How to prove using Groebner Bases that $x^2 +y^2 = −1, x^3 +y^3 = −1, x^5 +y^5 = −1$ is inconsistent in $\Bbb C\;^2$?

Question

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

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

Answer 1

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.