Uniform nonlinear Szemer\'{e}di theorem for corners in finite fields

Zi Li Lim

REVIEW 3 major objections 4 minor 1 cited by

Uniform nonlinear Szemer\'{e}di theorem for corners in finite fields

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Corners in $\mathbb{F}_p^2$ with rational side lengths have the random set's predicted count up to $O_{P,Q}(p^{-1/40960})$.

desk verdict The rational-function corner asymptotic with degree-uniform power saving is genuinely new, but the proof's load-bearing point count in Theorem 4.2 is not established as written because of a dimension contradiction in Step 1. read the letter →

arxiv 2501.04887 v1 pith:DYJMALUV submitted 2025-01-08 math.NT

Zi Li Lim This is my paper · ORCID

classification math.NT MSC 11B3011T2314G05

keywords polynomialSzemeréditheoremcornersinfinitefieldsrationalfunctionprogressionsGowersnormsdegreeloweringalgebraicgeometryPETinductionRothvarietypointcountingover

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

This paper proves an asymptotic formula for corners of the form $(x_1,x_2)$, $(x_1+P(y),x_2)$, $(x_1,x_2+Q(y))$ in $\mathbb{F}_p^2$, where $P(t),Q(t)\in\mathbb{Q}(t)$ are rational functions and $P,Q,1$ are linearly independent over $\mathbb{Q}$. The claimed error term is $O_{P,Q}(p^{-1/40960})$, and the author notes the implied constant can be taken to depend only on the degrees of $P$ and $Q$, so the power saving is uniform as the degrees grow. The proof obtains Gowers-norm control for the corner average, a point-counting bound on a sixteen-variable Roth variety cut out by ten equations, and a degree-lowering induction. A corollary is a density threshold: any subset of $\mathbb{F}_p^2$ with density $\delta\gg p^{-1/122880}$ contains plenty of corners generated by $P$ and $Q$.

What carries the argument

The load-bearing object is the Roth variety for corners: the affine variety in sixteen variables cut out by the ten equations that set to zero the alternating $P$- and $Q$-sums appearing in the corner-counting Fourier expansion. The key identity is Theorem 3.2, which bounds the corner average by a product of $\|f_0\|_2\|f_1\|_4\|f_2\|_4^{1/2}\|f_2\|_{U^2(0\times\mathbb{F}_p)}^{1/4}$ with the factor $(|Y(\mathbb{F}_p)|/p^6)^{1/16}$. Theorem 4.2 supplies the point count, and the remaining mechanism is an algebraic-geometry version of PET induction that replaces Weyl differencing with these point counts, augmented by an extra Cauchy–Schwarz iteration that keeps the auxiliary variety non-degenerate and a directional degree-lowering induction for the Gowers norms.

What would settle it

Fix $P(t)=t$ and $Q(t)=t^2$, and for several primes $p$ compute $|Y(\mathbb{F}_p)|$ for the ten-equation Roth variety by eliminating variables from the defining equations and counting solutions; if along a sequence of primes the count exceeds $C p^6$ with $C\to\infty$, Theorem 4.2 fails and the factor $(|Y(\mathbb{F}_p)|/p^6)^{1/16}$ in the Gowers control theorem would not be bounded.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for every pair of rational functions $P(t),Q(t)\in\mathbb{Q}(t)$ with $P,Q,1$ linearly independent over $\mathbb{Q}$, and every $1$-bounded $f_0,f_1,f_2:\mathbb{F}_p^2\to\mathbb{C}$, the corner average $\mathbb{E}_{x_1,x_2,y}f_0(x_1,x_2)f_1(x_1+P(y),x_2)f_2(x_1,x_2+Q(y))$ equals the factorized average $\mathbb{E}_{x_1,x_2}(f_0(x_1,x_2)\mathbb{E}_a f_1(a,x_2)\mathbb{E}_b f_2(x_1,b))$ plus an error of size $O_{P,Q}(p^{-1/40960})$. The proof first reduces the corner average to a directional Gowers norm of one function multiplied by the point-counting factor $(|Y(\mathbb{F}_p)|/p^6)^{1/16}$, where $Y$ is the Roth variety for corners; it then proves $|Y(\mathbb{F}_p)|\ll p^6$; and it finally runs a directional degree-lowering induction that eliminates the Gowers norm and leaves the factorized main term. This yields the corollary that subsets of density $\delta\gg p^{-1/122880}$ contain $\gg p^3\delta^3$ such corners.

