Consider two polynomial functions $f(x,y)$ and $g(x,y)$.

I would like to find the restriction of $f$ onto the algebraic curve $g(x,y)=0$.

My idea was to write a sort of Euclid algorithm $$ f(x,y) = g(x,y) p(x,y)+r(x,y) $$ but it happens I didn't find such decomposition with condition on the degree of $r$ and $g$ and what I read leads me to Gröbner basis... which I don't know

My question is the following: what mathematical subject should I study to tackle this problem ? If in Gröbner basis, what subdomain ?

Do you have any reference which present this problem not from the theoretical point of view but from the computational point of view (using maple or mathematica or sage) ?


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