Quadratic Forms in Prime Variables with small off-diagonal ranks

TL;DR

This paper refines the understanding of solutions to quadratic forms in prime variables by lowering the rank thresholds needed for solutions in degenerate cases, extending Zhao's and Green's previous results.

Contribution

It improves the lower bounds on off-diagonal rank for solutions in primes, specifically for ranks 6 and 8, expanding the applicability of existing techniques.

Findings

Lower bounds on off-diagonal rank for solutions in primes: 6 and 8.

Extension of Zhao's results to degenerate cases with smaller ranks.

Complementary to Green's breakthrough on non-degenerate rank 8 cases.

Abstract

The main goal of this note is to establish the limits of L. Zhao's techniques for counting solutions to quadratic forms in prime variables. Zhao considered forms with rank at least 9, and showed that these equations have solutions in primes provided there are no local obstructions. We consider in detail the degenerate cases of off-diagonal rank 1 and 2, and improve the rank lower bounds to at least 6 and at least 8 respectively. These results complement a recent breakthrough of Green on the non-degenerate rank 8 case.