Solubility of a family of conics with polynomial coefficients in many variables

Mathieu Da Silva

REVIEW 2 major objections 3 minor 28 references

Solubility of a family of conics with polynomial coefficients in many variables

T0 review · 2 major / 3 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read For conic families with polynomial coefficients in many variables, the paper proves the count of parameters with a rational point is asymptotic to c B/(log B)^{3/2}, with c equal to the conjectured leading constant.

desk verdict The uniform count for diagonal conics in arithmetic progressions is a real contribution, but the final bridge that turns it into a family result is false, so the main theorem fails. read the letter →

arxiv 2511.20282 v2 pith:4SB3PXM7 submitted 2025-11-25 math.NT

Mathieu Da Silva This is my paper · ORCID

classification math.NT MSC 11D0911P5511N3714G05

keywords conicbundlesrationalpointsasymptoticformulacirclemethodBirchsystemlocalsolubilityproductoverprimesDiophantinefamilies

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 establishes an asymptotic formula for the number of coefficient tuples y of bounded height for which the conic F0(y)x0^2 + F1(y)x1^2 = F2(y)x2^2 has a rational point, assuming the three homogeneous polynomials form a system with many variables (a Birch system) and have smooth, properly intersecting zero loci. The main theorem states that the count is asymptotic to c(π_F) B/(log B)^{3/2}, where c(π_F) is the explicit product of local densities that the general conjectural framework for fibrations predicts. The result matters because it pins down the leading constant — not just the power of the logarithm — for a natural family of varieties, giving strong support to the conjecture and a template for computing such constants. The strategy averages the local-solubility indicator over polynomial values via the circle method, then evaluates the resulting product over primes through a character-sum computation.

What carries the argument

The load-bearing object is the indicator ϑ_Q(t0,t1,t2), which is 1 exactly when the diagonal conic t0x0^2+t1x1^2=t2x2^2 has a nonzero rational point. The proof machinery is: a Birch system condition (enough variables relative to the degree so that a circle-method average is accurate), a recent circle-method theorem for averages of bounded arithmetic functions over polynomial values, an analytic averaging lemma that isolates the main coefficient in the average, and a product-over-primes computation of the local densities. An inclusion-exclusion step over common divisors converts sign-restricted coefficient counts into the full solubility count, yielding the final constant.

What would settle it

Compute, for a concrete homogeneous triple of polynomials and a point k where F0(k), F1(k), F2(k) are all positive and the conic F0(k)x0^2+F1(k)x1^2-F2(k)x2^2=0 has a rational point, the value of (1/2) Σ_ε 1_{min ε_jF_j(k)>0} ϑ_Q(ε0F0(k),ε1F1(k),ε2F2(k)). For example with F0=2X0, F1=X0, F2=3X0 and k=(1,0,...,0), the left side is 1/2 while the right side is 1; this would settle whether the proof's central identity holds.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.7: for homogeneous polynomials F0, F1, F2 of degree d that form a Birch system with smooth fibres and a complete-intersection common zero locus, the global solubility count N_glob(π_F,B) satisfies N_glob(π_F,B) ~ c(π_F) B/(log B)^{3/2}, with c(π_F) exactly the constant predicted by the fibration-families conjecture. Along the way, the paper proves a precise asymptotic for the proportion of diagonal ternary conics t0x0^2+t1x1^2+t2x2^2=0 with a rational point and with coefficients in arithmetic progressions, uniform in the modulus up to a power of log B. This arithmetic-progression theorem is the engine from which the main constant is assembled via local factors.

Load-bearing premise

The argument hinges on the identity that the count of rational points can be recovered by summing, over the three sign patterns that make the polynomial values all positive, the local solubility indicators and dividing by two; if that sign-bridge equality fails, the main theorem is not established.

Editorial extensions

If this is right

If correct, the conjectured leading constant for fibrations is realized in this family: the power of log is 3/2 and the constant is a product of local densities.
The uniform arithmetic-progression theorem for diagonal conics implies that the same constant governs counts for any modulus at most a small power of log B, allowing the circle-method averaging to be glued.
The paper computes the relevant Galois cohomology invariant of the fibration (a Z/2 quotient generated by an explicit quaternion algebra), which fixes the Tamagawa-type factor in the constant.
The result gives an additional example where the full constant, not just the logarithmic exponent, is verified for a family of conic bundles with polynomial coefficients in many variables.
The method shows how to derive the constant from local factors c_8 and c_p defined by volumes of solubility regions, suggesting these local factors are the correct universal building blocks.

