfrak{m}}{R/I^n}\big)\leqslant\sum_{i=0}^{n-1} \lambda_R\big(\HH{0}{\mathfrak{m}}{I^i/I^{i+1}}\big).\]
The right hand side is the sum transform of the function that defines the $j$-multiplicity, and hence for $n\gg 0$ it is equal to a polynomial of degree $d$ and leading coefficient $\frac{j(I)}{d!}$, see \cite{NU}. It follows that $\varepsilon(I)\leqslant j(I)$, so we can assume that $I$ has maximal analytic spread.

\subsection*{Step a:} Existence of the limit for the filtration $\{\overline{I^n}\}_{n\in\mathbb N}$:
\begin{align*}
\Gamma\big((\overline{I^n}:_R\mathfrak{m}^{\infty})/\overline{I^n}\big) &= \Gamma(\overline{I^n}:_R\mathfrak{m}^{\infty})\setminus \Gamma(\overline{I^n})\\
&= \Gamma(H_{1}^+\cap H_{2}^+\cap\cdots \cap H_{u}^+)\setminus \Gamma(H_{1}^+\cap H_{2}^+\cap\cdots \cap H_{w}^+)\\
&= \Gamma((H_1^+\cap\cdots\cap H_u^+)\cap (H_{u+1}^-\cup\cdots\cup H_{w}^-))\setminus  \bigcup_{i=u+1}^{w} \Gamma(\mathcal{F}_i)
\end{align*}
where the equality follows from the fact that $H_{1}^+\cap H_{2}^+\cap\cdots \cap H_{w}^+\cap H_{i}^-=\mathcal{F}_i$ for every $i$. From this we conclude
$$\Gamma\big((\overline{I^n}:_R\mathfrak{m}^{\infty})/\overline{I^n}\big)=\mathbb Z^d\cap \big(\operatorname{out}(I)\setminus \operatorname{bd}(I)\big).$$
Let $\mathcal{Q}=\operatorname{conv}(\operatorname{vert}(I))$ and $\mathcal{Q}'=\operatorname{conv}(\mathcal{Q},\operatorname{out}(I))$; it is easy to check that there is an equality $\mathcal{Q}'\setminus \mathcal{Q}=\operatorname{out}(I)\setminus \operatorname{bd}(I)$. By \cite[Lemma~3.3]{Sin}, the hyperplanes $\{nH_i\}$, for $1\leqslant i\leqslant {w}$, are the supporting hyperplanes of $\operatorname{conv}(I^n)$ for each $n\geqslant 1$. Then we also have $n\mathcal{Q}'\setminus n\mathcal{Q}=\operatorname{out}(I^n)\setminus \operatorname{bd}(I^n)$.

Hence,
\begin{align*}\lambda_R\big(\HH{0}{\mathfrak{m}}{R/\overline{I^n}}\big)&=\#\big(\mathbb Z^d\cap(\operatorname{out}(I^n)\setminus \operatorname{bd}(I^n))\big)\\
&=\#\big(\mathbb Z^d\cap(n\mathcal{Q}'\setminus n\mathcal{Q})\big)\\
&=E_{\mathcal{Q}'}(n)-E_{\mathcal{Q}}(n)\,,
\end{align*}
where the latter is the difference of two Ehrhart quasi-polynomials of the form
\[
\big(\operatorname{vol}(\mathcal{Q}')-\operatorname{vol}(\mathcal{Q})\big)n^d+O(n^{d-1})=\operatorname{vol}(\operatorname{out}(I))n^d+O(n^{d-1})\]
 (see proof of Lemma~\ref{poly}), and the result follows.

