# The peculiar (monic) polynomials, the zeros of which equal their coefficients

## TL;DR

This paper investigates peculiar monic polynomials whose zeros are equal to their coefficients, providing formulas and estimates for their counts under various coefficient constraints.

## Contribution

It introduces the concept of peculiar polynomials and simplifies their enumeration when coefficients are restricted to 0 or 1, offering new insights into their structure.

## Key findings

- Derived formulas for the number of peculiar polynomials of degree N
- Estimated counts of peculiar polynomials with specific coefficient conditions
- Simplified the problem for polynomials with coefficients 0 or 1

## Abstract

We evaluate the number of monic polynomials (of arbitrary degree $N$) the zeros of which equal their coefficients when these are allowed to take arbitrary complex values. In the following, we call polynomials with this property {\em peculiar\/} polynomials. We further show that the problem of determining the peculiar polynomials of degree $N$ simplifies when any of the coefficients is either 0 or 1. We proceed to estimate the numbers of peculiar polynomials of degree $N$ having one coefficient zero, or one coefficient equal to one, or neither.