Reading between the lines

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

The uniform arithmetic-progression theorem for diagonal conics could be imported to families of higher-dimensional quadrics or to fibrations over other bases, provided the circle-method averaging result is available.
The computation of the constant as a product of local densities suggests that the same local factors appear in any proof of the fibration conjecture, so the shape of c(π_F) is robust even if the present proof is later modified.
A direct numerical check of the sign-restricted identity for small-degree polynomial triples (e.g., triples of linear forms at a point with all values positive) would be a cheap way to validate the proof's central bridge before tackling the full theorem.

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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

Colleague,

I read Da Silva's arXiv:2511.20282 with the stress-test note in hand. The main theorem is not proved. The specific fault is equation (6.5) in Section 6, which claims a Möbius inversion that replaces ϑ_Q(F0(k),F1(k),F2(k)) by (1/2)Σ_ε ϑ_Q(ε0F0(k), ε1F1(k), ε2F2(k)) subject to min_j ε_jF_j(k)>0. For any k with all three F_j(k) nonzero, the ε-sum has exactly one term, ε = sign(F0,F1,F2), so the identity asserts ϑ_Q(F(k)) = (1/2)ϑ_Q(|F0|,|F1|,|F2|). That is false. The example F0=X0, F1=X1, F2=X0+X1−X2 at k=(1,1,3,0) gives F=(1,1,−1): the true conic x0^2+x1^2=−x2^2 has no rational point, while the all-positive conic has the point (1,0,1). This is not a measure-zero artifact; it happens on a positive-volume set of k. Worse, the sign substitution changes p-adic solubility, e.g., 2x^2−y^2−3z^2 is anisotropic over Q3 while 2x^2+y^2−3z^2 is soluble. So the computed asymptotic counts a different set of fibres.

What is good: Sections 2–5 contain real work. Theorem 4.2 gives a uniform asymptotic for diagonal ternary quadratics in arithmetic progressions with q up to (log B)^{1/7}, which extends the base family in Loughran–Rome–Sofos. Lemma 3.1 (multi-dimensional Selberg–Delange) is a new general tool. The constant computation in Proposition 2.3, assuming the final result, is plausible. The paper is clearly written and seriously engaged with the literature.

However, the load-bearing step fails. The final displayed constant also disagrees with the predicted constant (1.3) by a factor related to 1/ζ(n+1) and a 2, which corroborates that the reduction is not just a sign error but a conceptual one. I would not trust Theorem 1.7 in this form.

For a referee: this deserves a serious referee report, but the verdict should be reject with the door open: the author should either fix Section 6 (which may require a genuinely new idea to handle sign patterns) or strip the paper to its solid Sections 2–5 and resubmit as a standalone contribution on uniform counts of diagonal conics. I would not cite Theorem 1.7 in the next year, but I would note Theorem 4.2 as a possible citation if it survives scrutiny.

Recommendation: send back for major revision, but make clear the main theorem is unproved.

Referee Report

2 major / 3 minor

Summary. The paper studies the density of y ∈ P^n(Q) of bounded height for which the diagonal conic C_{F,y}: F_0(y)x_0^2 + F_1(y)x_1^2 = F_2(y)x_2^2 has a rational point. It claims an asymptotic N_glob(π_F,B) ~ c(π_F) B/(log B)^{3/2}, with the constant c(π_F) predicted by Loughran–Rome–Sofos. The proof combines a circle-method theorem of Destagnol–Lyczak–Sofos for averages over polynomial values with a new estimate (Theorem 4.2) for the number of soluble diagonal conics whose coefficients lie in arithmetic progressions, followed by a long computation of the resulting constant.

Significance. If the main theorem were correct, it would be a substantial confirmation of the Loughran–Rome–Sofos conjecture for a natural family of conic bundles, including the full leading constant. The paper contains a large amount of technical analytic number theory: a general Selberg–Delange lemma (Lemma 3.1), a careful application of the Destagnol–Lyczak–Sofos circle-method result, and an extensive computation of local constants. These components may be of independent interest. However, the central bridge connecting the sign-conditioned circle-method counts to the actual solubility count is invalid, so the claimed theorem is not established.

major comments (2)

