$K[x_{1},\ldots ,x_{n}]$ is polynomial ring. $I = (f_{1},\ldots ,f_{m})$ - it's ideal, where $(f_{1},\ldots ,f_{n})$ is finite set of polinomials. Task is to programmically determine if quotient(factor) ring over this ideal $K[x_{1},\ldots ,x_{n}]/I$ is finitely generated.
I'm trying to write a program, that solves this task at least for some cases of K (for example rational or complex numbers) but now I'm stuck with it. The only suggestion I have that Groebner basis should help somehow. Any ideas how to create algorithm what solve this task for given input $I = (f_{1},\ldots ,f_{m})$ in R, C or any other field or ring?
Solved: finite generated (finite number of points) when last polynomial in Groebner basis contains only one variable (e.g. $x^3$)