We know that Hilbert's Nullstellensatz is valid for $\Bbb{C}[X]$, as $\Bbb{C}$ is a closed field.
Let us consider the ideal $(x+y,x-y)\subset \Bbb{C}[x,y]$. Clearly, $Z((x+y,x-y))=\{(0,0)\}$. Now note that $x^2+y^3=0$ also satisfies $(0,0)$. Hence, by Hilbert's Nullstellensatz, we should have $(x^2+y^3)^n=f(x,y)(x+y)+g(x,y)(x-y)$ for some $n\in\Bbb{N}$. Is there an explicit way to determine $f(x,y)$ and $g(x,y)$?