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Let $S$ be an algebraic set generated by polynomials in $\mathbb{C}[x_1, \ldots, x_n]$. Define the "degree" of $S$ as $$ \min( \deg(f_1) + \deg(f_2) + \ldots + \deg(f_m) : f_1, f_2, \ldots, f_m \text{ generates } S ), $$ where $\deg(f_i)$ is the largest degree of a nonzero monomial in $f_i$.

Is there such a definition in algebraic geometry?

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So the degree of a point in $\mathbb{A}^n$ would be $n$? –  quim Mar 9 '13 at 6:46
    
Yes, the degree on a point is $n$. –  jsliyuan Mar 9 '13 at 14:00
    
I have never seen this notion but it seems funny. It is not at all the usual degree, but why not? Can you do something with this notion? Like for the usual degree, compare degree of intersection of two algebraic sets with the degree of the two sets? Or relate it to something else? –  Jérémy Blanc Mar 9 '13 at 20:08
    
yeah, maybe a Bezout-like theorem –  IMeasy Mar 10 '13 at 11:53
    
The motivation is to define a concept "algebraic immunity" over $\mathbb{C}[x_1, \ldots, x_n]$. The "algebraic immunity" of $f$ on ring $R = \mathbb{F}_2[x_1, \ldots, x_n]/(\ldots, x_i^2 = x_i, \ldots)$ is defined as the minimum degree of some $g$ such that $f$ restricted to the zeros of $g$ is a constant. –  jsliyuan Mar 12 '13 at 15:09
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