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The truncated moment problem on reducible cubic curves I: Parabolic and Circular type relations

Seonguk Yoo, Aljaž Zalar

Abstract

In this article we study the bivariate truncated moment problem (TMP) of degree $2k$ on reducible cubic curves. First we show that every such TMP is equivalent after applying an affine linear transformation to one of 8 canonical forms of the curve. The case of the union of three parallel lines was solved in 2022 by the second author, while the degree 6 cases in 2017 by the first author. Second we characterize in terms of concrete numerical conditions the existence of the solution to the TMP on two of the remaining cases concretely, i.e., a union of a line and a circle $y(ay+x^2+y^2)=0, a\in \mathbb{R}\setminus \{0\}$, and a union of a line and a parabola $y(x-y^2)=0$. In both cases we also determine the number of atoms in a minimal representing measure.

The truncated moment problem on reducible cubic curves I: Parabolic and Circular type relations

Abstract

In this article we study the bivariate truncated moment problem (TMP) of degree on reducible cubic curves. First we show that every such TMP is equivalent after applying an affine linear transformation to one of 8 canonical forms of the curve. The case of the union of three parallel lines was solved in 2022 by the second author, while the degree 6 cases in 2017 by the first author. Second we characterize in terms of concrete numerical conditions the existence of the solution to the TMP on two of the remaining cases concretely, i.e., a union of a line and a circle , and a union of a line and a parabola . In both cases we also determine the number of atoms in a minimal representing measure.
Paper Structure (13 sections, 8 theorems, 87 equations)

This paper contains 13 sections, 8 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Readers Guide
  3. Preliminaries
  4. Moment matrix
  5. Affine linear transformations
  6. Generalized Schur complements
  7. Extension principle
  8. Partially positive semidefinite matrices and their completions
  9. Hamburger TMP
  10. TMP on the unit circle
  11. Parabolic TMP
  12. TMP on reducible cubics - case reduction
  13. Solving the TMP on canonical reducible cubic curves

Key Result

Lemma 2.1

Let $n,m\in \mathbb N$ and where $A\in S_n$, $B\in \mathbb R^{n\times m}$ and $C\in S_m$. If $\mathop{\mathrm{rank}}\nolimits M=\mathop{\mathrm{rank}}\nolimits A$, then the matrix equation where $W\in \mathbb R^{n\times m}$, is solvable and the solutions are precisely the solutions of the matrix equation $AW=B$. In particular, $W=A^{\dagger}B$ satisfies 140722-1055-eq.

Theorems & Definitions (14)

  • Lemma 2.1
  • proof
  • Theorem 2.2: Alb69
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5: CF91
  • Theorem 2.6: CF02
  • Theorem 2.7
  • ...and 4 more