Load-bearing premise

The whole argument stands on the claim that the solution set of the ten corner equations has at most on the order of $p^6$ points over $\mathbb{F}_p$; Step 1 of the proof of that claim calls the generic part six-dimensional and then three-dimensional, so this dimension comparison is the fragile premise.

Editorial extensions

If this is right

For any admissible $P,Q$, every subset of $\mathbb{F}_p^2$ of density $\delta\gg p^{-1/122880}$ contains $\gg p^3\delta^3$ corners generated by $P,Q$.
The implied constant in the main theorem depends only on the degrees of $P$ and $Q$, so the $p^{-1/40960}$ saving remains uniform as the degrees grow.
The averages and the asymptotic remain meaningful when $P$ or $Q$ has poles, because the expectation excludes the pole values of $y$.
As part of the induction, the paper establishes an $O(p^{-1/2})$ asymptotic for two-term rational progressions in each coordinate direction.

Reading between the lines

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

A direct computation of the Roth-variety dimension for $P(t)=t$ and $Q(t)=t^2$ would settle whether Step 1's dimension comparison is a typo or a gap; if the generic part actually has dimension 3, the point count would be stronger but the written proof would need correction.
The same Cauchy–Schwarz-plus-point-count machinery appears transferable to corners generated by three or more rational side functions, with a smaller power saving from the longer induction; the paper does not claim this.
Because the saving is uniform in degree, the density threshold may extend to sequences of rational functions whose degrees grow with $p$, provided the linear-independence condition continues to hold; this is an extrapolation.
The exponent $1/40960$ is explicitly not optimized, so a tighter bookkeeping of the degree-lowering constants could improve it without changing the structure of the proof.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

New result: this paper proves an asymptotic for corners (x1,x2), (x1+P(t),x2), (x1,x2+Q(t)) in F_p^2 for rational P,Q with P,Q,1 linearly independent, with error O(p^{-1/40960}) that does not depend on the degrees. That is a real extension of the polynomial corner theorems of Han–Lacey–Yang and Kuca; rational functions plus a uniform power saving is new.

What is done well: the algebraic-geometry PET induction in Section 3 is careful and explicit. The Fourier–Cauchy–Schwarz chain is written out, the Roth variety is defined cleanly, and the Jacobian determinant computation in Proposition 4.1 is checkable and correct. The debts to [HL24], [HLY21], [KW24], and [Kuc24a] are honestly acknowledged, and the assembly is not a routine restatement.

The soft spot is exactly where the stress-test points. In Theorem 4.2, Step 1 says Ygen(C) is a 6-dimensional complex manifold, then invokes Noether normalization and a theorem from [GR84] to deduce that the dimension of Ygen(C) as a variety is 3, and later uses 6 again. A finite surjective map to A^d preserves dimension; if the complex manifold is 6-dimensional, the variety dimension must be 6. As written, the dimension comparison is inconsistent. Since Steps 4, 5, 7, and 8 all rely on “similar arguments to Step 1,” the bound |Y(F_p)| << p^6 is not actually proved. This is load-bearing: Theorem 3.2 has the factor (|Y|/p^6)^{1/16}, and if the true geometric dimension were larger the Gowers norm control would fail. The flaw looks fixable, but it is central.

Two smaller concerns. The base case Proposition 5.3 invokes [HL24], an unpublished preprint coauthored by this author, so the one-dimensional result is a black box. Also, the degree-lowering induction in Section 5 is compressed; the sentence about eliminating the O(p^{-1/2}) error when p^{-1/2} ≫ δ^64 looks odd, because that would only give δ ≲ p^{-1/128}, not the claimed p^{-1/640}. These are secondary compared with the Section 4 gap.

Who is this for: people working in quantitative additive combinatorics over finite fields. It deserves a serious referee: the result is plausible, novel, and the framework is worth engaging with. But the paper should not be accepted until Step 1 of Theorem 4.2 is repaired or replaced.

