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?



ideals of polynomial ring with complex number coefficients
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. 



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>  "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).
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