# Polynomials with Multiple Zeros and Solvable Dynamical Systems including Models in the Plane with Polynomial Interactions

## TL;DR

This paper develops a comprehensive method to analyze time-dependent polynomials with multiple zeros, leading to the discovery of new solvable dynamical systems, especially focusing on degree 4 polynomials with specific zero multiplicities.

## Contribution

It introduces a general approach for polynomials with arbitrary zeros and multiplicities, extending previous methods to more complex cases and identifying new solvable dynamical systems.

## Key findings

- Extended the analysis to polynomials with multiple zeros of arbitrary multiplicity.
- Identified new classes of solvable dynamical systems for degree 4 polynomials.
- Provided a framework for analyzing nongeneric polynomials with multiple zeros.

## Abstract

The interplay among the time-evolution of the coefficients and the zeros of a generic time-dependent (monic) polynomial provides a convenient tool to identify certain classes of solvable dynamical systems. Recently this tool has been extended to the case of nongeneric polynomials characterized by the presence, for all time, of a single double zero; and subsequently significant progress has been made to extend this finding to the case of polynomials featuring a single zero of arbitrary multiplicity. In this paper we introduce an approach suitable to deal with the most general case, i. e. that of a nongeneric time-dependent polynomial with an arbitrary number of zeros each of which features, for all time, an arbitrary (time-independent) multiplicity. We then focus on the special case of a polynomial of degree 4 featuring only 2 different zeros and, by using a recently introduced additional twist of this approach, we thereby identify many new classes of solvable dynamical systems.