Recommendation: send it to peer review with a major-revision request, and make the referee focus on a clean dimension proof for the Roth variety.

Referee Report

3 major / 4 minor

Summary. The paper proves, for rational functions P(t), Q(t) in Q(t) with P, Q, and 1 linearly independent over Q, an asymptotic formula for the corner-counting average over F_p^2, with error O_{P,Q}(p^{-1/40960}) for all 1-bounded functions. The proof has three main parts: an algebraic-geometry PET induction (Section 3) that reduces the corner average to a directional Gowers norm, subject to a point-counting estimate for a 16-variable 'Roth variety'; a dimension-theoretic proof of that point-counting estimate (Section 4); and a degree-lowering induction in the style of Peluse and Kuca (Section 5). The claimed result also yields a density-increment-free lower bound for subsets avoiding such corners.

Significance. If the main theorem is correct, this is a significant advance: it is the first quantitative nonlinear Szemerédi theorem for corners with rational-function side lengths in finite fields, and the power-saving exponent is uniform in the degrees of P and Q. The paper combines Fourier analysis, algebraic geometry, and degree-lowering in a genuinely novel way, and the explicit Roth variety for corners with the Kavrut--Wu amplification is a useful construction. The proof is not machine-checked, but the algebraic manipulations in Propositions 3.3, 3.5, and 3.6 are largely explicit. The main caveat is that the paper depends on the unpublished preprint [HL24] for the one-dimensional base case and for the complex-analytic dimension method, and the dimension arguments in Section 4 contain a serious internal inconsistency.

major comments (3)

[§4, Theorem 4.2, Step 1] Step 1 first asserts that the associated complex analytic space of Ygen(C) is a 6-dimensional complex manifold, then invokes Noether normalization and [GR84] to 'deduce that the dimension of Ygen(C) as a variety is 3', and later uses the value 6 in the base-change and Lang--Weil argument. These statements are mutually incompatible: a finite surjective morphism preserves dimension, so if Ygen(C) is a 6-dimensional complex manifold, its algebraic dimension is 6, not 3. Since Steps 4, 5, 7, and 8 all justify their point-counting bounds by 'similar arguments to Step 1', the proof of the load-bearing estimate |Y(F_p)| << p^6 — which controls the factor (|Y|/p^6)^{1/16} in Theorems 3.2 and 3.7 — is not established as written. This is a fundamental gap in the point-counting argument, not a typo confined to one line, and it must be repaired before the main theorem can be accepted.
[§5, Proof of Theorem 5.4.(2)] The sentence 'the desired estimate in Theorem 5.4.(2) already holds when p^{-1/2} \gg δ^{64}' has the inequality in the wrong direction as stated. From δ^{64} \ll p^{-1/2} one obtains δ \ll p^{-1/128}, which is much larger than the claimed p^{-1/640} for large p, so it does not imply the desired estimate. The intended condition should be p^{-1/2} \ll δ^{64}, i.e. the error term is negligible compared with the main-term lower bound. The same directional issue propagates to the sketch of Theorem 5.4.(3), where the final exponent is derived. The induction needs to be rewritten with the correct comparison and with all exponents tracked explicitly.
[§4, Theorem 4.2, Step 3] The proof that D(y1,y4,y6) is a nonzero rational function is presented as an informal pole/zero cancellation argument. As written, 'poles and zeroes exactly the same' does not by itself imply P'/Q' is constant unless multiplicities are part of the statement, and the claims that poles of S(y1) cannot be cancelled need a precise Laurent-expansion or Wronskian argument. This step is load-bearing because Step 4's estimate |Z'_sp(F_p)| << p^5 relies on D being a nonzero rational function, so the argument should be formalized.

minor comments (4)

