# 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 $\Bbb C\;^2$?

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

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

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

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.