[Section 6, Eq. (6.5)] The displayed "Möbius inversion" identity is false. For a fixed primitive k, the condition min_j ε_j F_j(k)>0 is satisfied by the unique sign vector ε(k)=(sign F_0(k), sign F_1(k), sign F_2(k)). The summand is then (1/2)ϑ_Q(|F_0(k)|,|F_1(k)|,|F_2(k)|), not (1/2)ϑ_Q(F_0(k),F_1(k),F_2(k)). These differ on a set of positive density. Example within the theorem's hypotheses: n=3, F_0=X_0, F_1=X_1, F_2=X_0+X_1−X_2, k=(1,1,3,0). Then F(k)=(1,1,−1); the true conic x_0^2+x_1^2=−x_2^2 has no Q-point, while the selected ε=(+,+,−) gives x_0^2+x_1^2=x_2^2, which has (1,0,1). Hence the RHS of (6.5) receives a spurious contribution of 1/2. The failure is not a boundary effect; it occurs on the open region F_0>0,F_1>0,F_2<0. Moreover sign changes alter p-adic solubility (2x^2−y^2−3z^2=0 is anisotropic over Q_3, but 2x^2+y^2−3z^2=0 is soluble), so no local correction can be absorbed into the constant.
[Section 6, final display] After substituting the asymptotic for N^1_ε into (6.5), the proof needs to sum over all eight ε. The displayed evaluation instead uses Σ_{ε≠±(1,1,−1)} vol(min_j ε_jF_j≥0)=c_8, i.e. it discards the two sign patterns ±(1,1,−1). These are exactly the patterns that made the bridge (6.5) false: on the positive-density region F_0>0,F_1>0,F_2<0 (and its negative), the selected ε is either (1,1,−1) or (−1,−1,1), and the corresponding N^1_ε count is nonzero because it counts positive-coefficient conics t_0x_0^2+t_1x_1^2=t_2x_2^2 with rational points. Dropping these terms silently changes the count from N_loc to a smaller set and is inconsistent with the preceding equality. The constant found is thus not the constant for the original family.

minor comments (3)

[Section 4, Eq. (4.1)] The factor δ_p(n) for p∤q is introduced as 'the value of the factor corresponding to p in the Euler product in [16, th. 1.1]' before any computation. Since the subsequent Section 5 does derive this factor, the wording gives the impression that the conjectural constant is used as an input. Please rephrase or reorder so that the definition is presented as a summary of the calculation, not as an assumption.
[Section 6, Lemma 6.8] The bound z ≤ sqrt((1/14) log_2 B) is stated without justification. It should be shown, or at least briefly explained, that this choice ensures W_z ≤ (log B)^{1/7} so that Theorem 4.2 applies.
[Throughout] There are several typographical and language issues: leftover French phrases ('et', 'prouvée'), 'Belun' in Reference [26] should be 'Belin', and the notation c_8 vs c8 is used inconsistently. These are minor but should be corrected.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the final constant is computed from genuine local solubility densities; the one normalization imported from [16] is explicit and does not force the result. The serious problem at (6.5) is a correctness gap, not a circular reduction.

full rationale

The claimed derivation is not equivalent to its inputs. Theorem 4.2 is an auxiliary count for positive coefficient triples in arithmetic progressions; the local factor δ_p(n) for p∤q is explicitly chosen 'from the value of the factor corresponding to p in the Euler product in [16, th. 1.1]' (Remark after (4.1)). This is a transparent normalization, and it does not enter the final constant: in Lemma 6.8 the tail ∏_{p>z}(1-1/p)^{3/2}δ_p is 1+O(1/z), and Lemma 6.12 produces the genuine densities c_p=μ_p({t:ϑ_p(F(t))=1}). There is no self-citation: the author has no overlap with [5] or [16], and the external circle-method theorem [5, th. 2.4] is an independent input. The only load-bearing step that might look circular is (6.5), which claims N_loc equals a Möbius average of N1_ε defined via ϑ_Q(εF). For F0,F1>0>F2 the only contributing ε=(+,+,-) tests the indefinite conic F0x0^2+F1x1^2=(-F2)x2^2, whereas the original fibre is F0x0^2+F1x1^2=F2x2^2; these are not the same conic, so (6.5) is at best unproved and appears false. That invalidates the proof but is a correctness issue, not a reduction of the theorem to its own assumptions. Accordingly the circularity score is low.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new entities or fitted constants of scientific content. Its proof relies on standard analytic number theory (Selberg–Delange, large sieve, Hilbert symbols) and on two load-bearing external results ([5, th. 2.4] and the strategy/constants of [16]). One axiom is an asserted identity (6.5) that is false; this is the main structural weakness.