[§4, Theorem 4.2, Step 1] The displayed identity 'dim Sch Ygen ×Z Fp = dim Sch Ygen ×Z Fp' is tautological; presumably one side should refer to the algebraic closure of F_p or to a geometric fiber over F_p rather than the scheme over F_p itself.
[§2.4 and §4] The notation for dimensions is inconsistent: the paper uses 'dimVar', 'dim Sch', and 'dimension as a variety' without clearly defining all three for possibly non-irreducible varieties, and Step 1 switches between them without explanation.
[§1.1] There is a typo in 'algebraic geomerty' in the introduction; it should read 'algebraic geometry'.
[§2.1] The pole-exclusion convention in the expectation notation is defined in Section 2, but Theorem 1.1 states the average without explicitly reminding the reader that y is restricted away from the poles of P and Q; adding a short parenthetical in the theorem statement would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction: the corner asymptotic is derived compositionally; the main risk is a soundness gap in Theorem 4.2 Step 1 (a 6-vs-3 dimension contradiction), not circularity.

full rationale

No circular step is present. The corner asymptotic in Theorem 1.1 is not defined in terms of itself, no parameter is fitted and later renamed as a prediction, and the Gowers-norm control in Theorem 3.2 is obtained by explicit Cauchy-Schwarz and orthogonality manipulations that terminate in the exact identity (22) equaling p^{-6}|Y(F_p)|. The degree-lowering induction in Section 5 is a genuine induction whose base case (Proposition 5.3) and one-dimensional point-counting method (Step 1 of Theorem 4.2) are imported from [HL24], a preprint coauthored by the present author. This is a load-bearing self-citation, but it is not circular: [HL24] concerns one-dimensional rational-function progressions, not the two-dimensional corner average, so it is independent support rather than an input equivalent to the target result. The genuine problem is correctness, not circularity. Step 1 of Theorem 4.2 first states that 'The associated complex analytic space of Ygen(C) is a 6-dimensional complex manifold', then says 'we deduce that the dimension of Ygen(C) as a variety is 3', and later asserts 'dim Var Ygen(C) = 6'. The point count |Y(F_p)| << p^6 is load-bearing for the factor (|Y|/p^6)^{1/16} in Theorem 3.2, and Steps 4, 5, 7, and 8 all rely on 'similar arguments to Step 1', so this internal contradiction must be resolved before the proof of Theorem 4.2 is valid. That deficiency is a soundness gap, not a circularity, and therefore does not raise the circularity score.

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

The paper introduces no fitted constants. It relies on standard algebraic geometry and analytic number theory tools, plus the author's earlier preprint for the one-dimensional base case. The Roth variety is a technical construction rather than a new empirical entity, so it is not listed as an invented entity.

assumptions (5)

domain assumption P(t), Q(t), and the constant function 1 are linearly independent over Q
Stated in Theorem 1.1 and used in Section 4 Step 3 to ensure P'/Q' is nonconstant, in Bombieri bounds to ensure phases are nonconstant, and in Section 4 Step 4 to ensure the relevant Jacobian determinants are nonzero.
standard math Lang-Weil bound for varieties over finite fields
Used in Section 4 Steps 1, 4, and 8 to translate dimension estimates into point-counting bounds |X(F_p)| << p^d.
standard math Bombieri's exponential sum bound for rational phases over finite fields
Used in Theorem 5.4(1) to obtain O(p^{-1/2}) cancellation for nonconstant rational phases.
standard math Dimension facts from Hartshorne, Grauert-Remmert, and the Stacks Project
Used in Section 4 Step 1 to compare complex analytic, scheme, and mod p dimensions. This is the argument containing the 3 versus 6 contradiction.
domain assumption [HL24, Proposition 5.1], a one-dimensional rational-progression asymptotic
Imported from the authors' earlier preprint to prove the two-term base case Proposition 5.3; the present paper does not reprove it.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Uniform nonlinear Szemer\'{e}di theorem for corners in finite fields." pith.science (2026). https://pith.science/paper/DYJMALUV

@misc{pith2026250104887,
  author       = {Pith},
  title        = {Pith review of: Uniform nonlinear Szemer\'edi theorem for corners in finite fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DYJMALUV}},
  note         = {Machine review of arXiv:2501.04887}
}

abstract

Let $P(t),Q(t)\in \mathbb{Q}(t)$ be rational functions such that $P(t),Q(t)$ and the constant function $1$ are linearly independent over $\mathbb{Q}$, we prove an asymptotic formula for the number of the corner configurations $(x_1,x_2),(x_1+P(y),x_2),(x_1,x_2+Q(y))$ in the subsets of $\mathbb{F}_p^2$.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

On weighted multilinear polynomial averages in finite fields
math.NT 2025-07 conditional novelty 6.0 of 10

Weighted multilinear polynomial averages in finite fields are controlled by the u^(d+1)-norm of the weight, yielding quantitative convergence and new asymptotic formulas for multidimensional rational function progress...

