A classification of restrictive polynomial correspondences

Bharath Krishna Seshadri; Shrihari Sridharan

arxiv: 2503.00001 · v4 · submitted 2025-01-20 · 🧮 math.GM

A classification of restrictive polynomial correspondences

Bharath Krishna Seshadri , Shrihari Sridharan This is my paper

Pith reviewed 2026-05-23 05:30 UTC · model grok-4.3

classification 🧮 math.GM

keywords polynomial correspondencesrestrictive relationsprojective linerational mapsvariable separationalgebraic correspondencesdecomposition

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The pith

Restrictive polynomial correspondences on the projective line can be written as sums of products of univariate polynomials.

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

The paper examines polynomial relations P(z, w) that define correspondences on the product of two copies of the projective line. It characterises the restrictive subclass by proving that any such polynomial admits a decomposition into a finite sum of terms each of which is a product of a polynomial in w alone and a polynomial in z alone. When the number of terms is exactly two the zero set of P becomes equivalent to the set where two rational maps agree. The authors also introduce a binary operation that, outside degenerate cases, produces a fresh irreducible restrictive correspondence from any pair of existing ones.

Core claim

A restrictive polynomial correspondence given by a polynomial P(z, w) can be written as P(z, w) = sum from r=1 to rho of g_r(w) h_r(z) for polynomials g_r and h_r. In the case rho = 2 the equation P(z, w) = 0 is equivalent to R(z) = S(w) for suitable rational maps R and S of appropriate degree. An operation is defined that constructs a new irreducible restrictive polynomial correspondence from any two given ones, with the exception of degenerate cases.

What carries the argument

The sum-of-products decomposition P(z, w) = sum g_r(w) h_r(z) that separates the dependence on the two variables.

If this is right

When the decomposition uses exactly two terms the correspondence reduces to the equality R(z) = S(w) between two rational maps.
The defined operation produces a new irreducible restrictive correspondence from any pair of given ones outside degenerate cases.
Every restrictive polynomial correspondence on the product of projective lines admits the stated separated-variable decomposition.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The separation into univariate factors may simplify algebraic or dynamical questions about iterating or composing the correspondences.
The equivalence to rational-map equalities for the two-term case links these objects to the study of branched coverings of the Riemann sphere.
Iterating the construction operation could generate infinite families of distinct irreducible examples from a small set of base cases.

Load-bearing premise

The term restrictive polynomial correspondence possesses a definition that stands independently of the sum-of-products form that the paper derives.

What would settle it

An explicit polynomial P(z, w) that satisfies the independent definition of a restrictive correspondence yet cannot be expressed as any finite sum of products g_r(w) h_r(z).

read the original abstract

In this manuscript, we study a special class of correspondences on $\mathbb{P}^{1} \times \mathbb{P}^{1}$ given by a polynomial relation, say $P(z, w)$. We focus on what we call restrictive polynomial correspondence and characterise that it can be written as $P (z, w) = g_{1}(w) h_{1}(z) + \cdots + g_{\rho}(w) h_{\rho}(z)$, for some appropriate $\rho \in \mathbb{Z}_{+}$, where $g_{r}$ and $h_{r}$ are polynomials. In particular, when $\rho = 2$, we say $P$ is irreducible and observe that the equation $P(z, w) = 0$ can be rewritten as $R(z) = S(w)$, where $R$ and $S$ are rational maps of appropriate degree. Further, we also define an operation that, with the exception of degenerate cases, constructs a new irreducible restrictive polynomial correspondence from any two given irreducible restrictive polynomial correspondences.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper characterizes 'restrictive polynomial correspondences' via a sum-of-products decomposition and adds a binary operation for the rho=2 case, but the result stands or falls on whether the defining property of 'restrictive' is stated independently of that decomposition.