\subsection*{Step b:} Existence of the original limit:

By \cite[Theorem~7.29]{Vas}, $\overline{I^{n+1}}=I\overline{I^n}$ for $n\geqslant d$. Then we have the following exact sequences for $n\geqslant 0$:

\[0\rightarrow \overline{I^n}/I^n \rightarrow R/I^n \rightarrow R/\overline{I^n} \rightarrow 0 \,,\]

\[0\rightarrow I^n/\overline{I^{n+d}} \rightarrow R/\overline{I^{n+d}} \rightarrow R/I^n \rightarrow 0\,, \]
and the following inequalities

\begin{align}
\lambda_R\big(\HH{0}{\mathfrak{m}}{R/\overline{I^{n+d}}}\big) &\leqslant  \lambda_R\big(\HH{0}{\mathfrak{m}}{R/I^n}\big)\!+\!\lambda_R\big(\HH{0}{\mathfrak{m}}{I^n/\overline{I^{n+d}}}\big) \label{ineq2}\\
&\leqslant \lambda_R\big(\HH{0}{\mathfrak{m}}{R/\overline{I^n}}\big)\!+\!\lambda_R\big(\HH{0}{\mathfrak{m}}{\overline{I^n}/I^n}\big)\!+\!\lambda_R\big(\HH{0}{\mathfrak{m}}{I^n/\overline{I^{n+d}}}\big)\label{ineq3}.
\end{align}
Now, $\lambda_R\big(\HH{0}{\mathfrak{m}}{I^n/\overline{I^{n+d}}}\big)\leqslant \lambda_R\big(\HH{0}{\mathfrak{m}}{\overline{I^n}/\overline{I^{n+d}}}\big)\leqslant \sum_{i=0}^{d-1} \lambda_R\big(\HH{0}{\mathfrak{m}}{\overline{I^{n+i}}/\overline{I^{n+i+1}}}\big)$,  so
\[
\limsup_{n\rightarrow \infty}\frac{d!}{n^d}\lambda_R\big(\HH{0}{\mathfrak{m}}{I^n/\overline{I^{n+d}}}\big)\leqslant \sum_{i=0}^{d-1}\limsup_{n\rightarrow \infty} \frac{d!}{n^d}\lambda_R\big(\HH{0}{\mathfrak{m}}{\overline{I^{n+i}}/\overline{I^{n+i+1}}}\big)=0\,,\]
 where the last equality holds by Proposition~\ref{jbar}.

\noindent Therefore, $\lim _{n\rightarrow \infty}\frac{d!}{n^d}\lambda_R\big(\HH{0}{\mathfrak{m}}{I^n/\overline{I^{n+d}}}\big)=0$.

\noindent Similarly, for $n\geqslant d$,
\[\lambda_R\big(\HH{0}{\mathfrak{m}}{\overline{I^n}/I^n}\big)\leqslant \lambda_R\big(\HH{0}{\mathfrak{m}}{I^{n-d}/I^n}\big)\leqslant \sum_{i=0}^{d-1}\lambda_R\big(\HH{0}{\mathfrak{m}}{I^{n-d+i}/I^{n-d+i+1}}\big)\,,\]
and then $\lim _{n\rightarrow \infty}\frac{d!}{n^d}\lambda_R \big(\HH{0}{\mathfrak{m}}{\overline{I^n}/I^{n}}\big)=0$.

\noindent Using these two limits in $\eqref{ineq2}$ and $\eqref{ineq3}$, we obtain
\begin{align*}
d!\operatorname{vol}(\operatorname{out}(I))&=\liminf_{n\rightarrow \infty}\frac{d!}{n^d}\lambda_R\big(\HH{0}{\mathfrak{m}}{R/\overline{I^{n+d}}}\big)\\
 &\leqslant \liminf_{n\rightarrow \infty}\frac{d!}{n^d}\lambda_R\big(\HH{0}{\mathfrak{m}}{R/I^n}\big)\\
 &\leqslant \limsup_{n\rightarrow \infty}\frac{d!}{n^d}\lambda_R\big(\HH{0}{\mathfrak{m}}{R/I^n}\big)\\
 &\leqslant \limsup_{n\rightarrow \infty}\frac{d!}{n^d}\lambda_R\big(\HH{0}{\mathfrak{m}}{R/\overline{I^n}}\big)= d!\operatorname{vol}(\operatorname{out}(I))
\end{align*}
which finishes the proof.
\end{proof}

