{"id":"074cc09f-1f1c-42aa-b943-e3a95793b97d","arxiv_id":"2501.04887","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Corners generated by two independent rational functions in F_p^2 have the expected asymptotic count with uniform power saving p^{-1/40960}.","lead":"New theorem counts corner configurations (x1,x2), (x1+P(y),x2), (x1,x2+Q(y)) in finite field planes when P and Q are rational functions, with explicit error p^{-1/40960}. It generalizes polynomial Szemeredi corner estimates to rational functions and gives a density threshold that does not decay with the degrees of P and Q.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.2's point count |Y(F_p)| << p^6 is load-bearing; Step 1's dimension contradiction (6 vs 3) must be resolved before the Gowers norm control in Theorem 3.2 is justified.","rationale":"The reader's weakest_assumption identifies Theorem 4.2 as the load-bearing point-counting step, and I agree. The Gowers norm control in Theorem 3.2 is only nontrivial if |Y(F_p)| << p^6; the entire degree-lowering argument in Section 5 inherits this factor. Step 1 of the proof contains an explicit dimension contradiction (6-dimensional complex manifold versus dimension 3 as a variety) that must be a typo, but it signals that the dimension comparison via [GR84] is unreliable as written, and Steps 4, 5, 7, and 8 all rely on the same 'similar arguments to Step 1' comparison. I also examined the potential objection that the de-meaning step in Theorem 5.4.(2) leaves an O(p^{-1/2}) error: since p^{-1/2} is smaller than the claimed p^{-1/640}, this is harmless; in fact, in the mean-zero case the induction step gives the stronger bound delta << p^{-1/128} directly, making the later degree-lowering steps unnecessary but not damaging the conclusion. Thus the central claim is plausible conditional on a correct proof of Theorem 4.2. A concrete computational check for P=t, Q=t^2 would settle whether the dimension contradiction is merely a typo or reflects a real gap in the point-counting argument.","tokens_in":31664,"tokens_out":30766,"duration_ms":273830,"concrete_test":"Take P(t)=t, Q(t)=t^2. (1) Symbolically verify Proposition 4.1 by expanding the 10x10 Jacobian determinant. (2) Compute the dimension of Ygen(C) for this example over Q with a computer algebra system (e.g., Macaulay2); the expected value is 6. (3) For small primes p (say p=11,13,...,101), enumerate the F_p-points of the Roth variety after clearing denominators and check whether |Y(F_p)| <= C p^6 with a modest constant. If the dimension is 6 and the counts are O(p^6), the dimension contradiction is a fixable typo; if the counts exceed p^6 by a power of p, Theorem 4.2 fails and the main theorem is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim depends on the Gowers norm control in Theorem 3.2, whose error term contains the factor (|Y|/p^6)^{1/16}. This factor is only bounded if the Roth variety satisfies |Y(F_p)| << p^6, i.e. Theorem 4.2. The proof of Theorem 4.2 is the weakest point. Step 1 first states that 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 the value 6 again. A finite surjective map preserves dimension, so if the complex manifold is 6-dimensional the variety dimension must be 6; the appearance of 3 is therefore either a typo or a sign that the dimension-comparison argument is unreliable. Steps 4, 5, 7, and 8 all use the same 'similar arguments to Step 1' dimension comparison for auxiliary varieties, so the entire point-counting argument inherits this fragility. If, for some P,Q satisfying the hypotheses, the true generic dimension of Ygen were larger than 6, then |Y(F_p)| could exceed p^6, the factor (|Y|/p^6)^{1/16} would blow up, and the degree-lowering induction in Section 5 would collapse. Thus Theorem 4.2 is load-bearing and is not established as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":31954,"tokens_out":13235,"duration_ms":125158,"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":[{"comment":"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.","section":"§4, Theorem 4.2, Step 1"},{"comment":"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.","section":"§5, Proof of Theorem 5.4.(2)"},{"comment":"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.","section":"§4, Theorem 4.2, Step 3"}],"minor_comments":[{"comment":"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.","section":"§4, Theorem 4.2, Step 1"},{"comment":"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.","section":"§2.4 and §4"},{"comment":"There is a typo in 'algebraic geomerty' in the introduction; it should read 'algebraic geometry'.","section":"§1.1"},{"comment":"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.","section":"§2.1"}],"recommendation":"major_revision","confidential_remarks":"The paper relies essentially on [HL24], a preprint coauthored by the present author, for Proposition 5.3 (the one-dimensional base case for two-term progressions) and for the complex-analytic dimension comparison used in Step 1 of Theorem 4.2. The editor may wish to verify the current status of [HL24] and whether the base case is independently established elsewhere. In addition, the central point-counting estimate in Theorem 4.2 is not rigorously proved as written because of the dimension contradiction in Step 1, so the paper should not be accepted before that argument is corrected and checked."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nThe 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.\n\nTwo 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.\n\nWho 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.\n\nRecommendation: send it to peer review with a major-revision request, and make the referee focus on a clean dimension proof for the Roth variety.","headline":"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.","tokens_in":32445,"tokens_out":5581,"would_cite":false,"duration_ms":50170,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B30","11T23","14G05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Corners in $\\mathbb{F}_p^2$ with rational side lengths have the random set's predicted count up to $O_{P,Q}(p^{-1/40960})$.","keywords":["polynomial Szemerédi theorem","corners in finite fields","rational function progressions","Gowers norms","degree lowering","algebraic geometry PET induction","Roth variety","point counting over finite fields"],"falsifier":"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.","tokens_in":31455,"feed_emoji":"📐","tokens_out":15510,"duration_ms":139028,"temperature":0.7,"pith_summary":"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 $Q$.","feed_headline":"Rational-function corners hit expected counts to p^{-1/40960}","feed_subtitle":"Even with rational side lengths, any dense subset contains plenty of such corners.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Develops the algebraic-geometry PET induction and the dimension-comparison method that Step 1 adapts to the corner Roth variety.","marker":"[HL24]"},{"why":"Provides the Fourier expansion and Cauchy–Schwarz skeleton for the corner average that the proof follows up to Equation (17).","marker":"[HLY21]"},{"why":"Supplies the additional Cauchy–Schwarz iteration that amplifies cancellation and prevents the auxiliary variety from degenerating.","marker":"[KW24]"},{"why":"Develops the directional Gowers-norm degree lowering reproduced in Theorem 5.4.","marker":"[Kuc24a]"},{"why":"Introduces the degree-lowering method for polynomial Szemerédi theorems that the final induction adapts.","marker":"[Pel19]"},{"why":"Gives the single-variable exponential sum estimate with rational-function phases used in the eigenfunction base case.","marker":"[Bom66]"},{"why":"Supplies the complex-analytic dimension comparison invoked in Step 1 of the point-counting proof.","marker":"[GR84]"},{"why":"Supplies the point-counting bound that converts dimension estimates into the needed finite-field point-count bounds.","marker":"[L W54]"}],"fun_headline_variants":["Rational corner counts match expectation to p^{-1/40960}","Corners with rational steps: uniform Szemeredi in finite fields","Any dense subset has plenty of rational corners"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Rational corner counts match expectation to p^{-1/40960}","Corners with rational steps: uniform Szemeredi in finite fields","Any dense subset has plenty of rational corners"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000836,"raw_usage":{"total_tokens":3633,"prompt_tokens":920,"completion_tokens":2713,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":536,"completion_tokens_details":{"reasoning_tokens":2658}},"tokens_in":536,"tokens_out":2713,"duration_ms":20886,"temperature":1.0,"reasoning_tokens":2658,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:24:57.189025+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}