# Polynomial reduction on algebraic curve

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) ?