arxiv: 2604.10605 · v1 · submitted 2026-04-12 · 🧮 math.CV · math.AG

Recognition: unknown

Singularities of diagonals of Laurent series for rational functions

Dmitriy Pochekutov

Pith reviewed 2026-05-10 16:01 UTC · model grok-4.3

classification 🧮 math.CV math.AG

keywords diagonalsLaurent seriesrational functionsanalytic continuationLandau varietyNewton polyhedronsingularitiescomplex tori

0 comments

The pith

For rational functions with nondegenerate denominators, the complete diagonal of the Laurent series admits analytic continuation along paths in the r-torus that avoid the Landau variety.

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

The paper shows that the complete diagonal of a multivariate Laurent series expansion of a rational function extends analytically to the r-dimensional complex torus minus an explicit complex analytic set. This holds provided the denominator is nondegenerate for its Newton polyhedron, with the singular set L formed as the union of discriminants coming from the truncations of the denominator to the faces of that polyhedron. A sympathetic reader cares because these diagonals serve as generating functions in combinatorics and algebraic geometry, and locating their singularities precisely determines domains of holomorphy and coefficient asymptotics without needing the full series. The initial domain is a logarithmically convex region, and continuation succeeds along any path missing L.

Core claim

For a rational function whose denominator is nondegenerate for its Newton polyhedron, the complete diagonal, initially defined in a logarithmically convex domain, can be analytically continued along any path in the r-dimensional complex torus that avoids the Landau variety L. The variety L is constructed explicitly as the union of the discriminants of the truncations of the denominator polynomial to the faces of the Newton polyhedron.

What carries the argument

The Landau variety L, built as the union of discriminants of the face truncations of the denominator with respect to the faces of its Newton polyhedron; it acts as the explicit barrier set preventing further analytic continuation of the diagonal.

Load-bearing premise

The denominator is nondegenerate for its Newton polyhedron, which is needed for the discriminants of the face truncations to correctly identify all singularities and support the continuation statement.

What would settle it

An explicit low-dimensional rational function whose denominator satisfies the nondegeneracy condition, yet whose computed complete diagonal develops a singularity at a point inside the r-torus that lies outside the predicted Landau variety L, would disprove the claim.

Figures

Figures reproduced from arXiv: 2604.10605 by Dmitriy Pochekutov.

**Figure 1.** Figure 1: Partition of the component E by the amoebas Af1 , Af2 and the choice of points xε(t). The class of the real n-dimensional torus Γε,t := Λ−1 (xε(t)) in the homology group Hn(T nz \ Z ×(f · f1 · · · fr)) with integer coefficients does not depend on the choice of the point xε(t) in Eε due to convexity. Then in the complement T n z \ Z ×(f · f1 · · · fr) one can define the n-dimensional cycle Γt := X ε (ε1 · … view at source ↗

**Figure 2.** Figure 2: Amoeba (gray, left), its complement components (the bounded [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

read the original abstract

We study the complete diagonal of the Laurent series expansion of a rational function in $n$-complex variables. For a denominator that is nondegenerate for its Newton polyhedron, we prove that the complete diagonal, initially defined in a logarithmically convex domain, can be analytically continued along any path in the $r$-dimensional complex torus that avoids an explicitly defined complex analytic set $L$ called the Landau variety. This variety is constructed as the union of discriminants associated with specific truncations of the denominator to the faces of its Newton polyhedron.

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 gives an explicit Landau variety for singularities of complete diagonals of rational Laurent series, built from discriminants of Newton polyhedron face truncations under a nondegeneracy assumption.

read the letter

The main point is that for a rational function with denominator nondegenerate relative to its Newton polyhedron, the complete diagonal extends analytically along paths in the r-torus that miss an explicitly described complex analytic set L, where L is the union of discriminants coming from truncations of the denominator to the faces of the polyhedron. This organizes the continuation problem around concrete combinatorial data rather than leaving it abstract.

