On the multiplier spectrum of polynomials

TL;DR

This paper investigates the properties of the multiplier spectrum of polynomials, proving generic injectivity and characterizing the non-injective cases, revealing deep structural relations among polynomials with identical spectra.

Contribution

It provides a detailed proof of the generic injectivity of the multiplier spectrum morphism and describes the structure of polynomials sharing the same spectrum, including Ritt move relations.

Findings

Multiplier spectrum morphism is generically injective on the moduli space.

Polynomials with the same multiplier spectrum are mostly related by Ritt moves or are equivalent.

The non-injective locus is characterized, with isolated exceptions identified.

Abstract

We prove several results on the multiplier spectrum of polynomials. We provide a detailed proof of the theorem stating that the multiplier spectrum morphism is generically injective on the moduli space of polynomials. We obtain a description of the non-injective locus of the multiplier spectrum morphism for polynomials of degree $d\geq2$. Roughly speaking, we prove that, apart from isolated exceptions, polynomials with the same multiplier spectrum are intertwined. More precisely, we show that, up to iteration and isolated exceptions, the polynomials are either equivalent or related by Ritt moves. We also investigate the relationship between Ritt moves and multiplier spectra over arithmetic progressions.