# Roots multiplicity without companion matrices

## TL;DR

This paper introduces a new polynomial interpolation method for roots' multiplicities that avoids companion matrices, improving the efficiency of the square-free decomposition algorithm in both theory and practice.

## Contribution

It presents a novel approach to roots' multiplicity interpolation that enhances the Guersenzvaig--Szechtman algorithm without relying on companion matrices.

## Key findings

- More efficient square-free decomposition algorithm
- Effective polynomial interpolation method for roots' multiplicities
- Practical improvements demonstrated in computational experiments

## Abstract

We show a method for constructing a polynomial interpolating roots' multiplicities of another polynomial, that does not use companion matrices. This leads to a modification to Guersenzvaig--Szechtman square-free decomposition algorithm that is more efficient both in theory and in practice.