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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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    $\begingroup$ So the degree of a point in $\mathbb{A}^n$ would be $n$? $\endgroup$
    – quim
    Mar 9, 2013 at 6:46
  • $\begingroup$ Yes, the degree on a point is $n$. $\endgroup$
    – jsliyuan
    Mar 9, 2013 at 14:00
  • $\begingroup$ 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? $\endgroup$ Mar 9, 2013 at 20:08
  • $\begingroup$ yeah, maybe a Bezout-like theorem $\endgroup$
    – IMeasy
    Mar 10, 2013 at 11:53

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