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I am looking for a generalization to fundamental theorem of algebra for several complex variables functions or systems. If such theorem exists, it should concisely relates the number of zeros of multivariate polynomial system to the order of each function of the polynomial system.

if such theorem exist please provide the proof or a proper citation.

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  • $\begingroup$ Since the zero set of a complex polynomial in two variables could be homeomorphic to the 2-torus, and in particular could be uncountable, it is hard to see what one could hope to be true in this direction $\endgroup$
    – Yemon Choi
    Apr 9, 2015 at 23:26
  • $\begingroup$ i mentioned system of polynomial. so we have the same number of polynomial functions as the variables numbers. so it should have finite number of roots. $\endgroup$ Apr 9, 2015 at 23:49
  • $\begingroup$ Bezout's theorem. $\endgroup$ Apr 10, 2015 at 4:09

1 Answer 1

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You may be looking for Bezout's Theorem.

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  • $\begingroup$ would you please write the complete statement of the theorem in accordance to the fundamental theorem of algebra? $\endgroup$ Apr 10, 2015 at 0:17
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    $\begingroup$ @behradmahboobi Why don't you follow the link and see for yourself? $\endgroup$
    – Igor Rivin
    Apr 10, 2015 at 0:33
  • $\begingroup$ Links can break, and when they do, answers like this become useless. Less likely for Wikipedia entries, but still a good practice not to answer with just links. $\endgroup$
    – bob
    Jun 16, 2022 at 13:52

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