On the multiplier spectrum of polynomials

TL;DR

The paper investigates the multiplier spectrum of degree $d\ge 2$ polynomials, situating the analysis on the moduli space ${\rm MPoly}^d$ and addressing when the multiplier data uniquely determine a polynomial up to conjugacy. It develops a polynomial version of the Ji–Xie framework by introducing pre-simple polynomials, proving generic injectivity of the multiplier spectrum morphism on ${\rm MPoly}^d$, and describing the non-injective locus via intertwined polynomials (Ritt moves) and generalized Lattès maps. It further extends results to degrees $2$ and $3$ and proves a Zariski-dense orbit statement for split polynomial endomorphisms on $(\mathbb P^1)^2$, while also examining multiplier spectra along arithmetic progressions and drawing connections to the length spectrum through Favre–Gauthier and Pakovich–Lattès theory. The work clarifies how multiplier data constrain polynomial dynamics, identifies precise obstructions to injectivity, and proposes several directions for studying stable spectra and arithmetic progression phenomena with potential implications for broader dynamical systems questions.

Abstract

We prove several results on multiplier spectrum for 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 all degrees $d\geq2$. Roughly speaking, non-injectivity implies intertwining, as it signifies the equivalence of polynomials or Ritt moves, except at isolated points and up to iteration. We investigate the relation between Ritt moves and the multiplier spectrum over arithmetic progressions.

Theorems & Definitions (107)

• Definition 1.1
• Theorem 1.2: McMullen
• Theorem 1.3: Ji and Xie
• Theorem 1.4: JX23 and VH2412
• Theorem 1.5
• Proposition 1.6
• Conjecture 1.7
• Theorem 1.8
• Theorem 1.9
• Remark 1.10