Sums-of-Squares Formulas over Algebraically Closed Fields

TL;DR

This paper investigates whether the existence of sums-of-squares formulas depends on the base field, showing that over algebraically closed fields, this existence is independent of characteristic for sufficiently large primes, with explicit bounds provided.

Contribution

It reformulates the existence problem as an algebraic geometry question and proves the independence of characteristic for large primes, providing explicit bounds and theoretical computability.

Findings

Existence of sums-of-squares formulas is independent of characteristic for large enough primes.

Explicit bounds on the prime characteristic are established.

Existence over algebraically closed fields is theoretically computable.

Abstract

In this paper, we consider whether existence of a sums-of-squares formula depends on the base field. We reformulate the question of existence as a question in algebraic geometry. We show that, for large enough p, existence of sums-of-squares formulas over algebraically closed fields is independent of the characteristic. We make the bound on p explicit, and we prove that the existence of a sums-of-squares formula of fixed type over an algebraically closed field is theoretically (though not practically) computable.