The Hasse principle for diagonal forms restricted to a hypersurface of adjacent degree

TL;DR

This paper studies the Hasse principle for systems with a diagonal form of degree k and a general form of degree k-1, refining previous bounds to require fewer variables for the principle to hold.

Contribution

It improves the bounds on the number of variables needed for the Hasse principle to hold in specific Diophantine systems, refining prior results.

Findings

Bound improved from n > 2^k k to n > 2^{k-1}(2k-1)

Reduced variable requirement for degrees three and two from 24 to 20

Enhanced understanding of the Hasse principle in restricted diagonal systems

Abstract

We investigate the Hasse principle for Diophantine systems consisting of one diagonal form of degree $k$ and one general form of degree $k-1$. By refining the method of Brandes and Parsell (arXiv:2003.04350) in this specific setting, we improve the bound $n > 2^k k$ to $n > 2^{k-1}(2k-1)$; in particular, the requirement $n > 24$ in the case of degrees three and two is relaxed to $n > 20$.