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Multipliers and invariants of endomorphisms of projective space in dimension greater than 1

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Pith's one-line read Higher-dimensional dynamical maps can be told apart, up to finitely many choices, by the eigenvalues of the derivative at periodic points.

desk verdict A genuine extension of multiplier invariants to higher-dimensional moduli spaces with real new results, but the headline degree count in Theorem 5.13 rests on an unproved uniqueness claim that needs fixing before the quantitative statement is trusted. read the letter →

arxiv 1908.03184 v1 pith:TZO5POJ7 submitted 2019-08-08 math.DS math.NT

Benjamin Hutz This is my paper · ORCID
classification math.DSmath.NT MSC 37P4537P0537A35
keywords multiplierinvariantsmodulispaceofendomorphismsprojectivedynamicsisospectralfamiliessplitpolynomialtriangularLattèsmapsfixedpointmultipliers
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to show that fixed-point multiplier data—the eigenvalues of the derivative at periodic points—can serve as (almost) separating invariants for dynamical systems on higher-dimensional projective space, not just on the Riemann sphere. It defines regular functions $\sigma^{(n)}_{i,j}$ on the moduli space $\mathcal{M}^N_d$, proves they satisfy relations, and gives an elimination-theoretic method to compute them. The main results show that the fixed-point multiplier map $\tau^N_{d,1}$ is generically $((d-2)!)^N$-to-one on split polynomial endomorphisms and generically finite-to-one on triangular polynomial endomorphisms and on a family of monic quadratic maps of $\mathbb{P}^2$. It also constructs several isospectral families whose multiplier invariants are constant. These results support a proposed higher-dimensional analogue of the classical one-dimensional theorem that multiplier maps are quasi-finite away from Lattès maps.

What carries the argument

The central object is the $n$-multiplier spectrum. For each point $P$ of exact period $n$, form the characteristic polynomial $\gamma_{f^n,P}(t)$ of the multiplier matrix $d(f^n)_P$; collect all of them into $\Sigma_n(f)=\prod_{P\in\mathrm{Per}_n(f)}(w-\gamma_{f^n,P}(t))$, and define $\sigma^{(n)}_{i,j}$ as its coefficients. These coefficients are symmetric functions of the eigenvalues and are invariant under $\mathrm{PGL}_{N+1}$ conjugation, and Theorem 2.4 shows they lie in the ring of regular functions $\mathbb{Q}[\mathcal{M}^N_d]$. The proof mechanism for the finite-to-one results is separation: since split maps have diagonal multiplier matrices, the eigenvalues from $\Sigma_1(f)$ can be assigned to coordinate polynomials, reducing the problem to the one-dimensional polynomial case; for triangular maps the same assignment feeds a multivariate Lagrange interpolation step; and for monic quadratics explicit elimination computes the image hypersurface. The fixed-point index identity $\sum_{P\in\mathrm{Fix}(f)}\gamma_{f,P}(t)/\gamma_{f,P}(1)=(t^{N+1}-d^{N+1})/(t-d)$ supplies relations among the invariants.

What would settle it

Compute $\Sigma_1(F)$ for every monic split polynomial $F=(x^3+a x+b,\, y^3+c y+e)$ in two variables with distinct fixed points. Factor $\Sigma_1(F)$ and enumerate all ways to split the nine pairs of eigenvalues into two sets of three pairs that could be the fixed-point spectra of cubic polynomials. If any two different splits produce the same coefficient list $\sigma^{(1)}_{i,j}$, or if one split supports more than $((3-2)!)^2=1$ pair of coordinate polynomials, the claimed generic degree is wrong.

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Extended reading notes

Core claim

The central discovery is that the multiplier spectrum—the unordered collection of characteristic polynomials of the derivative maps at periodic points—defines a system of regular conjugation-invariant functions on the moduli space $\mathcal{M}^N_d$, and that this system controls conjugacy classes in several natural families. Concretely, for split polynomial endomorphisms (each coordinate polynomial in a single variable), the fixed-point multiplier map $\tau^N_{d,1}$ is generically $((d-2)!)^N$-to-one: the only ambiguity left after reading the multipliers is an arbitrary permutation of the $N$ coordinate polynomials, each of which is recoverable up to the familiar $(d-2)!$ choices from one-variable polynomial dynamics. For triangular polynomial endomorphisms and for monic quadratic maps of $\mathbb{P}^2$ of the form $f=[x^2+a_1xz+a_2yz-a_1z^2 : y^2+b_1xz+b_2yz-b_1z^2 : z^2]$, the same map is generically finite-to-one; for the monic quadratics the image is an explicit hypersurface in $\mathbb{A}^5$. The paper also shows that Lattès-type constructions—symmetric products, cartesian products, and Segre embeddings of a Lattès family with the power map—give isospectral families whose multiplier invariants are constant for all periods.

