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

Karim Johannes Becher, Marco Zaninelli

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 $\mathbb{R}(\!(t_0,\dots,t_n)\!)$.

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

Abstract

Given a positive integer , a sufficient condition on a field is given for bounding its Pythagoras number by . The condition is satisfied for by function fields of curves over iterated formal power series fields over , as well as by finite field extensions of . In both cases, one retrieves the upper bound on the Pythagoras number. The new method presented here might help to establish more generally as an upper bound for the Pythagoras number of function fields of curves over and for finite field extensions of .
Paper Structure (5 sections, 32 theorems, 27 equations)

This paper contains 5 sections, 32 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Square-effective rings
  3. Semilocal Bézout rings
  4. Function fields in one variable
  5. Geometric global fields

Key Result

Theorem 1.2

Let $n\in\mathbb{N}^+$ be such that $p(E)\leqslant 2^{n}$ for every finite field extension $E/K(X)$. Let $r\in\mathbb{N}$ and let $F/K(\!(t_1)\!)\dots(\!(t_r)\!)(\!(t_{r+1},t_{r+2})\!)$ be a finite field extension. Then $p(F)\leqslant 2^n+1$.

Theorems & Definitions (70)

  • Theorem 1.2: \ref{['C:final bound']}
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • ...and 60 more