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The Role of Gröbner Bases in the Study of Extremal Truncated Moment Problems

Raúl E. Curto, Marc R. Moore

TL;DR

This work extends the Curto–Yoo framework for extremal truncated moment problems to arbitrary orders $M(k)$ by leveraging Gröbner bases. It shows that the Gröbner basis of the ideal of the finite variety associated with a moment matrix captures all column relations and yields a concrete, computable moment condition equivalent to the existence of a representing measure, which must be finitely atomic. The central result provides equivalences between the existence of a representing measure, vanishing of the basis polynomials under the Riesz functional, and actual vanishing of these polynomials on the operator tuple $(Z,\overline{Z})$, when the problem is extremal. Numerically, once the Gröbner basis is computed, the necessary and sufficient conditions reduce to linear moment equations, enabling explicit verification in a wide class of harmonic-column-relations. Overall, the paper broadens the toolkit for TCMP by tying finite-algebraic geometry (varieties and Grobner bases) directly to moment-condition certificates.

Abstract

In a 2014 paper, R.E. Curto and S. Yoo proved that a moment matrix $M(3)$ with specific harmonic polynomials as column relations admits a representing measure if and only if a condition at the level of moments holds. \ In this paper, we generalize the 2014 result to arbitrary moment matrices $M(k)$ ($k \in \mathbb{Z}_{+}$), with column relations given by general harmonic polynomials. \ We accomplish this by proving that the Gröbner basis for the ideal generated by a finite variety associated with the moment matrix provides all the necessary column relations for the matrix as well as a suitable condition on the moments, which is equivalent to the existence of a representing measure.

The Role of Gröbner Bases in the Study of Extremal Truncated Moment Problems

TL;DR

This work extends the Curto–Yoo framework for extremal truncated moment problems to arbitrary orders by leveraging Gröbner bases. It shows that the Gröbner basis of the ideal of the finite variety associated with a moment matrix captures all column relations and yields a concrete, computable moment condition equivalent to the existence of a representing measure, which must be finitely atomic. The central result provides equivalences between the existence of a representing measure, vanishing of the basis polynomials under the Riesz functional, and actual vanishing of these polynomials on the operator tuple , when the problem is extremal. Numerically, once the Gröbner basis is computed, the necessary and sufficient conditions reduce to linear moment equations, enabling explicit verification in a wide class of harmonic-column-relations. Overall, the paper broadens the toolkit for TCMP by tying finite-algebraic geometry (varieties and Grobner bases) directly to moment-condition certificates.

Abstract

In a 2014 paper, R.E. Curto and S. Yoo proved that a moment matrix with specific harmonic polynomials as column relations admits a representing measure if and only if a condition at the level of moments holds. \ In this paper, we generalize the 2014 result to arbitrary moment matrices (), with column relations given by general harmonic polynomials. \ We accomplish this by proving that the Gröbner basis for the ideal generated by a finite variety associated with the moment matrix provides all the necessary column relations for the matrix as well as a suitable condition on the moments, which is equivalent to the existence of a representing measure.
Paper Structure (17 sections, 12 theorems, 26 equations, 7 figures)

This paper contains 17 sections, 12 theorems, 26 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Notation and Basic Terminology
  3. Basic Theory of Truncated Moment Problems
  4. Cubic column relations in the extremal TMP
  5. Commutative Algebra
  6. Preliminary Results
  7. Main Results
  8. Numerical Results
  9. Examples
  10. Reproducing the main result of Curto and Yoo
  11. A different cubic result
  12. A quartic example
  13. Acknowledgments
  14. Declarations
  15. Conflicts of interest/competing interests
  16. ...and 2 more sections

Key Result

Theorem 2.1

(CY14) Let $M(3)\geq 0\,$, with $M(2)>0$ and $q_{7}(Z,\bar{Z})=0$. Let $q_{LC}(z,\overline{z}):=i(z-i \overline{z})(\overline{z}z-u)$. Then there exists a representing measure for $M(3)$ if and only if

Figures (7)

  • Figure 1: The $7$-point set $\mathcal{Z}(q_{7})$. (The circle has radius $\sqrt{u}$.)
  • Figure 2: Complex plot of $q_7(z,\overline{z}) = z^3 - i 8 z + 5 \overline{z}$. We can clearly see the seven points of $\mathcal{V}$ in the plot of $q_7$.
  • Figure 3: Complex contour plot of the polynomial found in the Gröbner basis, which is exactly $q_{LC}(z,\overline{z}) = i(z - i\overline{z})(\overline{z}z - 5)$.
  • Figure 4: Complex contour plot of $g_1(z,\overline{z})$ (left) and of the polynomial found in the Gröbner basis, $g_2(z,\overline{z})$ (right)
  • Figure 5: Complex contour plot of $g_1(z,\overline{z})$ (left) and $g_2(z,\overline{z})$ (right).
  • ...and 2 more figures

Theorems & Definitions (17)

  • Theorem 2.1
  • Proposition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7: Eisenbud95, Theorem 15.3
  • Theorem 2.8: CLO15 Theorem 5.3.6
  • Proposition 2.9: CLO15 Proposition 5.3.7
  • Theorem 3.1
  • ...and 7 more