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On the Archimedean Positivstellensatz in Real Algebraic Geometry

Konrad Schmüdgen

Abstract

A variant of the Archimedean Positivstellensatz is proved which is based on Archimedean semirings or quadratic modules of generating subalgebras. It allows one to obtain representations of strictly positive polynomials on compact semi-algebraic by means of smaller sets of squares or polynomials. A large number of examples is developed in detail.

On the Archimedean Positivstellensatz in Real Algebraic Geometry

Abstract

A variant of the Archimedean Positivstellensatz is proved which is based on Archimedean semirings or quadratic modules of generating subalgebras. It allows one to obtain representations of strictly positive polynomials on compact semi-algebraic by means of smaller sets of squares or polynomials. A large number of examples is developed in detail.
Paper Structure (3 sections, 11 theorems, 51 equations)

This paper contains 3 sections, 11 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Proofs of Theorems \ref{['positivst']} and \ref{['daggerpre']}
  3. Examples

Key Result

Theorem 2

Suppose that the algebra $A$ is generated by the subalgebras $A_1,\dots,A_m$ of $A$. We assume that $C_i$ is an Archimedean semiring or an Archimedean quadratic module of the algebra $A_i$ for $i=1,\dots,m$. Let $M$ be a unital cone of $A$. Then for any element $a\in A$ the following are equivalent:

Theorems & Definitions (31)

  • Definition 1
  • Theorem 2
  • Corollary 3
  • proof
  • Corollary 4
  • Theorem 5
  • Remark 6
  • Proposition 7
  • proof
  • Proposition 8
  • ...and 21 more