# Sums of squares in function fields over Henselian local fields

## TL;DR

This paper establishes upper bounds for the level and Pythagoras number in function fields over Henselian local rings, notably proving the Pythagoras number of real Laurent series fields is at most 2^{n-1}, answering a prior open question.

## Contribution

It provides new bounds on the Pythagoras number for function fields over Henselian local rings, including a specific bound for real Laurent series fields.

## Key findings

- Pythagoras number of $
eal((x_1, dots,x_n))$ is at most $2^{n-1}$
- Upper bounds for level and Pythagoras number in function fields over Henselian rings
- Positive answer to a question by Choi, Dai, Lam, and Reznick.

## Abstract

We give upper bounds for the level and the Pythagoras number of function fields over fraction fields of integral Henselian excellent local rings. In particular, we show that the Pythagoras number of $\mathbb{R}((x_1,\dots,x_n))$ is $\leq 2^{n-1}$, which answers positively a question of Choi, Dai, Lam and Reznick.