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Constructive approach to the truncated moment problem on reducible cubic curves: Hyperbolic type relations

Seonguk Yoo, Aljaž Zalar

TL;DR

This work delivers a constructive solution to the bivariate truncated moment problem of even degree on reducible cubic curves where the conic part is hyperbolic. By affinely transforming to canonical hyperbola forms and decomposing the moment sequence into line-supported and conic-supported components, the authors derive explicit, verifiable conditions for the existence of a $oldsymbol{Z}(p)$-representing measure, and construct such measures when possible. They treat hyperbolic type 1 (no real self-intersections), type 2 (single real self-intersection), and type 3 (double real self-intersection), providing rank-based criteria and bounds on the number of atoms in minimal representing measures. The results extend previous constructive TMP solutions for parallel lines, circles, and parabolas to hyperbolic curves, contributing to a fuller, computationally actionable theory of TMP on reducible plane curves with practical implications for moment-based modeling and approximation.

Abstract

In this paper, we solve constructively the bivariate truncated moment problem (TMP) of even degree on reducible cubic curves, where the conic part is a hyperbola. According to the classification from our previous work, these represent three out of nine possible canonical forms of reducible cubic curves after applying an affine linear transformation. The TMP on the union of three parallel lines, the circular and the parabolic type TMP were solved constructively in our previous work, while in this paper we consider three cases of hyperbolic type, i.e., a type without real self-intersection points, a type with a simple real self-intersection point and a type with a double real self-intersection point. In all cases, we also establish bounds on the number of atoms in a minimal representing measure.

Constructive approach to the truncated moment problem on reducible cubic curves: Hyperbolic type relations

TL;DR

This work delivers a constructive solution to the bivariate truncated moment problem of even degree on reducible cubic curves where the conic part is hyperbolic. By affinely transforming to canonical hyperbola forms and decomposing the moment sequence into line-supported and conic-supported components, the authors derive explicit, verifiable conditions for the existence of a -representing measure, and construct such measures when possible. They treat hyperbolic type 1 (no real self-intersections), type 2 (single real self-intersection), and type 3 (double real self-intersection), providing rank-based criteria and bounds on the number of atoms in minimal representing measures. The results extend previous constructive TMP solutions for parallel lines, circles, and parabolas to hyperbolic curves, contributing to a fuller, computationally actionable theory of TMP on reducible plane curves with practical implications for moment-based modeling and approximation.

Abstract

In this paper, we solve constructively the bivariate truncated moment problem (TMP) of even degree on reducible cubic curves, where the conic part is a hyperbola. According to the classification from our previous work, these represent three out of nine possible canonical forms of reducible cubic curves after applying an affine linear transformation. The TMP on the union of three parallel lines, the circular and the parabolic type TMP were solved constructively in our previous work, while in this paper we consider three cases of hyperbolic type, i.e., a type without real self-intersection points, a type with a simple real self-intersection point and a type with a double real self-intersection point. In all cases, we also establish bounds on the number of atoms in a minimal representing measure.
Paper Structure (11 sections, 9 theorems, 29 equations)

This paper contains 11 sections, 9 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Reader's Guide
  3. Preliminaries
  4. Moment matrix
  5. Affine linear transformations
  6. Generalized Schur complements
  7. Partially positive semidefinite matrices and their completions
  8. Extension principle
  9. (Strong) Hamburger TMP
  10. Hyperbolic TMP
  11. Common approach to all cases

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 (13)

  • Lemma 2.1
  • Theorem 2.2: Alb69
  • Lemma 2.3: YZ24
  • Proposition 2.4: Fia95 or Zal22a
  • Theorem 2.5: CF91
  • Theorem 2.6
  • Remark 2.7
  • Theorem 2.8
  • Remark 2.9
  • Corollary 2.10
  • ...and 3 more