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Nonnegative Polynomials and Moment Problems on Algebraic Curves

Lorenzo Baldi, Grigoriy Blekherman, Rainer Sinn

Abstract

The cone of nonnegative polynomials is of fundamental importance in real algebraic geometry, but its facial structure is understood in very few cases. We initiate a systematic study of the facial structure of the cone of nonnegative polynomials $\pos$ on a smooth real projective curve $X$. We show that there is a duality between its faces and totally real effective divisors on $X$. This allows us to fully describe the face lattice in case $X$ has genus one. We compute the Carathéodory number of the dual moment cone $\pos^\vee$ for an elliptic normal curve $X$, which measures the complexity of quadrature rules of measures supported on $X$. Interestingly, the topology of the real locus of $X$ influences the Carathéodory number of $\pos^\vee$. We apply our results to truncated moment problems on affine cubic curves, where we deduce sharp bounds on the flat extension degree.

Nonnegative Polynomials and Moment Problems on Algebraic Curves

Abstract

The cone of nonnegative polynomials is of fundamental importance in real algebraic geometry, but its facial structure is understood in very few cases. We initiate a systematic study of the facial structure of the cone of nonnegative polynomials on a smooth real projective curve . We show that there is a duality between its faces and totally real effective divisors on . This allows us to fully describe the face lattice in case has genus one. We compute the Carathéodory number of the dual moment cone for an elliptic normal curve , which measures the complexity of quadrature rules of measures supported on . Interestingly, the topology of the real locus of influences the Carathéodory number of . We apply our results to truncated moment problems on affine cubic curves, where we deduce sharp bounds on the flat extension degree.
Paper Structure (25 sections, 37 theorems, 67 equations)

This paper contains 25 sections, 37 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Main results and related works
  3. Preliminaries and notations
  4. Real algebraic curves
  5. Convex geometry
  6. Nonnegative forms and the moment problem
  7. The convex cone of nonnegative forms on projective curves
  8. Bivariate forms
  9. Faces and divisors
  10. Dimension of faces
  11. Extreme rays of the nonnegative cone
  12. Extreme rays and the Jacobian
  13. Real rooted extreme rays
  14. Nonnegative forms on genus one curves
  15. Elliptic normal curves
  16. ...and 10 more sections

Key Result

Theorem A

Let $(\mathcal{F}\setminus\{\, \{0\}\, \}, \subset)$ be the poset of positive-dimensional faces of $\mathop{\mathrm{\mathrm{P}}}\nolimits_{X,2}$, ordered by inclusion. Then $\Phi$ and $\Psi$ form a Galois connection between $(\mathcal{F}\setminus\{\, \{0\}\, \}, \, \subset \,)$ and the face divisors Moreover, we have $(\Psi \circ \Phi)(\mathrm{F}) = \mathrm{F}_{\mathrm{div}(\mathrm{F})} = \mathrm{

Theorems & Definitions (81)

  • Theorem A: see \ref{['thm:char_faces']} and \ref{['cor:galois']}
  • Theorem B: see \ref{['cor:families_torsion']}
  • Theorem C: see \ref{['thm:car']}
  • Theorem D: see \ref{['prop:almost_flat']}
  • Example 3.1.1: see also schulzeConesLocallyNonNegative2021
  • Lemma 3.2.1
  • proof
  • Definition 3.2.2
  • Definition 3.2.3
  • Lemma 3.2.4
  • ...and 71 more