Let $\mathbb{C}[x,y]$ be the polynomial ring with variables $x,y$ and coefficient in $\mathbb{C}$.

Let $f,g\in \mathbb{C}[x,y]$.

Let $(f,g)$ be the ideal of $\mathbb{C}[x,y]$ generated by $f,g$.

Given $h\in \mathbb{C}[x,y]$, how to determine whether $h\in (f,g)$ or not?

I have tried some examples by the online programming "sagemath".

Are there any methods that can give a proof?

complexnumbers for the coefficients? If you had suggestedrationalcoefficients, I would not be asking this question, but even to determine thezeronessof a complex number that was arrived at by some analytic process seems to me to pose a serious problem. $\endgroup$ – Lubin Apr 27 '15 at 17:37