I have that R is the k-algebra (k is a field) finitely generated by S={f1,...,fm}⊂k[x1,⋯,xn] and this set of polynomials is minimal with respect to inclusion (i.e., e do not have redundant elements). However, I know that the fi's are algebraically dependent. Can be the number m=|S| unique? What are the conditions?
When you say that the k-algebra is generated, then this typically means generated as a vector space. If this is what you mean, then this answers your questions: m is the dimension. If you mean something else, please clarify.
I think that, in general, very little can be said about the number m.
As examples, take the Veronese subring R(d) of k[x1, ..., xn] generated (minimally) by all monomials of degree d. If d >1 then these generators are algebraically dependent, and as d grows the number of monomials grows too.
In general, let R be the k-subalgebra of a polynomial ring k[x1, ..., xn] generated minimally by polynomials {f1, ..., fm}. If we choose e.g. the fi's homogeneous, then m is the embedding dimension of R, whereas the transcendence degree of R over k is the Krull dimension of R. The only relation among these numbers is dim(R) ≤ min{n,m}.
Algebra R is isomorphic K[x1, ...,xm] since the fi's are algebraically independent Consequently m - invariant of algebra R.
Thanks for the answers. I'm working with a fixed-point subring of the action of a certain group. It seems that in general there is no way of finding such an m.
Generally, this number is not unique. For example, {x} and {x2,x+x2} are both minimal sets of generators of k[x]. For the subalgebra k of k[x1,...,xn] every minimal set of generators has 0 elements (duh). I suspect that this is the only example where the number of elements in a minimal set of generators is unique.
The situation is different in the graded case. The polynomial ring k[x1,...,xn] has a grading. If R is a graded sub-algebra then every minimal set of homogeneous generators of R has d elements, where d is the dimension of the k-vector space m/m2 and m is the ideal generated by all homogeneous polynomials in R of positive degree. If R is not finitely generated then d is infinite. Every minimal set of (not necessarily homogeneous) generators of R must have at least d elements.
Dear Harm,
"Every minimal set of (not necessarily homogeneous) generators of R must have at least d elements."
could you please give me some source of this result? In my case, the algebra R is almost graded so maybe I can get good lower bounds on the minimal number of generators.
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