On the multiplier spectrum of polynomials

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.

💡 Research Summary

The paper investigates the multiplier spectrum of complex polynomials of degree (d\ge2). For a rational map (f:\mathbb{P}^1\to\mathbb{P}^1) the multiplier at a periodic point (z_0) of exact period (n) is defined as (\rho_f(z_0)=(f^{\circ n})’(z_0)). Collecting the multipliers of all fixed points of the iterates (f^{\circ n}) and forming the elementary symmetric functions yields a point (\mathbf{S}n(f)\in\mathbb{C}^{N{d,n}}) where (N_{d,n}=d^n+1). The maps (\tau_{d,n}:M_d\to\mathbb{A}^{N_{d,1}}\times\cdots\times\mathbb{A}^{N_{d,n}}) record these data up to level (n). A Noetherian argument shows that there exists a minimal integer (m_d) such that the combined map (\tau_d:=\tau_{d,m_d}) already determines the full infinite sequence (\mathbf{S}(f)=(\mathbf{S}_1(f),\mathbf{S}_2(f),\dots)).

McMullen proved that, away from the flexible Lattès locus, (\tau_d) is quasi‑finite. Ji and Xie later strengthened this to generic injectivity: there is a Zariski‑open set (U\subset M_d) on which (\tau_d) is a bijection onto its image. The present work restricts these results to the polynomial subspace (\mathrm{Poly}_d\subset\mathrm{Rat}_d) and its moduli space (\mathrm{MPoly}d). Theorem 1.4 (citing Ji‑Xie and Huguin) asserts that the induced map (\tilde\tau_d:\mathrm{MPoly}d\to\mathbb{A}^{N{d,1}+\dots+N{d,m_d}}) is also generically injective.

The authors then turn to the complementary non‑injective locus. Two mechanisms can produce distinct polynomial classes with identical multiplier spectra: (i) elementary transformations, where (f=h_1\circ h_2) and (g=h_2\circ h_1) for non‑constant rational maps (h_1,h_2); and (ii) the “intertwining” relation, which is equivalent to the existence of an algebraic curve (Z\subset\mathbb{A}^2) invariant under ((f,g)). Intertwining can be described via Ritt moves: there exist polynomials (R,h_1,h_2) and an integer (n>0) such that (f^{\circ n}\circ h_1=h_1\circ R) and (g^{\circ n}\circ h_2=h_2\circ R). In particular, intertwining forces (\deg f=\deg g).

A key technical notion introduced is that of a “pre‑simple” polynomial: a degree‑(d) polynomial with exactly (d-1) distinct critical values in (\mathbb{C}). For (d\ge4) a Zariski‑open dense set of polynomials is pre‑simple, and Pakovich’s methods for simple rational maps apply. Theorem 1.5 shows that for a generic pre‑simple polynomial (f), any other pre‑simple polynomial (g) that intertwines with (f) must actually be conjugate to (f) in (\mathrm{MPoly}_d). For degrees 2 and 3, Proposition 1.7 (derived from Pakovich’s work) gives the analogous statement without the pre‑simple hypothesis.

Having established generic injectivity, the paper describes the non‑injective locus (PNI_d\subset\mathrm{MPoly}_d\times\mathrm{MPoly}_d) consisting of distinct pairs with identical multiplier spectra. Theorem 1.9 and its more precise version Theorem 1.10 consider an irreducible curve (C) defined over (\mathbb{Q}) inside (\mathrm{MPoly}_d\times\mathrm{MPoly}_d) whose projections are non‑constant. If (C(\mathbb{C})) contains only finitely many points outside (PNI_d), then every point of (C) corresponds to a pair of polynomials that are intertwined. Moreover, after possibly iterating both maps, one of two explicit algebraic relations holds: either the iterates become elementary‑equivalent, or they become equivalent to monomials composed with a common polynomial (V) after suitable rescaling. This result links the arithmetic of the parameter curve with the dynamical phenomenon of non‑injectivity.

The authors discuss known sources of non‑injectivity: flexible Lattès maps (which never arise for polynomials) and elementary equivalence. They formulate Conjecture 1.8 (originally due to Pakovich) specialized to polynomials: any two non‑conjugate isospectral polynomials must be elementary‑equivalent. For prime degrees this conjecture is known to hold because elementary equivalence collapses to conjugacy. For composite degrees, Huguin’s computer‑assisted work shows that (\tilde\tau_4) is not injective, and the non‑injective pairs are exactly those related by an elementary transformation.

The paper concludes with several open problems: a full classification of the non‑injective locus for all degrees, extensions of the intertwining description to higher‑dimensional dynamical systems, and a deeper understanding of how Ritt moves interact with multiplier spectra over arithmetic progressions. Overall, the work provides a comprehensive picture of how the multiplier spectrum serves as a near‑complete invariant for complex polynomials, clarifying precisely where and why it fails to be injective.