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Positive semidefinite analytic functions on real analytic surfaces

José F. Fernando

Abstract

Let $X\subset{\mathbb R}^n$ be a (global) real analytic surface. Then every positive semidefinite meromorphic function on $X$ is a sum of $10$ squares of meromorphic functions on $X$. As a consequence, we provide a real Nullstellensatz for (global) real analytic surfaces.

Positive semidefinite analytic functions on real analytic surfaces

Abstract

Let be a (global) real analytic surface. Then every positive semidefinite meromorphic function on is a sum of squares of meromorphic functions on . As a consequence, we provide a real Nullstellensatz for (global) real analytic surfaces.
Paper Structure (29 sections, 18 theorems, 82 equations)

This paper contains 29 sections, 18 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Analytic functions.
  3. Meromorphic functions.
  4. Main result
  5. Real Nullstellensatz
  6. Preliminaries
  7. General terminology
  8. Reduced analytic spaces gmt
  9. Anti-involution and complexifications gmt
  10. Real part space
  11. Complexification and $C$-analytic spaces gmt
  12. Irreducible components of a $C$-analytic space
  13. Full approximation
  14. $C$-analytic sets
  15. Well reduced structure
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $X$ be an analytic surface germ and $f\in{\EuScript O}(X)$ a positive semidefinite analytic function germ. Then there are analytic function germs $h_0,h_1,h_2,h_3,h_4\in{\EuScript O}(X)$ such that $h_0^2f=h_1^2+h_2^2+h_3^2+h_4^2$ and $h_0$ is a sum of squares in ${\EuScript O}(X)$ with $\{h_0=0\

Theorems & Definitions (32)

  • Theorem 1.1: abfr1
  • Theorem 1.2: abfr1
  • Theorem 1.3: Hilbert's 17th problem
  • Corollary 1.4: Real Nullstellensatz
  • Theorem 2.1: Full approximation
  • proof
  • Corollary 2.3
  • proof
  • Lemma 3.1
  • proof
  • ...and 22 more