One may suppose that M is the kernel of a morphism S^n to S^m of free S-modules, and that S is a polynomial ring in x_1,...,x_k with complex coefficients.
Dear Alex:
Do you wish to add some constraints to your question? Otherwise...?
Best, Aron
Dear Alex:
If the symmetric algebra S(M) has the "expected" dimension k+rank(M) then the minimal number of generators is bounded by k+1rank(M)+1. This can be imporved by 1 if M is a module of linear type.
In general, the dimension of S(M) is the expected one if S(M) is an unmixed algebra.
Also, quite generally, dim S(M) coincides with the Forster number, wich is the maximum over suitable sum of local information.
Dear Aron,
Thank you very much for your answer. My interest, as one can easily guess, is in the case when M is the S-module AR(f)=D_0(f) of all Jacobian syzygies of a homogeneous polynomial f in 3 or 4 variables. Is your answer explicit in this case?
Best wishes, Alex
Dear Alex:
I believe I have something in the case the polynomial is rather the determinant of a large matrix, but mostly about the number of independent linear syzygies. Let me think a bit about the situation of few variables.
Best,
Aron
Dear Alex:
I know that a possible set of existing generators comes from Groebner basis for a free
submodule M over S^n. It can be minimal under certain condition (pertaining specific set of syzygies called S-pairs). Now the question is: Is the Groebner basis of M makes a minimal set (in general)? The most comprehensive work in this regard is the book written by Adam and Loustanau. Recently I started to work on some encoding procedure to determine the infectivity of a given ideal via polynomial expressions.
I think it is uninteresting topic. Hope the above comments would help.
Best,
Rostam Sabeti
Great Lakes association for algebra and computation
The simplest instance where the above question is interesting the following: let S be the polynomial ring in x,y,z and let I be a SATURATED ideal in S. Is there a bound on the number of generators for I ?
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