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$K[x_{1},\ldots ,x_{n}]$ 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. $x^3$)

user745
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    This algebra is always finitely generated over $K$. What do you mean by finitely generated? – Slup Jul 02 '16 at 23:58
  • Are you trying to find a Noether normalization or something? – Hoot Jul 03 '16 at 00:04
  • Finitely generated as what? An algebra? Over what? $K$? $K[x_1, \cdots, x_n]$? As Slup says, what do you mean by finitely generated? – Hamed Jul 03 '16 at 00:07
  • $K[x_{1}\ldots x_{n}]/I$ is finitely generated if any element of it can be represented as linear combination of finite number of elements. I guess if I is trivial zero ideal with only {e} element $K[x_{1}\ldots x_{n}]/I$ is not finitely generated. Don't sure for other cases may be statement will be always true. If it is so, could you please write some kind of proof for it? – user745 Jul 03 '16 at 07:36
  • Those quotients are usually not finite-dimensional vector spaces over $K$. The ring would have to have Krull dimension $0$, so is describing a finite scheme of points in the plane. – Hoot Jul 03 '16 at 09:24
  • For example let's take R as K, so it is a field and we know it's Krull dimension. How will it help us to determine is $R[x_1,/ldots , x_n]/I$ is finitely generated for our I? If I is zero ideal, it is not finitely generated, isn't it? – user745 Jul 03 '16 at 11:11

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"Finite generated" was about finite number of points. If line, plane, etc - not finite generated. Solved for this definition: just look at the last polinomial at Groebner basis, if it is contains only one variable (e.g. $x^3$) it is finitely generated.

user745
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