# A notion of entropy on the roots of polynomials

## TL;DR

This paper introduces a new entropy concept for polynomials, demonstrating its properties and how it increases under finite free addition, leading to a novel inequality involving polynomial roots and derivatives.

## Contribution

It defines a canonical entropy for polynomials and proves its monotonicity under finite free addition, establishing a new inequality for polynomial roots and derivatives.

## Key findings

- Entropy increases smoothly with finite free addition
- Established a new inequality involving polynomial roots and derivatives
- Provided a canonical entropy measure for polynomials

## Abstract

We introduce a canonical notion of entropy for polynomials analogue to that of random variables in probability. We prove that entropy increases smoothly with respect to finite free addition. In particular we get the new inequality : $ \Dis(p-tp')>\Dis(p) $ for a polynomial $p$, its derivative $p'$ and t any non zero real.