Load-bearing premise

The load-bearing premise is that, given the full list of eigenvalue pairs of a split polynomial, there is only one way to assign those pairs to the $N$ coordinate polynomials so that they form valid multiplier spectra; if more than one assignment works, the claimed degree $((d-2)!)^N$ would be an overcount, and a similar independence assumption is needed for the triangular interpolation step.

Editorial extensions

If this is right

  • If the theorems are correct, the multiplier invariants give an explicit, computable coordinate system on large parts of $\mathcal{M}^N_d$, so conjugacy classes can be compared by finite multiplier data rather than by searching for conjugacies.
  • For split polynomial endomorphisms, including period-2 multiplier invariants should make $\tau^N_{d,n}$ generically one-to-one, assuming the one-dimensional conjecture on period-2 multipliers; the paper states this as a corollary of that conjecture.
  • The monic quadratic family of $\mathbb{P}^2$ shows that $\tau^2_{2,1}$ has image of codimension one in $\mathbb{A}^5$, with an explicit hypersurface equation that can be used to test membership and compute fibers.
  • Isospectral families provide higher-dimensional analogues of Lattès maps: symmetric products, cartesian products, and Segre images of Lattès families have constant multiplier invariants, so any quasi-finiteness statement must exclude them just as dimension 1 excludes Lattès maps.
  • The conjecture that $\tau^N_{d,n}$ is quasi-finite for large $n$ reduces the classification problem to the finite ambiguity encoded by the multiplier spectra, making the moduli space more accessible to arithmetic and computational study.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A concrete test of the split-polynomial degree formula: enumerate all monic split degree-3 polynomials in two variables and compare fibers of the map sending the polynomial to the coefficients of $\Sigma_1(f)$; finding a fiber with two different valid partitions of the eigenvalue pairs would lower the generic degree below $((d-2)!)^N$.
  • The generic degree of $\tau^2_{2,1}$ on the monic quadratic family remains open; partial computations in the paper suggest degree 8 on an open set and degree 12 on closed subsets, so a direct fiber-count would settle it.
  • The isospectral property for Segre embeddings seems to persist for more general isospectral factors; the paper notes computations suggest this but were beyond available machine power, so a proof for arbitrary isospectral pairs is a natural extension.
  • If the uniqueness-of-partition assumption fails, the corrected statement would likely involve a combinatorial factor counting compatible partitions of the multiplier spectrum, computable explicitly for small $N$ and $d$.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 5 minor

Summary. The paper defines multiplier invariants for endomorphisms of projective space of degree d, collected as coefficients σ_{i,j}^{(n)} of a polynomial Σ_n(f) built from characteristic polynomials of multiplier matrices at periodic points. It proves these are regular functions on the moduli space M_d^N, derives some relations among them, gives an elimination-theoretic algorithm for computing them without finding periodic points, and then addresses the extent to which multiplier data determine the conjugacy class. The main results are a conjecture generalizing McMullen's theorem, several constructions of isospectral families (symmetric products, Cartesian products, Segre products), and three finite-to-one statements: for split polynomial endomorphisms (a generic degree ((d−2)!)^N is claimed), for triangular polynomial endomorphisms, and for monic quadratic endomorphisms of P^2. The paper also contains an explicit computational description of the multiplier map on the monic quadratic family.

