Zeros of Systems of ${\mathfrak p}$-adic Quadratic Forms

TL;DR

This paper proves that a system of quadratic forms over a p-adic field has a non-trivial common zero when the number of variables exceeds four times the number of forms, given a sufficiently large residue field.

Contribution

It establishes a new bound on the number of variables needed for the existence of common zeros in systems of p-adic quadratic forms, linking it to the size of the residue field.

Findings

Non-trivial common zeros exist when variables > 4r

Residue field size condition: at least (2r)^r

Provides explicit bounds for zeros in p-adic quadratic systems

Abstract

It is shown that a system of $r$ quadratic forms over a ${\mathfrak p}$-adic field has a non-trivial common zero as soon as the number of variables exceeds $4r$, providing that the residue class field has cardinality at least $(2r)^r$.