Cohomology, stratifications and parametric Gröbner bases in characteristic zero

✍ Scribed by Uli Walther

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 193 KB
• Volume: 35
• Category: Article
• ISSN: 0747-7171
• DOI: 10.1016/s0747-7171(03)00019-1

✦ Synopsis

Let P K (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 c n,d such that if f ∈ P K (n, d) then the algebraic de Rham cohomology group H i dR (K n \Var( f )) has rank at most c n,d . We also show the existence of a bound c n,d,l for the ranks of de Rham cohomology groups of complements of varieties in n-space defined by the vanishing of l polynomials in P K (n, d). In fact, if β i : P K (n, d) l → N is the ith Betti number of the complement of the corresponding variety, we establish the existence of a Q-algebraic stratification on P K (n, d) l such that β i 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.