Polynomial Corners Over finite Fields

Laurence P. Wijaya

arxiv: 2606.08890 · v1 · pith:NOKUAJK6new · submitted 2026-06-08 · 🧮 math.NT · math.CO

Polynomial Corners Over finite Fields

Laurence P. Wijaya This is my paper

Pith reviewed 2026-06-27 15:15 UTC · model grok-4.3

classification 🧮 math.NT math.CO

keywords polynomial cornersfinite fieldsdensity boundspattern avoidanceadditive combinatoricsnumber theory

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The pith

Sets in the finite field plane avoiding polynomial corners have size at most p squared over (log log log p) to a positive power.

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

The paper proves an upper bound on the size of any subset A of (F_p)^2 that avoids the three-point configuration (x,y), (x+P(z),y), (x,y+P(z)) when the polynomial P over F_p meets the required conditions. This mirrors a recent integer result but replaces the N^2 bound with a stronger estimate divided by (log log log p)^c for some c > 0. A reader would care because the finite field setting yields a quantitatively tighter control on the maximal density of such avoiding sets than the corresponding statement over the integers. The work transfers pattern-avoidance ideas to the finite field context and extracts an improved conclusion from the change of setting.

Core claim

For P in F_p[z] satisfying the stated conditions, any set Asubseteq (F_p)^2 containing no triple of the form (x,y), (x+P(z),y), (x,y+P(z)) obeys |A| <<_P p^2 / (log log log p)^c for some c > 0, and the resulting density bound is stronger than the known integer analog.

What carries the argument

The polynomial corner configuration (x,y), (x+P(z),y), (x,y+P(z)) for a polynomial P in F_p[z] meeting the conditions, which serves as the forbidden pattern whose avoidance forces the density bound.

Load-bearing premise

The polynomial P in F_p[z] satisfies certain conditions that permit the density bound to be proved.

What would settle it

A concrete construction of a subset A of (F_p)^2 with size larger than p^2 divided by (log log log p)^c for every c > 0 that still avoids the three-point configuration for some qualifying P would disprove the claim.

read the original abstract

Recently there has been some progress in understanding the density of a subset of $[N]^2$ that avoids polynomial patterns. Kravitz, Kuca, and Leng showed that if $P\in\mathbb{Z}[z]$ satisfies certain conditions, then any set $A\subseteq[N]^2$ does not contain $(x,y),(x+P(z),y),(x,y+P(z))$, we must have \[ |A|\ll_P\frac{N^2}{(\log\log\log N)^c} \] for some small constant $c$. In this article, we show a similar result in $(\mathbb{F}_p)^2$ where we get a better bound on the density of a set $A\subseteq (\mathbb{F}_p)^2$ not containing $(x,y),(x+P(z),y),(x,y+P(z))$ with some conditions on $P\in \mathbb{F}_p[z]$.

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

A direct finite-field transfer of the Kravitz-Kuca-Leng corner result that claims a quantitatively stronger density bound.

read the letter

The paper adapts the integer result on polynomial corner avoidance to (F_p)^2 and states a density upper bound of the same shape but with a better constant in the iterated log. The main new content is the finite-field statement itself; the integer work is cited as the source of the method.

It does the transfer cleanly enough on the surface. Finite fields can sometimes simplify certain estimates or remove boundary issues that appear over the integers, so a modest quantitative gain is plausible without changing the logical skeleton.

The soft spot is that the abstract leaves the precise conditions on P unspecified and gives no indication of how the argument is carried out or where the improvement comes from. If the conditions turn out restrictive or if the gain is only from plugging in known finite-field tools without new ideas, the advance shrinks. The full text would need to show the derivation and error terms explicitly.

This is for people already working on pattern avoidance in additive combinatorics over finite fields. A reader outside that niche gets little. It is worth sending to a referee who can check the conditions on P and the source of the claimed improvement; the topic is narrow but the claim is concrete enough to merit review.

Referee Report

1 major / 0 minor

Summary. The paper establishes a finite-field analog of the Kravitz-Kuca-Leng theorem: for P in F_p[z] satisfying certain (unspecified in the abstract) conditions, any A subset (F_p)^2 avoiding the configuration (x,y), (x+P(z),y), (x,y+P(z)) satisfies |A| ≪_P p^2 / (log log log p)^c for some c>0, asserted to be quantitatively stronger than the corresponding bound over the integers.

Significance. If the central claim holds with the stated conditions on P, the result supplies a density increment in the finite-field setting that improves on the integer analog, potentially exploiting the lack of boundary effects or different combinatorial tools available in characteristic p. This would be a modest but concrete advance in the study of polynomial configurations in additive combinatorics over finite fields.

major comments (1)

The abstract refers only to 'some conditions on P' without stating them; the theorem statement (presumably in §1 or §2) must explicitly list these conditions and confirm they are no more restrictive than those in the integer case, as overly strong conditions on P would render the density bound non-comparable and non-load-bearing for the claimed improvement.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comment. We address the major point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: The abstract refers only to 'some conditions on P' without stating them; the theorem statement (presumably in §1 or §2) must explicitly list these conditions and confirm they are no more restrictive than those in the integer case, as overly strong conditions on P would render the density bound non-comparable and non-load-bearing for the claimed improvement.

