Even-carry polynomials and cohomology of line bundles on the incidence correspondence in positive characteristic

Evan M. O'Dorney

arxiv: 2404.04166 · v3 · submitted 2024-04-05 · 🧮 math.AG · math.AC· math.NT· math.RT

Even-carry polynomials and cohomology of line bundles on the incidence correspondence in positive characteristic

Evan M. O'Dorney This is my paper

Pith reviewed 2026-05-24 02:30 UTC · model grok-4.3

classification 🧮 math.AG math.ACmath.NTmath.RT

keywords even-carry polynomialsNim polynomialsincidence correspondenceline bundlescohomologypositive characteristicrecurrence relations

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The pith

Even-carry polynomials give an explicit formula for the cohomology of line bundles on the n=3 incidence correspondence in any characteristic.

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

The paper introduces even-carry polynomials as a generalization of Nim polynomials. These allow solving the recurrence relation previously established by Liu-Gao-Raicu for the cohomology of line bundles on the incidence correspondence when n equals 3. This produces an explicit description valid for any prime characteristic p, in contrast to earlier results that were explicit only for p=2. A sympathetic reader cares because the base-p digit dependence of the cohomology was previously mysterious, and the new polynomials remove that mystery for the n=3 case. The work also formulates conjectures for the general-n situation.

Core claim

We devise a suitable generalization of Nim polynomials, which we call even-carry polynomials, by which we can solve the recurrence of Liu--Gao--Raicu to yield an explicit formula for the cohomology for n = 3 and general p.

What carries the argument

even-carry polynomials, a generalization of Nim polynomials that encodes base-p carry information to solve the Liu-Gao-Raicu recurrence for cohomology dimensions.

If this is right

Cohomology dimensions for any line bundle on the n=3 incidence correspondence are given by a closed-form expression built from even-carry polynomials.
The expression depends combinatorially on the base-p expansions of the degrees d and e.
The p=2 case handled by Nim polynomials extends uniformly to arbitrary primes p.

Where Pith is reading between the lines

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

If even-carry polynomials admit a version for higher n, they would resolve the conjectures stated for general n.
The Nim connection raises the possibility that other combinatorial games encode sheaf cohomology data in positive characteristic.

Load-bearing premise

The recurrence relation previously derived by Liu-Gao-Raicu correctly encodes the cohomology groups of the line bundles on the incidence correspondence for n=3 in every characteristic p.

What would settle it

Direct computation of the cohomology groups for small fixed p, d, and e via Čech cohomology or computer algebra, checking whether the resulting dimensions match the values produced by the even-carry polynomial formula.

read the original abstract

We consider the cohomology groups of line bundles $\mathcal{L}$ on the \emph{incidence correspondence}, that is, a general hypersurface $X \subset \mathbb{P}^{n-1} \times \mathbb{P}^{n-1}$ of degrees $(1,1)$. Whereas the characteristic $0$ situation is completely understood, the cohomology in characteristic $p$ depends in a mysterious way on the base-$p$ digits of the degrees $(d, e)$ of $\mathcal{L}$. Gao and Raicu (following Linyuan Liu) prove a recursive description of the cohomology for $n = 3$, which relates to Nim polynomials when $p = 2$. In this paper, we devise a suitable generalization of Nim polynomials, which we call \emph{even-carry polynomials,} by which we can solve the recurrence of Liu--Gao--Raicu to yield an explicit formula for the cohomology for $n = 3$ and general $p$. We also make some conjectures on the general form of the cohomology for general $n$ and $p$, for which a recurrence relation was recently derived by Kyomuhangi--Marangone--Raicu--Reed.

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

O'Dorney turns the Liu-Gao-Raicu recurrence into an explicit closed-form formula for n=3 cohomology by defining even-carry polynomials.

read the letter

The paper's main contribution is an explicit formula for the cohomology of line bundles on the incidence correspondence when n=3, obtained by solving the existing recurrence with a new family of polynomials. These even-carry polynomials generalize the Nim polynomials that handled the p=2 case, and they produce a direct expression in terms of the base-p digits of the degrees instead of requiring iteration of the recurrence. That step is the concrete advance here. The construction follows the recurrence relations directly and yields a usable closed form for any prime p. Readers who already accept the Liu-Gao-Raicu recurrence will find the formula immediately applicable for explicit calculations. The combinatorial definition appears straightforward and avoids introducing extra parameters or fitting. The soft spot is the dependence on the input recurrence. The paper does not re-derive or geometrically re-check that recurrence for general p, so any characteristic-specific failure in the earlier work would carry through. The conjectures for larger n are stated but left open, which is fine given the paper's stated scope. This work is aimed at specialists in positive-characteristic cohomology on flag varieties and incidence correspondences who need concrete dimensions rather than recursive descriptions. It is narrow but fills a specific computational gap inside an established program. The paper deserves a serious referee because the explicit formula is a verifiable step forward that others in the subfield can test or apply directly.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces even-carry polynomials, a generalization of Nim polynomials, to solve the recurrence of Liu--Gao--Raicu for the cohomology groups of line bundles on the incidence correspondence X ⊂ ℙ² × ℙ² (n=3 case) in arbitrary positive characteristic p. This yields an explicit formula for the cohomology in terms of the base-p digits of the bidegree (d,e). The paper also states conjectures on the form of the cohomology for general n.

