[Math] ideals of polynomial ring with complex number coefficients

Question

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?

Answer 1

You should use "Gröbner basis", (Groebner) . see the book by "Cox D., Little J., O'Shea D.": named "Ideals, Varieties, and Algorithms", for example. In page.82 they have:

Corollary.2. Let $G = \{g_1, \cdots , g_t\}$ be a Groebner basis for an ideal $I \subset k[x_1, \cdots , x_n]$ and let $f \in k[x_1, \cdots , x_n]$. Then $f \in I$ if and only if the remainder on division of $f$ by $G$ is zero.

"Buchberger’s Algorithm", (page.88 of the book), helps you to produce the Groebner basis. Also "CoCoA" can compute Gröbner basis with the command "GBasis(I)" (for special field).