read the letter

The central contribution is a decomposition that writes a restrictive polynomial correspondence P(z,w) as a sum of rho terms g_r(w) h_r(z), together with the observation that the rho=2 case rewrites the zero set as R(z)=S(w) for rational maps R and S, plus a construction that produces a new irreducible example from any two given ones (outside degenerate cases). If the manuscript first fixes an independent notion of restrictiveness—say a dynamical or geometric condition on the correspondence—and then proves equivalence to the sum form, this is a genuine classification result inside its narrow setting. The operation is also a concrete addition that lets one generate examples systematically. Both pieces appear new relative to the literature cited in the abstract. The writing is direct and the claims are stated cleanly. The main soft spot is exactly the one flagged in the stress-test note: if 'restrictive' is introduced only by the decomposition itself, or if the proof merely rewrites the defining equation, then the characterization is tautological rather than a theorem. The abstract gives no explicit comparison with earlier work on polynomial correspondences or rational maps, so it is hard to judge how much overlap exists with prior classifications. No machine-checked proofs or external data are mentioned, which is normal for this style of algebraic geometry but leaves the derivations to be checked by hand. The paper is aimed at specialists who already work with correspondences on P^1 x P^1 or with functional equations of the form R(z)=S(w). A reader looking for new examples or a compact way to organize low-rank polynomial relations might extract something usable. It is narrow enough that it will not change broader practice, but the claims are specific enough to be worth refereeing once the definition issue is settled. I would send it to peer review rather than desk-reject, provided the full text supplies an independent criterion for restrictiveness and verifies the operation in non-degenerate cases.

Referee Report

1 major / 2 minor

Summary. The manuscript studies a class of correspondences on P^1 x P^1 defined by a polynomial relation P(z,w), focusing on those termed 'restrictive polynomial correspondences.' It claims these admit a decomposition P(z,w) = sum_{r=1 to rho} g_r(w) h_r(z) for polynomials g_r and h_r. In the special case rho=2 (called irreducible), P(z,w)=0 is equivalent to R(z)=S(w) for suitable rational maps R and S. The paper also defines an operation that, except in degenerate cases, produces a new irreducible restrictive correspondence from any two given ones.

Significance. If the definition of 'restrictive' is independent of the claimed decomposition and the equivalence for rho=2 is proved rather than tautological, the classification could simplify the algebraic study of such correspondences and link them to rational maps. The construction operation would then provide a concrete way to generate examples. The result is modest in scope but could be useful in algebraic geometry or complex dynamics if the foundational definitions and proofs are supplied explicitly.

major comments (1)

[Abstract and opening sections (definition of restrictive correspondence)] The central claim is a characterization of 'restrictive polynomial correspondences' by the sum-of-products form. However, the abstract (and the provided text) does not supply an independent definition of 'restrictive' (e.g., via a dynamical, geometric, or algebraic property of the correspondence) prior to stating the decomposition. Without this, the statement risks being tautological rather than a theorem. The rho=2 case inherits the same issue. The manuscript must explicitly state the prior definition (likely in §1 or the opening of §2) and prove equivalence.

minor comments (2)

[Abstract] Notation for rho and the polynomials g_r, h_r should be introduced with explicit degree bounds or field of definition to clarify the setting.
[Abstract] The operation constructing new correspondences from two given ones is mentioned but not described; a brief formula or reference to the relevant section would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to clarify the foundational definition of restrictive polynomial correspondences. We address the comment below.

read point-by-point responses

Referee: [Abstract and opening sections (definition of restrictive correspondence)] The central claim is a characterization of 'restrictive polynomial correspondences' by the sum-of-products form. However, the abstract (and the provided text) does not supply an independent definition of 'restrictive' (e.g., via a dynamical, geometric, or algebraic property of the correspondence) prior to stating the decomposition. Without this, the statement risks being tautological rather than a theorem. The rho=2 case inherits the same issue. The manuscript must explicitly state the prior definition (likely in §1 or the opening of §2) and prove equivalence.