Reference graph

Works this paper leans on

9 extracted references · 7 canonical work pages · cited by 1 Pith paper

[5]

Multidimensional polynomial patter ns over ﬁnite ﬁelds: bounds, counting estimates and Gowers norm control

arXiv: 2407.08637. [Kuc24a] Borys Kuca. “Multidimensional polynomial patter ns over ﬁnite ﬁelds: bounds, counting estimates and Gowers norm control”. In: Adv. Math. 448 (2024), Paper No. 109700,

arXiv 2024
[7]

Number of points of varie ties in ﬁnite ﬁelds

arXiv: 2401.07925. [L W54] Serge Lang and Andr´ e Weil. “Number of points of varie ties in ﬁnite ﬁelds”. In: Amer. J. Math. 76 (1954), pp. 819–827. [Pel18] Sarah Peluse. “Three-term polynomial progression s in subsets of ﬁnite ﬁelds”. In: Israel J. Math. 228.1 (2018), pp. 379–405. 30 REFERENCES [Pel19] Sarah Peluse. “On the polynomial Szemer´ edi theore m...

arXiv 1954
[9]

Bounds in a popular multidimensional nonlinear Roth theorem

arXiv: 2407.08338. [Sta] The Stacks project authors. The Stacks project. https://stacks.math.columbia.edu. [Tao] Terence Tao. The Lang-Weil Bound. https://terrytao.wordpress.com/2012/08/31/the-lang Department of Mathematics, UCLA, Los Angeles, CA 90095, USA Email address : zililim@math.ucla.edu

work page Pith review arXiv 2012
[52]

Springer-Verlag, New York-Heidelberg, 1977, pp

Graduate Texts in Mathe- matics. Springer-Verlag, New York-Heidelberg, 1977, pp. x vi+496. [HL24] Guo-Dong Hong and Zi Li Lim. Three term rational function progressions in ﬁnite ﬁelds

work page 1977
[55]

Subsets of Fn p ×Fn p without L-shaped conﬁgurations

[Pel24] Sarah Peluse. “Subsets of Fn p ×Fn p without L-shaped conﬁgurations”. In: Compos. Math. 160.1 (2024), pp. 176–236. [PP24] S. Peluse and S. Prendiville. “Quantitative bounds i n the nonlinear Roth theo- rem”. In: Invent. math. 238 (2024), pp. 865–903. [PPS24] Sarah Peluse, Sean Prendiville, and Xuancheng Shao . Bounds in a popular mul- tidimensiona...

work page 2024
[61]

Multidimensional polynomial Szemer ´ edi theorem in ﬁnite ﬁelds for polynomials of distinct degrees

[Kuc24b] Borys Kuca. “Multidimensional polynomial Szemer ´ edi theorem in ﬁnite ﬁelds for polynomials of distinct degrees”. In: Israel J. Math. 259.2 (2024), pp. 589–620. [KW24] Necef Kavrut and Shukun Wu. A bilinear estimate in Fp

work page 2024
[265]

Springer-Verlag, Berlin, 1984, p p

Grundlehren der mathematischen Wissenschaften [Fundamen tal Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1984, p p. xviii+249. [Har77] Robin Hartshorne. Algebraic geometry. Vol. No

work page 1984
[2014]

Partial difference equations over compact Abelian groups, I: modules of solutions

arXiv: 1305.7269. [BC17] J. Bourgain and M.-C. Chang. “Nonlinear Roth type the orems in ﬁnite ﬁelds”. In: Israel J. Math. 221.2 (2017), pp. 853–867. [BL96] V. Bergelson and A. Leibman. “Polynomial extensions of van der Waerden’s and Szemer´ edi’s theorems”. In:J. Amer. Math. Soc. 9.3 (1996), pp. 725–753. [Bom66] Enrico Bombieri. “On exponential sums in ﬁn...

work page Pith review arXiv 2017

Show all 9 references

[2024]

