Effective de Rham Cohomology - The Hypersurface Case

TL;DR

This paper establishes an effective degree bound for generators of algebraic de Rham cohomology of smooth affine hypersurfaces, aiding computational methods and addressing questions in differential equations.

Contribution

It provides a new explicit degree bound for generators of de Rham cohomology of smooth hypersurfaces, linking algebraic geometry with computational and differential equation applications.

Findings

De Rham cohomology of hypersurfaces can be generated by forms of degree d^{O(pn)}.

The result improves understanding of the algebraic structure of cohomology.

Applications include algorithmic computation and differential equations theory.

Abstract

We prove an effective bound for the degrees of generators of the algebraic de Rham cohomology of smooth affine hypersurfaces. In particular, we show that the de Rham cohomology H_dR^p(X) of a smooth hypersurface X of degree d in C^n can be generated by differential forms of degree d^O(pn). This result is relevant for the algorithmic computation of the cohomology, but is also motivated by questions in the theory of ordinary differential equations related to the infinitesimal Hilbert 16th problem.