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Sums of squares on hypersurfaces

Kacper Błachut, Tomasz Kowalczyk

Abstract

We show that the Pythagoras number of rings of type $\mathbb{R}[x,y, \sqrt{f(x,y)}]$ is infinite, provided that the polynomial $f(x,y)$ satisfies some mild conditions.

Sums of squares on hypersurfaces

Abstract

We show that the Pythagoras number of rings of type is infinite, provided that the polynomial satisfies some mild conditions.
Paper Structure (3 sections, 9 theorems, 20 equations)

This paper contains 3 sections, 9 theorems, 20 equations.

Table of Contents

  1. Preliminaries
  2. Main result
  3. Applications and further problems

Key Result

Proposition 1.7

Let $\varphi : R_1 \rightarrow R_2$ be a homomorphism of rings. Then for any $x\in R_1$, the length of $x$ is greater than or equal to the length of $\varphi(x)$ in $R_2$. If $\varphi$ is an epimorphism, then $p(R_1)\geq p(R_2)$.

Theorems & Definitions (20)

  • Definition 1
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Example 1.4
  • Example 1.5
  • Example 1.6
  • Proposition 1.7
  • Theorem 1.8
  • Corollary 1.9
  • ...and 10 more