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

2025/10/16 by Yoo, Seonguk, Zalar, Aljaž
#47A20 #47A57 #47N40 #FOS: Mathematics #Functional Analysis (math.FA) #Primary 44A60 #Secondary 15A04

paper · doi:10.48550/arxiv.2510.15131

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.

Citations

• Quadrature rules with few nodes supported on algebraic curves
• Positive polynomials and the truncated moment problem on plane cubics
• The pure Y=X^d truncated moment problem
• Bivariate Truncated Moment Sequences with the Column Relation XY=X^m + q(X), with q of degree m-1
• Nonnegative Polynomials and Moment Problems on Algebraic Curves
• The truncated moment problem on reducible cubic curves I: Parabolic and Circular type relations
• The truncated moment problem on curves y=q(x) and yx^ℓ=1
• Optimization approaches to quadrature: new characterizations of Gaussian\n quadrature on the line and quadrature with few nodes on plane algebraic\n curves, on the plane and in higher dimensions
• Solution of the truncated hyperbolic moment problem

Related

• The truncated moment problem on reducible cubic curves I: Parabolic and Circular type relations
• Positive polynomials and the truncated moment problem on plane cubics
• Bivariate Truncated Moment Sequences with the Column Relation XY=X^m + q(X), with q of degree m-1
• Further results on the real cubic truncated moment problem
• The Tracial Moment Problem on Quadratic Varieties