Significance. If the finite-to-one results are correct, the paper makes a meaningful step toward a higher-dimensional analogue of McMullen's theorem and provides useful tools for studying the moduli space M_d^N. The construction of the invariants is natural, the isospectral families via symmetric and Cartesian products are elegant, and the explicit computation on monic quadratics gives a rare concrete data point. The paper is also honest about its own gaps: the proof of Theorem 5.17 explicitly states that the independence of the interpolation equations remains a question, and the proof of Theorem 5.13 asserts a partition-uniqueness step without proof. These gaps are load-bearing for the stated degree and finite-to-one claims, so the paper's central new assertions require additional work before they can be accepted.

major comments (5)
  1. [§5.2.1, Theorem 5.13] The claimed degree ((d−2)!)^N depends on the unsupported assertion that, from the unordered N-tuples of eigenvalues of the multiplier matrices, one can split the eigenvalues into the multiplier spectra of the coordinate polynomials in only one way. The data form a complete multipartite hypergraph with vertex sets the coordinate spectra, and generic uniqueness of the factorization is not evident; with repeated eigenvalues or with the special eigenvalues contributed by fixed points at infinity, the partition can fail to be unique. If multiple partitions occur on a Zariski-open set, the fiber contains more than ((d−2)!)^N maps and the theorem's degree is an overcount, even if finite-to-one-ness survives. This step must be proved or the statement weakened to an upper bound.
  2. [§5.2.2, Theorem 5.17] The interpolation argument assumes that the linear equations determining the coefficients of F_k from their values at the fixed points of (F_1,...,F_{k−1}) are generically independent. The proof shows that a dependency would place the fixed points on a degree-j hypersurface and that this is a closed condition, but it does not prove that the union of these closed conditions is a proper subset of the moduli space; the example of the powering map avoids one such condition, but that does not rule out another closed condition covering the entire space. In addition, the earlier step of selecting, among the total eigenvalue data, the finite subsets that could be the multiplier spectrum of F_1 is again a partition-uniqueness problem that is not addressed.
  3. [§2, Theorem 2.4(1)] The proof that σ_{i,j}^{(n)} is regular on Hom_d^N is too terse. The assertion that the only possible poles come from partial derivatives of the dehomogenized map, and that these yield only powers of the resultant, needs a careful argument: the fixed points themselves are algebraic over the coefficient field and their coordinates can have denominators, and one must show these cancel in the symmetric functions. This is load-bearing because regularity on M_d^N is one of the paper's foundational claims (Theorem A).
  4. [§5.2.3, Theorem 5.19] The hypersurface equation for the image of τ_{2,1}^2 restricted to monic polynomials is asserted after a Sage computation, but no code, script, or certificate is provided. Since Corollary 5.20 and the claim that the image is a hypersurface rest entirely on this calculation, the computation should be reproducible (for example, by including the Sage code and the elimination Groebner basis) or verified by an independent method.
  5. [§2, paragraph after Definition 2.1] The stated number of points of period n is D_n = (d^{n(N+1)}−1)/(d−1), but the number of fixed points of f^n on P^N is ((d^n)^{N+1}−1)/(d^n−1). For N=1, d=2, n=2 the formula gives 15, while f^2 has 5 fixed points counted with multiplicity. This error propagates into the indexing of Σ_n in equations (1)–(2) and into the proof of Theorem 3.1 for n>1.
minor comments (5)
  1. [Example 5.14] The example states that the invariants for F=(x^2+c, y^2+d) are generated by σ_{2,2}=8(c+d)+60 and σ_{2,3}=16(c+d)+24, both of which depend only on c+d; this cannot determine the pair (c,d) up to permutation. Please provide the full set of invariants or correct the assertion.
  2. [§3, Theorem 3.1] The proof of independence of {σ_{1,j},...,σ_{j,j}} is terse; the claim that each σ_{b,j} contains a partition not found in σ_{a,j} for a<b≤j requires a more formal statement about partitions of j into at most i parts.
  3. [§4, Algorithm 4.3] The algorithm is described as a way to compute Σ_1(f), but the treatment of multiplicities is not fully justified; the remark that Groebner bases lose multiplicity information and the claimed fix via Chow forms need a precise correctness statement.
  4. [§5.2.1] The phrase 'Fujimura's results summarized in [6]' is confusing because earlier in the paper Fujimura–Nishizawa [7] is cited for the polynomial multiplier coordinate; please clarify which reference is meant.
  5. [Throughout] There are numerous typos and OCR artifacts (e.g., 'endomorph ism' in the abstract, 'indeterminant' for 'indeterminate', stray spacing in 'Latt` es'), which should be corrected in a final version.

Circularity Check

1 steps flagged · score 2.0 of 10

No definitional or fitted-input circularity; one load-bearing self-citation in the isospectral-family section, while the finite-to-one reconstruction arguments are derived from external one-dimensional results.

