Surely. Maybe you can give me a counterexample. Let's just consider GF(2). Give me a set of generators with polynomials of degree <=2 over n variables, such that, 1 is a member of the ideal (i.e. generators generate the whole ring of polynomials) - so if I understand this correctly, Grobner basis is just 1, and, during the construction of the Grobner basis, you need to construct a polynomial of an arbitrarily high degree (or let's say degree 7). And there is no way to avoid this (or same degree) polynomial during the basis construction by finding a lower degree polynomial first.
See my problem? Somehow, I feel these syzygies need to be constructed from the degree 2 polynomials (the initial set), but then we can probably only work with the polynomials of the low degree instead, from which those syzygies are constructed.
To me, this is a very interesting question. Because if we can find the Grobner basis (even for just those ideals that coincide with the whole ring) in GF(2) in a way that we only need to construct polynomials of bounded degree, then there is a polynomial algorithm for SAT. Since most people believe there isn't such algorithm, I would really like to see a counterexample to this situation.
See my problem? Somehow, I feel these syzygies need to be constructed from the degree 2 polynomials (the initial set), but then we can probably only work with the polynomials of the low degree instead, from which those syzygies are constructed.
To me, this is a very interesting question. Because if we can find the Grobner basis (even for just those ideals that coincide with the whole ring) in GF(2) in a way that we only need to construct polynomials of bounded degree, then there is a polynomial algorithm for SAT. Since most people believe there isn't such algorithm, I would really like to see a counterexample to this situation.