I've spent some time recently looking at some Groebner bases for some specific ideals coming from problems in computer vision. The generators are not sparse, and they all have the same degree (specifically, degree 2). I'm actually surprised by the relatively small number of basis elements needed in my examples. So it's got me thinking of some general questions about the number of elements in any reduced Groebner basis.
For concreteness, let's say that we've got an ideal $I$ generated by $m$ polynomials, each of degree $d$, in a polynomial ring in $n$ indeterminates over a field $k$, with a deglex term order. If it matters, my ideals will not be zero-dimensional.
- If the generators of $I$ are "sufficiently general", what can we say about the expected number of elements in a reduced Groebner basis?
- Can we say anything about bounds on the number of elements in such a reduced Groebner basis, in terms of $m$ and $d$? In general, I've heard that the bound is "doubly-exponential", but in what?
- For my needs, the polynomials are not very sparse. But if we did have many of the coefficients in our generators being $0$, can we use that to get some different bounds?