# Polynomials Whose Coefficients Coincide with Their Zeros

## TL;DR

This paper investigates polynomials whose coefficients are equal to their zeros, combining algebraic geometry and dynamical systems to analyze their properties, count their instances, and explore their applications in differential equations and N-body problems.

## Contribution

It introduces new methods for analyzing Ulam polynomials, estimates their quantity, and explores their role in differential operators and solvable N-body systems.

## Key findings

- Number of Ulam polynomials of degree N estimated.
- Only trivial Ulam polynomials are eigenfunctions of hypergeometric operators.
- A family of solvable N-body problems with equilibria at Ulam polynomial zeros.

## Abstract

In this paper we consider monic polynomials such that their coefficients coincide with their zeros. These polynomials were first introduced by S. Ulam. We combine methods of algebraic geometry and dynamical systems to prove several results. We obtain estimates on the number of Ulam polynomials of degree $N$. We provide additional methods to obtain algebraic identities satisfied by the zeros of Ulam polynomials, beyond the straightforward comparison of their zeros and coefficients. To address the question about existence of orthogonal Ulam polynomial sequences, we show that the only Ulam polynomial eigenfunctions of hypergeometric type differential operators are the trivial Ulam polynomials $\{x^N\}_{N=0}^\infty$. We propose a family of solvable $N$-body problems such that their stable equilibria are the zeros of certain Ulam polynomials.