Effective de Rham Cohomology - The General Case

TL;DR

This paper generalizes bounds on polynomial degrees of differential forms representing de Rham cohomology classes for smooth affine varieties, aiding computational and theoretical aspects of algebraic geometry and differential equations.

Contribution

It extends degree bounds from hypersurfaces to arbitrary codimension smooth affine varieties, providing explicit polynomial degree estimates.

Findings

Degree bounds for de Rham cohomology forms are (pD)^{O(pm)}.

Results are applicable to algorithmic cohomology computation.

Implications for differential equations and Hilbert's 16th problem.

Abstract

Grothendieck has proved that each class in the de Rham cohomology of a smooth complex affine variety can be represented by a differential form with polynomial coefficients. After having proved a single exponential bound for the degrees of these forms in the case of a hypersurface, here we generalize this result to arbitrary codimension. More precisely, we show that the p-th de Rham cohomology of a smooth affine variety of dimension m and degree D can be represented by differential forms of degree (pD)^{O(pm)}. 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.