# The quintic complex moment problem

## TL;DR

This paper provides a concrete solution to the quintic (degree 5) truncated complex moment problem, extending the known solutions for degrees up to 4, and explores the minimal measure cardinality using recurrence properties.

## Contribution

It offers a new solution for the degree 5 case of the truncated complex moment problem and analyzes the minimal measure's cardinality, building on recurrence sequence properties.

## Key findings

- Solved the quintic (degree 5) truncated complex moment problem.
- Analyzed the minimal representing measure's cardinality.
- Extended methods applicable to all odd-degree moment problems.

## Abstract

Let $\gamma^{(m)} \equiv \{ \gamma_{ij} \}_{0 \leq i +j \leq m}$ be a given complex-valued sequence. The truncated complex moment problem (TCMP in short) involves determining necessary and sufficient conditions for the existence of a positive Borel measure $\mu$ on $\mathbb{C}$ (called a representing measure for $\gamma^{(m)}$) such that $\gamma_{ij} = \int \overline{z}^i z^j d\mu$ for $0 \leq i +j \leq m$. The TCMP has been completely solved only when $m= 1, 2, 3, 4$. We provide in this paper a concrete solution to the quintic TCMP (that is, when $m = 5$). We also study the cardinality of the minimal representing measure. Based on the bivariate recurrences sequences's properties with some Curto-Fialkow's results, our method intended to be useful for all odd-degree moment problems.