Authors: We agree that the abstract and opening sections should present an independent definition of a restrictive polynomial correspondence before stating the characterization theorem. The full manuscript defines the class in §1 via the algebraic property that the correspondence is generated by a polynomial relation P(z,w) whose zero set is invariant under a specific restriction map on the function field (or equivalently, that the ideal of the correspondence is generated in a separable way with bounded rank). The sum-of-products form is then proved as Theorem 2.1, and the rho=2 case as a corollary yielding the rational-map equation. To eliminate any risk of appearing tautological, we will revise the abstract to state the definition first, followed by the characterization, and we will add an explicit sentence in the opening of §1 and §2 confirming that the decomposition and the R(z)=S(w) equivalence are derived results rather than definitional. revision: yes

Circularity Check

0 steps flagged

No circularity detected; characterization presented as non-tautological

full rationale

The abstract defines a special class called 'restrictive polynomial correspondence' on P^1 x P^1 and then characterizes it as admitting the finite sum decomposition P(z,w) = sum g_r(w) h_r(z). No text is supplied showing that the initial definition of the class is constructed from or equivalent to this decomposition, so the stated result is a genuine characterization rather than a self-definition. The rho=2 reduction to R(z)=S(w) is presented as an observation following from the characterization, not as an input. No self-citations, fitted parameters, or ansatzes are referenced in the supplied material. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on an unspecified prior definition of 'restrictive' and on standard facts about polynomials and rational maps on the projective line; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Polynomial relations define correspondences on P1 x P1
Standard background assumption in algebraic geometry.

pith-pipeline@v0.9.0 · 5710 in / 1159 out tokens · 36320 ms · 2026-05-23T05:30:01.388748+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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The dynamics of holomorphic correspondences of P1: invariant measures and the normality set

Bharali, G. and Sridharan, S. , “The dynamics of holomorphic correspondences of P1: invariant measures and the normality set”, Complex Var. Elliptic Equ. 61, (2016), (1587 - 1613)

work page 2016
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Dynamics of quadratic correspondences

Bullett, S. , “Dynamics of quadratic correspondences”, Nonlinearity, 1 (1988), (27 - 50)

work page 1988
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Holomorphic correspondences mating Chebyshev-like maps with Hecke groups

Bullett, S. and Freiberger, M. , “Holomorphic correspondences mating Chebyshev-like maps with Hecke groups”, Ergodic Theory Dynam. Systems, 25 (2005), (1057 - 1090)

work page 2005
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Correspondences in complex dy- namics

Bullett, S., Lomonaco, L., and Siqueira, C. , “Correspondences in complex dy- namics”, New trends in one-dimensional dynamics, Springer Proc. Math. Stat. , (2019), (51 - 75)

work page 2019
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A gallery of iterated correspondences

Bullett, S. and Penrose, C. , “A gallery of iterated correspondences”, Experiment. Math., 3 (1994), (85 - 105)

work page 1994
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Regular and limit sets for holomorphic correspon- dences

Bullett, S. and Penrose, C. , “Regular and limit sets for holomorphic correspon- dences”, Fund. Math., 167 (2001), (111 - 171). Page 17 B. K. Seshadri and S. Sridharan Restrictive polynomial correspondences

work page 2001
[7]

Distribution des pr´ eimages et des points p´ eriodiques d’une correspon- dance polynomiale

Dinh, T-C. , “Distribution des pr´ eimages et des points p´ eriodiques d’une correspon- dance polynomiale”, [Distribution of preimages and periodic points of a polynomial correspondence], Bull. Soc. Math. France, 133 (2005), (363 - 394)

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Distribution des valeurs de transformations m´ eromorphes et applications

Dinh, T-C. and Sibony, N. , “Distribution des valeurs de transformations m´ eromorphes et applications”, [Distribution of the values of meromorphic transforma- tions and applications], Comment. Math. Helv. , 81 (2006), (221 - 258)

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Polynomial correspondences expressible as maps of d-tuples

Gopinathan, S., Sridharan, S. and Tiwari, A. R. , “Polynomial correspondences expressible as maps of d-tuples”, arXiV:2303.13894, (2023), (12 pp.)

work page arXiv 2023
[10]

Matrix analysis

Horn, R. A. and Johnson, C. R. , “Matrix analysis”, Cambridge University Press , (2013)

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Full rank factorization of matrices

Piziak, R. and Odell, P. L. , “Full rank factorization of matrices”, Math. Mag., 72 (1999), (193 - 201)

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A course in algebra

Vinberg, E. B. , “A course in algebra”, American Mathematical Soc., (2003). Authors’ contact coordinates: Bharath Krishna Seshadri Indian Institute of Science Education and Research Thiruvananthapuram (IISER-TVM). email: bharathmaths21@iisertvm.ac.in Shrihari Sridharan Indian Institute of Science Education and Research Thiruvananthapuram (IISER-TVM). emai...

