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Representation of positive semidefinite elements as sum of squares in 2-dimensional local rings

José F. Fernando

Abstract

A classical problem in real geometry concerns the representation of positive semidefinite elements of a ring $A$ as sums of squares of elements of $A$. If $A$ is an excellent ring of dimension $\geq3$, it is already known that it contains positive semidefinite elements that cannot be represented as sums of squares in $A$. The one dimensional local case has been afforded by Scheiderer (mainly when its residue field is real closed). In this work we focus on the $2$-dimensional case and determine (under some mild conditions) which local excellent henselian rings $A$ of embedding dimension $3$ have the property that every positive semidefinite element of $A$ is a sum of squares of elements of $A$.

Representation of positive semidefinite elements as sum of squares in 2-dimensional local rings

Abstract

A classical problem in real geometry concerns the representation of positive semidefinite elements of a ring as sums of squares of elements of . If is an excellent ring of dimension , it is already known that it contains positive semidefinite elements that cannot be represented as sums of squares in . The one dimensional local case has been afforded by Scheiderer (mainly when its residue field is real closed). In this work we focus on the -dimensional case and determine (under some mild conditions) which local excellent henselian rings of embedding dimension have the property that every positive semidefinite element of is a sum of squares of elements of .
Paper Structure (24 sections, 49 theorems, 271 equations)

This paper contains 24 sections, 49 theorems, 271 equations.

Table of Contents

  1. Introduction
  2. Basic tools when dealing with formal rings
  3. Formal rings
  4. Equivalence of series and finite determinacy tools
  5. Exchanging positiveness
  6. Applications of Weierstrass polynomials and Puiseux series
  7. Applications of curve selection lemma and Puiseux series
  8. List of candidates
  9. Characterization of positive semidefinite elements.
  10. Proof of Theorem \ref{['gp=s']}
  11. The non-principal case
  12. Polynomial density and polynomial reduction
  13. Positive definite elements.
  14. Polynomial density
  15. Strong Artin's approximation and bounded Pythagoras numbers.
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $A$ be an excellent ring of real dimension $\geq3$. Then ${\mathcal{P}}(A)\neq{\Sigma{A}^2}$.

Theorems & Definitions (94)

  • Theorem 1.1: frs1
  • Theorem 1.2: sch2
  • Theorem 1.3: sch2
  • Theorem 1.4: f2, f4f5
  • Theorem 1.5: List of candidates
  • Remark 1.6
  • Theorem 1.7: frs2
  • Theorem 1.8: Affirmative cases
  • Corollary 1.9: Full characterization
  • Corollary 1.11
  • ...and 84 more