Cohomology, stratifications and parametric Gröbner bases in characteristic zero

Let PK(n,d) be the set of polynomials in n variables of degree at most d over the field K of characteristic zero. We show that there is a number cn,d such that if f PK(n,d) then the algebraic de Rhamcohomology group HdRi(Kn Var(f)) has rank at most cn,d. We also show the existence of a bound cn,d,l for the ranks of de Rhamcohomology groups of complements of varieties in n-space defined by the vanishing of l polynomials in PK(n,d). In fact, if is the ithBetti number of the complement of the corresponding variety, we establish the existence of a -algebraic stratification on PK(n,d)l such that βi is constant on each stratum. The stratifications arise naturally from parametric Gröbner basis computations; we prove for parameter-insensitive weight orders in Weyl algebras the existence of specializing Gröbner bases