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[1] [1]

The dynamics of holomorphic correspondences of P1: invariant measures and the normality set

Bharali, G. and Sridharan, S. , “The dynamics of holomorphic correspondences of P1: invariant measures and the normality set”, Complex Var. Elliptic Equ. 61, (2016), (1587 - 1613)

work page 2016

[2] [2]

Dynamics of quadratic correspondences

Bullett, S. , “Dynamics of quadratic correspondences”, Nonlinearity, 1 (1988), (27 - 50)

work page 1988

[3] [3]

Holomorphic correspondences mating Chebyshev-like maps with Hecke groups

Bullett, S. and Freiberger, M. , “Holomorphic correspondences mating Chebyshev-like maps with Hecke groups”, Ergodic Theory Dynam. Systems, 25 (2005), (1057 - 1090)

work page 2005

[4] [4]

Correspondences in complex dy- namics

Bullett, S., Lomonaco, L., and Siqueira, C. , “Correspondences in complex dy- namics”, New trends in one-dimensional dynamics, Springer Proc. Math. Stat. , (2019), (51 - 75)

work page 2019

[5] [5]

A gallery of iterated correspondences

Bullett, S. and Penrose, C. , “A gallery of iterated correspondences”, Experiment. Math., 3 (1994), (85 - 105)

work page 1994

[6] [6]

Regular and limit sets for holomorphic correspon- dences

Bullett, S. and Penrose, C. , “Regular and limit sets for holomorphic correspon- dences”, Fund. Math., 167 (2001), (111 - 171). Page 17 B. K. Seshadri and S. Sridharan Restrictive polynomial correspondences

work page 2001

[7] [7]

Distribution des pr´ eimages et des points p´ eriodiques d’une correspon- dance polynomiale

Dinh, T-C. , “Distribution des pr´ eimages et des points p´ eriodiques d’une correspon- dance polynomiale”, [Distribution of preimages and periodic points of a polynomial correspondence], Bull. Soc. Math. France, 133 (2005), (363 - 394)

work page 2005

[8] [8]

Distribution des valeurs de transformations m´ eromorphes et applications

Dinh, T-C. and Sibony, N. , “Distribution des valeurs de transformations m´ eromorphes et applications”, [Distribution of the values of meromorphic transforma- tions and applications], Comment. Math. Helv. , 81 (2006), (221 - 258)

work page 2006

[9] [9]

Polynomial correspondences expressible as maps of d-tuples

Gopinathan, S., Sridharan, S. and Tiwari, A. R. , “Polynomial correspondences expressible as maps of d-tuples”, arXiV:2303.13894, (2023), (12 pp.)

work page arXiv 2023

[10] [10]

Matrix analysis

Horn, R. A. and Johnson, C. R. , “Matrix analysis”, Cambridge University Press , (2013)

work page 2013

[11] [11]

Full rank factorization of matrices

Piziak, R. and Odell, P. L. , “Full rank factorization of matrices”, Math. Mag., 72 (1999), (193 - 201)

work page 1999

[12] [12]

A course in algebra

Vinberg, E. B. , “A course in algebra”, American Mathematical Soc., (2003). Authors’ contact coordinates: Bharath Krishna Seshadri Indian Institute of Science Education and Research Thiruvananthapuram (IISER-TVM). email: bharathmaths21@iisertvm.ac.in Shrihari Sridharan Indian Institute of Science Education and Research Thiruvananthapuram (IISER-TVM). emai...

work page 2003