## Intersection of curves

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

Let $f(x,y)=0$ and $g(x,y)=0$ be curves in $\mathbb R^2$. Assume that the origin $(0,0)\in \mathbb R^2$ is a $d$-fold point of $f$ and an $e$-fold point of $g$, respectively. Let $f_d(x,y)$ be the sum of the terms of degree $d$ in $f(x,y)$, $g_e(x,y)$ be the sum of the terms of degree $e$ in $g(x,y)$. If $f_d(x,y)$ and $g_e(x,y)$ have a common factor of positive degree, then the intersection multiplicity $I_O(f,g)>de.$

-

5
mfn -- 1. what is the question? 2. there is a much better chance of getting a good answer if you make the question easier to read (hint: dollars). – algori Oct 12 2011 at 3:51

7
@algori: the dollar hint makes you sound like you're asking for a bribe. Of course, if you'd asked for latex instead, who knows what it would make it sound like... – Thierry Zell Oct 12 2011 at 12:11

## 1 Answer

This is proved in Fulton's "Algebraic Curves", available online here. The precise reference is Section 3.3, property (5) on p. 37.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 17, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9082209467887878, "perplexity_flag": "head"}