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How can it be proved using Groebner Bases that the following system of equations is inconsistent in $\Bbb C\;^2$ ?

$x^2 +y^2 = −1, x^3 +y^3 = −1, x^5 +y^5 = −1$

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    $\begingroup$ I assume the numbers here are supposed to be powers? $\endgroup$
    – AshSeifert
    Commented May 20, 2023 at 0:43
  • $\begingroup$ I have taken the liberty of implementing the suggestion of @Ash. $\endgroup$ Commented May 20, 2023 at 1:06
  • $\begingroup$ @AshSeifert Yes, numbers are supoposed to be powers. Sorry, I did not add the correct formatting $\endgroup$
    – Alfonso_MA
    Commented May 20, 2023 at 9:50

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A system $S$ of polynomial equations in $\mathbb{C}^n$ is inconsistent if and only if there is no solution in $\mathbb{C}^n$. Translated into algebro-geometric terms, this holds if and only if the affine variety $V(S)$ is empty. By the Nullstellensatz, this is true if and only if the system $S$ generates $\mathbb{C}[x_1,x_2,…,x_n]$. This is something you can compute via Groebner bases.

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