  1. self citation load bearing [Theorem 5.3, Section 5.1.1 (proof of isospectrality for k-symmetric products)]
    "The multipliers of the symmetric product F depend only on the multipliers of f [8]."

    This sentence is the whole proof of Theorem 5.3. Reference [8] is Gauthier–Hutz–Kaschner, a paper with the present author as coauthor, so the isospectrality of symmetric products is imported from the authors' own prior work rather than proved in this paper. The paper does not reproduce or independently verify the cited multiplier property, making this a load-bearing self-citation for that theorem. It is not, however, used in the main finite-to-one reconstruction results (Theorems 5.13, 5.17, 5.20), whose arguments rest on external one-dimensional results (Fujimura, Milnor, Sugiyama) and on explicit elimination computations, so the core derivation is not circular.

full rationale

The invariant construction (Section 2) is tied to multipliers by definition, but the claim that the sigma functions are regular on M_d^N is proved via Galois invariance and resultants, not by assuming any reconstruction. Theorem 5.13 reconstructs split polynomial endomorphisms from Sigma_1(f) by factoring eigenvalues and applying Fujimura's one-dimensional multiplier-spectrum theorem componentwise; the only questionable point is the unproved assertion that the eigenvalue sets split uniquely into coordinate spectra. That is a possible correctness gap (an overcount if false), not a circularity, because it does not presuppose the ((d-2)!)^N degree it is trying to establish. Theorem 5.17 has a similar unproven-genericity step in its interpolation argument, again a gap rather than a circular reduction. Theorem 5.19 and Corollary 5.20 derive the image hypersurface and finite-to-one-ness by explicit elimination, with no fitted parameters. The only self-citation that carries a result is Theorem 5.3's appeal to [8], which supports one isospectral-family construction but is not needed for the finite-to-one claims. No equation is defined in terms of a target invariant, and no fitted input is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted; the paper is a pure mathematical construction. The main external inputs are standard results in dynamics and algebraic geometry, several of them cited from the literature, including one self-citation for the symmetric product multiplier property.

assumptions (5)
  • domain assumption Ueda's fixed point formula (Proposition 3.2) for holomorphic endomorphisms of P^N
    Used to derive relations among the σ_{i,j} in Section 3; cited to [30].
  • domain assumption Fujimura's result: for a polynomial of degree d, a given set of fixed point multiplier invariants corresponds to (d-2)! conjugacy classes
    Used in Theorems 5.13 and 5.17 to bound the number of possible coordinate polynomials; cited to [6].
  • domain assumption Minimair's theorem that the resultant of an iterate is a power of the resultant
    Used in Theorem 2.4 to show the σ are regular functions; cited to [23].
  • domain assumption The multipliers of the symmetric product of a map depend only on the multipliers of the original map
    Used in Theorem 5.3 for isospectrality; cited to [8], which includes an author of this paper.
  • standard math Standard properties of Gröbner bases and elimination theory
    Used in Section 4 algorithms and Theorem 5.19.

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Pith. "Pith review of Multipliers and invariants of endomorphisms of projective space in dimension greater than 1." pith.science (2026). https://pith.science/paper/TZO5POJ7

@misc{pith2026190803184,
  author       = {Pith},
  title        = {Pith review of: Multipliers and invariants of endomorphisms of projective space in dimension greater than 1},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TZO5POJ7}},
  note         = {Machine review of arXiv:1908.03184}
}
abstract

There is a natural conjugation action on the set of endomorphism of $\P^N$ of fixed degree $d \geq 2$. The quotient by this action forms the moduli of degree $d$ endomorphisms of $\P^N$, denoted $\mathcal{M}_d^N$. We construct invariant functions on this moduli space coming from to set of multiplier matrices of the periodic points. The basic properties of these functions are demonstrated such as that they are in the ring of regular functions of $\mathcal{M}_d^N$, methods of computing them, as well as the existence of relations. The main part of the article examines to what extend these invariant functions determine the conjugacy class in the moduli space. Several different types of isospectral families are constructed and a generalization of McMullen's theorem on the multiplier mapping of dimension 1 is proposed. Finally, this generalization is shown to hold when restricted to several specific families in $\mathcal{M}_d^N$.

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