Significance. If the underlying recurrence holds for all p, the explicit formula resolves the n=3 case completely, extending the known characteristic-zero results and the p=2 Nim-polynomial case. The combinatorial construction supplies a concrete, explicit solution rather than a recursive description, which is a clear advance for this class of problems in positive-characteristic algebraic geometry.

major comments (2)

[§1 (statement of the recurrence and main theorem)] The central claim is an explicit formula for the cohomology groups themselves (abstract and §1). This formula is obtained by solving the Liu--Gao--Raicu recurrence; the manuscript does not re-derive or independently verify that recurrence for arbitrary p. A load-bearing check would be to confirm that the even-carry formula reproduces known cohomology dimensions (e.g., via direct computation or the original geometric arguments) for at least one odd prime p and small bidegrees not covered by the p=2 case.
[§3] §3 (definition of even-carry polynomials): the claim that these polynomials solve the recurrence for general p requires an explicit inductive verification or closed-form identity showing that the even-carry expression satisfies every step of the Liu--Gao--Raicu relation; without such a displayed identity (or machine-checked check), the passage from the combinatorial object to the cohomology formula remains a derivation gap.

minor comments (2)

[§3] Notation for the even-carry polynomials (e.g., the precise dependence on the base-p digits) should be stated once in a single displayed equation before the main theorem, rather than introduced piecemeal.
[final section] The conjectures for general n (final section) would benefit from a short table comparing the conjectural formula against the known recurrence for small n and p.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where the exposition can be strengthened. We respond to each major comment below and will incorporate the suggested clarifications and verifications in a revised version.

read point-by-point responses

Referee: [§1 (statement of the recurrence and main theorem)] The central claim is an explicit formula for the cohomology groups themselves (abstract and §1). This formula is obtained by solving the Liu--Gao--Raicu recurrence; the manuscript does not re-derive or independently verify that recurrence for arbitrary p. A load-bearing check would be to confirm that the even-carry formula reproduces known cohomology dimensions (e.g., via direct computation or the original geometric arguments) for at least one odd prime p and small bidegrees not covered by the p=2 case.

Authors: The recurrence is a theorem of Liu--Gao--Raicu that we cite; our contribution is the explicit solution via even-carry polynomials. We agree that an independent consistency check for an odd prime strengthens the claim. In the revision we will add explicit dimension computations for p=3 and small bidegrees (outside the p=2 Nim case) and compare them with the even-carry formula. revision: yes
Referee: [§3] §3 (definition of even-carry polynomials): the claim that these polynomials solve the recurrence for general p requires an explicit inductive verification or closed-form identity showing that the even-carry expression satisfies every step of the Liu--Gao--Raicu relation; without such a displayed identity (or machine-checked check), the passage from the combinatorial object to the cohomology formula remains a derivation gap.

Authors: We accept that a displayed inductive verification or key identity would remove any ambiguity. In the revised manuscript we will insert a self-contained inductive argument in §3 showing that the even-carry polynomials satisfy the full Liu--Gao--Raicu recurrence for arbitrary p. revision: yes

Circularity Check

0 steps flagged

No circularity; explicit formula derived by solving externally cited recurrence

full rationale

The paper's central derivation takes the Liu--Gao--Raicu recurrence as an external input (cited from prior independent work) and solves it using newly defined even-carry polynomials for the n=3 case. No step reduces the output formula or polynomials to a fitted parameter, self-defined quantity, or self-citation chain by the paper's own equations. The recurrence itself is not re-derived here, so the solution method stands as independent content against the given recurrence. This matches the default expectation of no significant circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the correctness of the input recurrence and on the new combinatorial definition of even-carry polynomials; no numerical free parameters are mentioned.

axioms (1)

domain assumption The recurrence relation derived by Liu-Gao-Raicu accurately describes the cohomology groups for n=3 in every characteristic p.
The paper uses this recurrence as the starting point that the even-carry polynomials are designed to solve.

invented entities (1)

even-carry polynomials no independent evidence
purpose: To generalize Nim polynomials and convert the recurrence into an explicit formula for arbitrary prime p.
New combinatorial objects introduced by the paper; no independent evidence outside the paper is supplied in the abstract.