\section{Examples}
\begin{figure}
\begin{minipage}[b]{1\linewidth}
\centering
\includegraphics[scale=.5]{fig2col.pdf}
\caption{Regions for the ideal $(y^4,x^2y,xy^2)$}
\label{fig2}
\end{minipage}\\
\begin{minipage}[b]{1\linewidth}
\centering
\includegraphics[scale=.4]{fig31col.pdf}
\caption{Regions for the ideal $(x^6,y^6,z^6,x^2yz,xy^2z,xyz^2)$}
\label{fig3}
\end{minipage}
\end{figure}

\begin{xmp}Let $I=(y^4,x^2y,xy^2)$. We compute the $j$-multiplicity as two times the area of the green region in Figure~\ref{fig2}, obtaining $j(I)=7$. For the $\varepsilon$-multiplicity, we take two times the area of the portion of the green region that lies above the dotted line, obtaining $\varepsilon(I)=5$.
\end{xmp}

\begin{xmp} Let $I=(x^6,y^6,z^6,x^2yz,xy^2z,xyz^2)$. This monomial ideal is $\mathfrak{m}$-primary, so we can compute its Hilbert-Samuel multiplicity as the volume underneath the bounded faces of its Newton polyhedron, which is depicted in Figure~\ref{fig3}. We decompose this region as three regions under the yellow faces corresponding to monomial ideals ${I_1=(x^6,y^6,x^2yz,xy^2z)}$, $I_2=(y^6,z^6,xy^2z,xyz^2)$, $I_3=(x^6,z^6,x^2yz,xyz^2)$, and one region under the red face corresponding to $I_4=(x^2yz,xy^2z,xyz^2)$. We compute $j(I_1)=j(I_2)=j(I_3)=42$, $j(I_4)=4$, so that $e(I)=j(I)=130$.
\end{xmp}

\begin{xmp} \cite[Example~2.4]{CHS} Let $I=(xy,yz,zx)$. The region $\operatorname{out}(I)$ is the tetrahedron with vertices $\{(1,0,1),(1,1,0),(0,1,1),(\frac12,\frac12,\frac12)\}$, and its volume is $\frac{1}{12}$. Thus, $\varepsilon(I)=\frac12$.
\end{xmp}

\begin{xmp} For a graph $G$ on the vertex set $\{1,\dots,d\}$, the \emph{edge ideal} of $G$ is the monomial ideal
\[ I_G:= (x_i x_j \,|\, \{i,j\}\, \text{is an edge of} \ G) \subset k[x_1,\dots,x_d] \,. \]
The region $\operatorname{bd}(I_G)$ is known in the literature as the \emph{edge polytope} of $G$, see \cite{OH}. By Theorem~\ref{main}, the $j$-multiplicity $j(I_G)$ is $(d-1)!\cdot h$ times the volume of the edge polytope of $G$, where $h$ is the distance from the origin to the plane $\sum x_i=1$. As a particular example, let $C_d$ be the cycle on $d$ vertices. Then,
\[ j(I_{C_d}) = \begin{cases} 0 &\mbox{if } d \ \mbox{is even} \\
2 & \mbox{if } d \ \mbox{is odd}.\end{cases}
\]

\end{xmp}

\section*{Acknowledgements}
The authors would like to thank their Ph.D. advisors, Anurag Singh and Bernd Ulrich. Additionally, we thank Alexandra Seceleanu for her help in finding an appropriate reference for Lemma~$\ref{close}$, and for many helpful comments on an earlier draft of this paper. Both authors also wish to thank Bernd Sturmfels for his enthusiastic instruction on monomial ideals. We are also grateful to the referee for his or her thorough and helpful corrections. This material is based upon the work supported by the National Science Foundation under Grant No. 0932078 000, while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall semester of 2012. The first author was also supported in part by NSF grants DMS 0758474 and DMS 1162585, and the second author by NSF grant DMS 0901613.

redpajama_set_name: RedPajamaArXiv


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