Simplify multivariate polynomial modulo ideal generators

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Suppose we have some multivariate polynomials $f, g_1, g_2, g_3 \in F[x,y,z, ... ]$ where $F$ is some field, e.g $\mathbb{Q}$ or some finite field. I'm asking for an algorithm that "simplifies" $f$ assuming $g_1 = g_2 = g_3 = 0$. That is, I want to find a simple/short representative of $f$ in the quotient ring $F[x,y,z,...]/\langle g_1, g_2, g_3 \rangle$. I suspect it has something to do with Grobner bases but I haven't found anything concrete yet.

Thank you.