pith-pipeline@v0.9.0 · 5750 in / 1482 out tokens · 24626 ms · 2026-05-24T02:30:02.760349+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

Cohomology of Line Bundles on the Incidence Correspondence

Zhao Gao. Cohomology of Line Bundles on the Incidence Correspondence . ProQuest LLC, Ann Arbor, MI, 2022. Thesis (Ph.D.)–University of Notre Dame

work page 2022
[2]

Cohomology of line bundles on the incidence correspondence

Zhao Gao and Claudiu Raicu. Cohomology of line bundles on the incidence correspondence. Trans. Amer. Math. Soc. Ser. B, 11:64–97, 2024

work page 2024
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The Pelletier-Ressayre hidden symmetr y for Littlewood-Richardson coeﬃcients

Darij Grinberg. The Pelletier-Ressayre hidden symmetr y for Littlewood-Richardson coeﬃcients. Comb. Theory, 1:Paper No. 16, 44, 2021

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Han and P

C. Han and P. Monsky. Some surprising Hilbert-Kunz funct ions. Math. Z. , 214(1):119–135, 1993

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Lengths of local cohomology of thicken ings, 2019

Jennifer Kenkel. Lengths of local cohomology of thicken ings, 2019. arXiv:1912.02917

work page arXiv 2019
[6]

On the decomposition of tensor products of the representations of the classical groups: by means of t he universal characters

Kazuhiko Koike. On the decomposition of tensor products of the representations of the classical groups: by means of t he universal characters. Adv. Math. , 74(1):57–86, 1989

work page 1989
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Cohomologie des ﬁbr´ es en droites sur SL 3/B en caract´ eristique positive : deux ﬁltrations et cons´ equences

Linyuan Liu. Cohomologie des ﬁbr´ es en droites sur SL 3/B en caract´ eristique positive : deux ﬁltrations et cons´ equences. J. Eur. Math. Soc. , 2023. Published online ﬁrst

work page 2023
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Regularity and cohomology of determinan tal thickenings

Claudiu Raicu. Regularity and cohomology of determinan tal thickenings. Proc. Lond. Math. Soc. (3) , 116(2):248–280, 2018

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Modular Schur functions

Grant W alker. Modular Schur functions. Trans. Amer. Math. Soc. , 346(2):569–604, 1994. Evan M. O’Dorney, Department of Mathematical Sciences, Car negie Mellon University, Pittsburgh, PA

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[1] [1]

Cohomology of Line Bundles on the Incidence Correspondence

Zhao Gao. Cohomology of Line Bundles on the Incidence Correspondence . ProQuest LLC, Ann Arbor, MI, 2022. Thesis (Ph.D.)–University of Notre Dame

work page 2022

[2] [2]

Cohomology of line bundles on the incidence correspondence

Zhao Gao and Claudiu Raicu. Cohomology of line bundles on the incidence correspondence. Trans. Amer. Math. Soc. Ser. B, 11:64–97, 2024

work page 2024

[3] [3]

The Pelletier-Ressayre hidden symmetr y for Littlewood-Richardson coeﬃcients

Darij Grinberg. The Pelletier-Ressayre hidden symmetr y for Littlewood-Richardson coeﬃcients. Comb. Theory, 1:Paper No. 16, 44, 2021

work page 2021

[4] [4]

Han and P

C. Han and P. Monsky. Some surprising Hilbert-Kunz funct ions. Math. Z. , 214(1):119–135, 1993

work page 1993

[5] [5]

Lengths of local cohomology of thicken ings, 2019

Jennifer Kenkel. Lengths of local cohomology of thicken ings, 2019. arXiv:1912.02917

work page arXiv 2019

[6] [6]

On the decomposition of tensor products of the representations of the classical groups: by means of t he universal characters

Kazuhiko Koike. On the decomposition of tensor products of the representations of the classical groups: by means of t he universal characters. Adv. Math. , 74(1):57–86, 1989

work page 1989

[7] [7]

Cohomologie des ﬁbr´ es en droites sur SL 3/B en caract´ eristique positive : deux ﬁltrations et cons´ equences

Linyuan Liu. Cohomologie des ﬁbr´ es en droites sur SL 3/B en caract´ eristique positive : deux ﬁltrations et cons´ equences. J. Eur. Math. Soc. , 2023. Published online ﬁrst

work page 2023

[8] [8]

Regularity and cohomology of determinan tal thickenings

Claudiu Raicu. Regularity and cohomology of determinan tal thickenings. Proc. Lond. Math. Soc. (3) , 116(2):248–280, 2018

work page 2018

[9] [9]

Modular Schur functions

Grant W alker. Modular Schur functions. Trans. Amer. Math. Soc. , 346(2):569–604, 1994. Evan M. O’Dorney, Department of Mathematical Sciences, Car negie Mellon University, Pittsburgh, PA

work page 1994