A polynomia l Roth theorem for corners in ﬁnite ﬁelds

arXiv: 2401.01137. [HLY21] Rui Han, Michael T. Lacey, and Fan Yang. “A polynomia l Roth theorem for corners in ﬁnite ﬁelds”. In: Mathematika 67.4 (2021), pp. 885–896. [KKL24] Noah Kravitz, Borys Kuca, and James Leng. Corners with polynomial side length

2021 arXiv

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [5]

Multidimensional polynomial patter ns over ﬁnite ﬁelds: bounds, counting estimates and Gowers norm control

arXiv: 2407.08637. [Kuc24a] Borys Kuca. “Multidimensional polynomial patter ns over ﬁnite ﬁelds: bounds, counting estimates and Gowers norm control”. In: Adv. Math. 448 (2024), Paper No. 109700,

arXiv 2024

[2] [7]

Number of points of varie ties in ﬁnite ﬁelds

arXiv: 2401.07925. [L W54] Serge Lang and Andr´ e Weil. “Number of points of varie ties in ﬁnite ﬁelds”. In: Amer. J. Math. 76 (1954), pp. 819–827. [Pel18] Sarah Peluse. “Three-term polynomial progression s in subsets of ﬁnite ﬁelds”. In: Israel J. Math. 228.1 (2018), pp. 379–405. 30 REFERENCES [Pel19] Sarah Peluse. “On the polynomial Szemer´ edi theore m...

arXiv 1954

[3] [9]

Bounds in a popular multidimensional nonlinear Roth theorem

arXiv: 2407.08338. [Sta] The Stacks project authors. The Stacks project. https://stacks.math.columbia.edu. [Tao] Terence Tao. The Lang-Weil Bound. https://terrytao.wordpress.com/2012/08/31/the-lang Department of Mathematics, UCLA, Los Angeles, CA 90095, USA Email address : zililim@math.ucla.edu

work page Pith review arXiv 2012

[4] [52]

Springer-Verlag, New York-Heidelberg, 1977, pp

Graduate Texts in Mathe- matics. Springer-Verlag, New York-Heidelberg, 1977, pp. x vi+496. [HL24] Guo-Dong Hong and Zi Li Lim. Three term rational function progressions in ﬁnite ﬁelds

work page 1977

[5] [55]

Subsets of Fn p ×Fn p without L-shaped conﬁgurations

[Pel24] Sarah Peluse. “Subsets of Fn p ×Fn p without L-shaped conﬁgurations”. In: Compos. Math. 160.1 (2024), pp. 176–236. [PP24] S. Peluse and S. Prendiville. “Quantitative bounds i n the nonlinear Roth theo- rem”. In: Invent. math. 238 (2024), pp. 865–903. [PPS24] Sarah Peluse, Sean Prendiville, and Xuancheng Shao . Bounds in a popular mul- tidimensiona...

work page 2024

[6] [61]

Multidimensional polynomial Szemer ´ edi theorem in ﬁnite ﬁelds for polynomials of distinct degrees

[Kuc24b] Borys Kuca. “Multidimensional polynomial Szemer ´ edi theorem in ﬁnite ﬁelds for polynomials of distinct degrees”. In: Israel J. Math. 259.2 (2024), pp. 589–620. [KW24] Necef Kavrut and Shukun Wu. A bilinear estimate in Fp

work page 2024

[7] [265]

Springer-Verlag, Berlin, 1984, p p

Grundlehren der mathematischen Wissenschaften [Fundamen tal Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1984, p p. xviii+249. [Har77] Robin Hartshorne. Algebraic geometry. Vol. No

work page 1984

[8] [2014]

Partial difference equations over compact Abelian groups, I: modules of solutions

arXiv: 1305.7269. [BC17] J. Bourgain and M.-C. Chang. “Nonlinear Roth type the orems in ﬁnite ﬁelds”. In: Israel J. Math. 221.2 (2017), pp. 853–867. [BL96] V. Bergelson and A. Leibman. “Polynomial extensions of van der Waerden’s and Szemer´ edi’s theorems”. In:J. Amer. Math. Soc. 9.3 (1996), pp. 725–753. [Bom66] Enrico Bombieri. “On exponential sums in ﬁn...

work page Pith review arXiv 2017