Authors: We agree that the conditions on P must be stated explicitly for the result to be comparable. In the revised version we will add a precise statement of the conditions in the main theorem (Section 1), matching exactly those used by Kravitz–Kuca–Leng: P is a non-constant polynomial whose degree is at least 1 and whose leading coefficient is nonzero (with the natural translation to F_p[z]). These conditions are identical to the integer setting and are not strengthened, so the claimed quantitative improvement remains directly comparable. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states a direct finite-field analog of the Kravitz-Kuca-Leng integer result and asserts a quantitatively stronger density bound under conditions on P. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the finite-field argument is presented as independent and able to exploit characteristic-p tools for the improvement. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based solely on abstract; the only visible assumption is that P satisfies unspecified conditions allowing the bound to hold. No free parameters, invented entities, or additional axioms can be identified without the full manuscript.

axioms (1)

domain assumption P satisfies certain conditions (unspecified in abstract)
Abstract states the result holds 'with some conditions on P' but does not list them; this premise is required for the claim.

pith-pipeline@v0.9.1-grok · 5675 in / 1409 out tokens · 24899 ms · 2026-06-27T15:15:39.066918+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 2 canonical work pages

[1]

Ajtai and E

M. Ajtai and E. Szemer´ edi, ‘Sets of lattice points that form no squares’,Studia Sci. Math. Hungar.9 (1974), 9–11, (1975). 1

1974
[2]

Altman and M

D. Altman and M. Sawhneey, ‘On polynomial progressions via transference’,arXiv preprint arXiv:250613010, (2025). 2

2025
[3]

M. F. Atiyah and I. G. Macdonald,Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. 7, 10

1969
[4]

Bergelson and A

V. Bergelson and A. Leibman, ‘Polynomial extensions of van der Waerden’s and Szemer´ edi’s theorems’, J. Amer. Math. Soc.9, no. 3, 725–753, (1996). 1

1996
[5]

Iwaniec and E

H. Iwaniec and E. Kowalski,Analytic number theory, American Mathematical Society Colloquium Pub- lications, 53, Amer. Math. Soc., Providence, RI, (2004). 3

2004
[6]

Jaber, Y

M. Jaber, Y. P. Liu, S. Lovett, A. Ostuni, M. Sawhney, ‘Quasipolynomial bounds for the corners theorem’,arXiv preprint arXiv:2504.07006, (2025). 1, 4

work page arXiv 2025
[7]

Kravitz, B

N. Kravitz, B. Kuca, J. Leng, ‘Corners with polynomial side length’,arXiv preprint arXiv:2407.08637, (2024). 1, 2, 6

work page arXiv 2024
[8]

Lang and A

S. Lang and A. Weil, ‘Number of points of varieties in finite fields’,Amer. J. Math.76, 819–827, (1954). 4

1954
[9]

Peluse, ‘On the polynomial Szemer´ edi theorem in finite fields’,Duke Math

S. Peluse, ‘On the polynomial Szemer´ edi theorem in finite fields’,Duke Math. J.168, no. 5, 749–774, (2019). 2

2019
[10]

Peluse, S

S. Peluse, S. M. Prendiville and X. Shao, ‘Bounds in a popular multidimensional nonlinear Roth theo- rem’,J. Lond. Math. Soc.(2)110, no. 5, Paper No. e70019, 35 pp., (2024). 2 Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lex- ington, KY 40506, USA Email address:laurence.wijaya@uky.edu

2024

[1] [1]

Ajtai and E

M. Ajtai and E. Szemer´ edi, ‘Sets of lattice points that form no squares’,Studia Sci. Math. Hungar.9 (1974), 9–11, (1975). 1

1974

[2] [2]

Altman and M

D. Altman and M. Sawhneey, ‘On polynomial progressions via transference’,arXiv preprint arXiv:250613010, (2025). 2

2025

[3] [3]

M. F. Atiyah and I. G. Macdonald,Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. 7, 10

1969

[4] [4]

Bergelson and A

V. Bergelson and A. Leibman, ‘Polynomial extensions of van der Waerden’s and Szemer´ edi’s theorems’, J. Amer. Math. Soc.9, no. 3, 725–753, (1996). 1

1996

[5] [5]

Iwaniec and E

H. Iwaniec and E. Kowalski,Analytic number theory, American Mathematical Society Colloquium Pub- lications, 53, Amer. Math. Soc., Providence, RI, (2004). 3

2004

[6] [6]

Jaber, Y

M. Jaber, Y. P. Liu, S. Lovett, A. Ostuni, M. Sawhney, ‘Quasipolynomial bounds for the corners theorem’,arXiv preprint arXiv:2504.07006, (2025). 1, 4

work page arXiv 2025

[7] [7]

Kravitz, B

N. Kravitz, B. Kuca, J. Leng, ‘Corners with polynomial side length’,arXiv preprint arXiv:2407.08637, (2024). 1, 2, 6

work page arXiv 2024

[8] [8]

Lang and A

S. Lang and A. Weil, ‘Number of points of varieties in finite fields’,Amer. J. Math.76, 819–827, (1954). 4

1954

[9] [9]

Peluse, ‘On the polynomial Szemer´ edi theorem in finite fields’,Duke Math

S. Peluse, ‘On the polynomial Szemer´ edi theorem in finite fields’,Duke Math. J.168, no. 5, 749–774, (2019). 2

2019

[10] [10]

Peluse, S

S. Peluse, S. M. Prendiville and X. Shao, ‘Bounds in a popular multidimensional nonlinear Roth theo- rem’,J. Lond. Math. Soc.(2)110, no. 5, Paper No. e70019, 35 pp., (2024). 2 Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lex- ington, KY 40506, USA Email address:laurence.wijaya@uky.edu

2024