free parameters (3)

m_p(z) = ⌊z⌋ + 2·1_{p=2} (Def. 6.1) = m_2(z)=⌊z⌋+2, m_p(z)=⌊z⌋ for 3≤p≤z
Hand-chosen (following [5]) to make ε(z)→0 while keeping W_z small; a proof parameter of the method, not fitted to data.
Exponent constraints q≤(log B)^{1/7} (Thm 4.2) and z≤(log_2 B)^{1/2}/√14 (Lemma 6.8) = 1/7, 1/14
Chosen to balance the error terms in Sections 4 and 6; hand-tuned constants of the proof.
Truncation C=3 in Corollary 4.13 / Prop. 4.14 = 3
Any C>5/2 works; chosen for convenience in the error-term bookkeeping.

assumptions (6)

standard math Multi-dimensional Selberg–Delange estimates (Tenenbaum [25, II.5])
Lemma 3.1 rests on the Selberg–Delange method with explicit Taylor coefficients a_u(z), b_v(z); cited from [25].
domain assumption Destagnol–Lyczak–Sofos circle-method theorem [5, th. 2.4] on averages of arithmetic functions over polynomial values (Birch system)
Theorem 6.2 is the engine converting AP-uniform conic counts into polynomial-value averages; it is a 2025 preprint and is not independently verified here.
standard math Hasse–Minkowski and Hilbert-symbol criteria for ternary quadratic forms ([22, ch. 3])
Used in Lemma 4.3 and Prop. 4.4 to factor ϑ_Q into local factors; standard.
domain assumption Large-sieve bounds of Friedlander–Iwaniec [6] and Wilson [28, lemma 5.1]
Corollary 6.7 and the large/small conductor estimates in Prop. 4.12 rest on these imported bounds.
standard math Geometric inputs: purity exact sequence [3, 3.7.2], [20, 2.6], exact sequence (2.1) from [14, (2.12)]
Proposition 2.2's computation of the subordinate Brauer group depends on these cited results.
ad hoc to paper Sign-splitting identity (1/2)Σ_ε 1_{min_j ε_jF_j>0}ϑ_Q(εF) = ϑ_Q(F) (eq. (6.5))
This identity is asserted without proof and is false as stated (see red flags); Section 6 depends on it to pass from sign-restricted counts to N_loc.

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Pith. "Pith review of Solubility of a family of conics with polynomial coefficients in many variables." pith.science (2026). https://pith.science/paper/4SB3PXM7

@misc{pith2026251120282,
  author       = {Pith},
  title        = {Pith review of: Solubility of a family of conics with polynomial coefficients in many variables},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4SB3PXM7}},
  note         = {Machine review of arXiv:2511.20282}
}

abstract

We study the proportion of conics given by $(\mathcal{C}_{\mathbf{F}, \mathbf{y}}) : F_0(\mathbf{y})x_0^2 + F_1(\mathbf{y})x_1^2 = F_2( \mathbf{y})x_2^2 $ which have a rational point $\mathbf{x} = (x_0 :x_1:x_2) \in \mathbb{P}^2(\mathbb{Q})$, where $\mathbf{y} = (y_0 : \dots : y_n)\in \mathbb{P}^n(\mathbb{Q})$ and $F_0,F_1,F_2 \in \mathbb{Z}[X_0,\ldots, X_n]$ are homogeneous polynomials in many variables of the same degree $d$. We provide an asymptotic formula for the number of $\mathbf{y}$ of bounded height such that the corresponding conic $(\mathcal{C}_{\mathbf{F}, \mathbf{y}})$ has a rational point. In particular, our result agrees with the Loughran--Smeets and the Loughran--Rome--Sofos conjectures. Our strategy is based on a recent result of Destagnol--Lyczak--Sofos relying on the circle method to estimate the average of an arithmetic function over polynomials in many variables. To this end, we study the proportion of conics $t_0x_0^2 + t_1x_1^2 + t_2x_2^2 = 0$ having a rational point, and coefficients $t_0,t_1,t_2$ in arithmetic progressions.

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

B. J. Birch, Forms in many variables.Proc. Roy. Soc. London Ser. A,265(1962), 245–263

1962

[2] [2]

T. D. Browning,Cubic forms and the circle method.Birkhäuser (2021), xiv–166

2021

[3] [3]

Colliot-Thélène, A

J.-L. Colliot-Thélène, A. N. