# Pseudo-Polynomial Time Algorithm for Computing Moments of Polynomials in Free Semicircular Elements

## TL;DR

This paper presents a polynomial-time algorithm for computing moments of polynomials in free semicircular elements, improving upon the exponential time naive approach by utilizing a rearranged Schützenberger's algorithm.

## Contribution

The paper introduces a novel polynomial-time algorithm for calculating moments in free probability, advancing computational methods in the field.

## Key findings

- Efficient polynomial-time algorithm for moments calculation
- Rearranged Schützenberger's algorithm used for optimization
- Significant reduction from exponential to polynomial complexity

## Abstract

We consider about calculating $M$th moments of a given polynomial in free independent semicircular elements in free probability theory. By a naive approach, this calculation requires exponential time with respect to $M$. We explicitly give an algorithm for calculating them in polynomial time by rearranging Sch\"utzenberger's algorithm.