Referee Report

1 major / 2 minor

Summary. The paper proves that, for a rational function in n complex variables whose denominator is nondegenerate with respect to its Newton polyhedron, the complete diagonal of the associated Laurent series (initially defined in a logarithmically convex domain) admits analytic continuation along any path in the r-dimensional complex torus that avoids an explicitly constructed complex analytic set L, the Landau variety. L is defined as the union of the discriminants of the truncations of the denominator to the faces of the Newton polyhedron.

Significance. If the central claim holds, the result supplies an explicit, geometrically natural description of the singularity locus for complete diagonals of rational Laurent series, building on Newton polyhedra, amoebas, and discriminants. This is a useful advance for determining domains of analyticity in multivariate generating functions, with direct relevance to asymptotic combinatorics and coefficient extraction. The construction is parameter-free once the nondegeneracy hypothesis is fixed and the polyhedron is given, which is a methodological strength.

major comments (1)

[proof of the main theorem] The nondegeneracy hypothesis is stated to ensure that the discriminants of the face truncations suffice to define L and that no singularities lie outside L. The proof of analytic continuation must therefore verify that this hypothesis excludes additional obstructions arising from the torus compactification or from higher-order degeneracies; a concrete check or reference to the relevant analytic-continuation step (e.g., the argument that paths avoiding L can be lifted without encountering poles) would confirm that L is complete.

minor comments (2)

The relation between the number of variables n and the dimension r of the torus should be stated explicitly at the outset (r is presumably the number of independent diagonal directions).
A short illustrative example computing L for a low-dimensional nondegenerate denominator would help readers verify the construction of the discriminants.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and the recommendation for minor revision. We address the single major comment below by clarifying the role of the nondegeneracy hypothesis in the analytic continuation argument.

read point-by-point responses

Referee: The nondegeneracy hypothesis is stated to ensure that the discriminants of the face truncations suffice to define L and that no singularities lie outside L. The proof of analytic continuation must therefore verify that this hypothesis excludes additional obstructions arising from the torus compactification or from higher-order degeneracies; a concrete check or reference to the relevant analytic-continuation step (e.g., the argument that paths avoiding L can be lifted without encountering poles) would confirm that L is complete.

Authors: We agree that an explicit verification of this point improves the exposition. In the revised manuscript we have added a dedicated paragraph immediately after the statement of the main result (Theorem 1.1). There we recall that the nondegeneracy assumption on the denominator with respect to its Newton polyhedron implies that every truncation to a face remains nondegenerate. Consequently, any potential singularity arising from the torus compactification or from higher-order degeneracies would force a vanishing of the face discriminant, contradicting the hypothesis. We then refer directly to the lifting argument in Lemma 3.4, which shows that any path in the r-torus avoiding L lifts to a path in the universal cover that encounters no poles of the rational function. This establishes that L is the complete singularity locus. revision: yes

Circularity Check

0 steps flagged

No circularity detected in the derivation.

full rationale

The paper states a theorem proving analytic continuation of the complete diagonal along paths in the r-torus avoiding an explicitly constructed Landau variety L, defined as the union of discriminants of face truncations of the denominator under the nondegeneracy hypothesis w.r.t. its Newton polyhedron. This construction and the continuation statement invoke standard facts from complex analysis, toric geometry, and algebraic geometry (discriminants, Newton polyhedra) without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. The nondegeneracy condition is an explicit hypothesis required for the construction to hold, not a derived or renamed input. The central claim therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 1 invented entities

The central claim rests on the nondegeneracy assumption for the Newton polyhedron and on background results from several complex variables concerning analytic continuation along paths avoiding analytic sets and the fact that discriminants define analytic hypersurfaces. No free parameters are introduced. The Landau variety is defined rather than postulated as an independent entity.

axioms (3)

standard math Analytic continuation of holomorphic functions along paths in complex manifolds is possible when avoiding analytic sets
Invoked implicitly when stating that continuation holds outside L
standard math Discriminants of polynomials define complex analytic sets
Used to construct L from truncations to faces
domain assumption Newton polyhedra and their faces control the behavior of Laurent series at infinity
Central to the nondegeneracy hypothesis and truncation construction

invented entities (1)

