Diophantine and cohomological dimensions

TL;DR

This paper establishes explicit linear bounds relating the p-cohomological dimension of a field to its Diophantine dimension, providing new insights into the relationship between these two measures for fields.

Contribution

It introduces explicit linear bounds connecting the p-cohomological dimension and Diophantine dimension of fields, especially for fields with Diophantine dimension up to 4.

Findings

For fields with Diophantine dimension ≤ 4, the 3-cohomological dimension is ≤ the Diophantine dimension.

Provides explicit linear bounds on p-cohomological dimension based on Diophantine dimension.

Shows a specific relationship between Diophantine and cohomological dimensions for certain fields.

Abstract

We give explicit linear bounds on the p-cohomological dimension of a field in terms of its Diophantine dimension. In particular, we show that for a field of Diophantine dimension at most 4, the 3-cohomological dimension is less than or equal to the Diophantine dimension.