Bounding the Pythagoras number of a field by $2^n+1$

TL;DR

This paper introduces a new method to bound the Pythagoras number of certain fields by $2^n+1$, with specific cases achieving the bound of 3, and suggests potential for broader applications.

Contribution

A novel approach is developed to establish upper bounds on the Pythagoras number for fields related to real function fields and their extensions.

Findings

Bound of 3 for specific fields when n=1

Method potentially extends to higher n for function fields over real power series

Provides a new tool for analyzing Pythagoras numbers in algebraic fields

Abstract

Given a positive integer $n$, a sufficient condition on a field is given for bounding its Pythagoras number by $2^n+1$. The condition is satisfied for $n=1$ by function fields of curves over iterated formal power series fields over $\mathbb{R}$, as well as by finite field extensions of $\mathbb{R}(\!(t_0,t_1)\!)$. In both cases, one retrieves the upper bound $3$ on the Pythagoras number. The new method presented here might help to establish more generally $2^n+1$ as an upper bound for the Pythagoras number of function fields of curves over $\mathbb{R}(\!(t_1,\dots,t_n)\!)$ and for finite field extensions of $\mathbb{R}(\!(t_0,\dots,t_n)\!)$.