Landau variety L no independent evidence
purpose: To serve as the explicit obstacle set for analytic continuation of the diagonal
Defined constructively as union of discriminants of face truncations; no independent falsifiable prediction is given beyond the continuation statement itself

pith-pipeline@v0.9.0 · 5376 in / 1677 out tokens · 36766 ms · 2026-05-10T16:01:43.684763+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references

[1]

Gelfand, M

I. Gelfand, M. Kapranov, and A. Zelevinsky.Discriminants, resultants and multidimensional determinants. Birkh¨ auser, 1994

1994
[2]

Forsberg, M

M. Forsberg, M. Passare, and A. Tsikh. Laurent determinants and ar- rangements of hyperplane amoebas.Advances in Mathematics, 151(1):45– 70, 2000

2000
[3]

Melczer.An Invitation to Analytic Combinatorics: From One to Several Variables

S. Melczer.An Invitation to Analytic Combinatorics: From One to Several Variables. Texts & Monographs in Symbolic Computation. Springer, 2021

2021
[4]

Stanley.Enumerative combinatorics, Volume 2

R. Stanley.Enumerative combinatorics, Volume 2. Cambridge Univer- sity Press, 1999. 12

1999
[5]

Leinartas and T

E. Leinartas and T. Nekrasova. Generating function of the solution of a difference equation and the Newton polyhedron of the charac- teristic polynomial.The Bulletin of Irkutsk State University. Series Mathematics, 40:3–14, 2022

2022
[6]

Bostan, S

A. Bostan, S. Boukraa, G. Christol, S. Hassani, and J.-M. Maillard. Ising n-fold integrals as diagonals of rational functions and integrality of series expansions.Journal of Physics A: Mathematical and Theoretical, 46(18):185–202, 2013

2013
[7]

Batyrev and M

V. Batyrev and M. Kreuzer. Constructing new Calabi-Yau 3-folds and their mirrors via conifold transitions.Advances in Theoretical and Mathematical Physics, 14(03):879–898, 2010

2010
[8]

A. Bishop. On groups whose cogrowth series is the diagonal of a rational series.International Journal of Algebra and Computation, 34(08):1209– 1224, 2024

2024
[9]

Pochekutov

D. Pochekutov. Diagonals of the Laurent series of rational functions. Siberian Math. Journal, 50(6):1081–1091, 2009

2009
[10]

A. K. Tsikh K. V. Safonov. Singularities of the Grothendieck parametric residue and diagonals of a double power series.Soviet Math. (Iz. VUZ), 28(4):65–74, 1984

1984
[11]

Pham.Singularities of integrals

F. Pham.Singularities of integrals. Springer London, 2011

2011
[12]

V. A. Vasiliev.Applied Picard-Lefschetz Theory. AMS, 2002

2002
[13]

Savin and B

A. Savin and B. Sternin.Introduction to Complex Theory of Differential Equations. Birkh¨ auser Cham, 2017

2017
[14]

Pochekutov

D. Pochekutov. Analytic continuation of diagonals of laurent series for rational functions.Journal of Siberian Federal University. Mathematics & Physics, 14(3):360–368, 2021

2021
[15]

D. Yu. Pochekutov and A. V. Senashov. Toric morphisms and diag- onals of the Laurent series of rational functions.Siberian Electronic Mathematical Reports, 19(2):651–661, 2022. 13

2022
[16]

Orders of branch points of complete diagonals for Laurent series of rational functions of two variables

Pochekutov D.Yu. Orders of branch points of complete diagonals for Laurent series of rational functions of two variables. (In Russ.).Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika., 1(9):58–69, 2025

2025
[17]

C. W. Curtis and I. Reiner.Representation Theory of Finite Groups and Associative Algebras. Interscience Publishers, 1962

1962
[18]

Khovanskii

A.G. Khovanskii. Newton polyhedra and toroidal varieties.Functional Analysis and Its Applications, 11(4):289–296, 1977. 14

1977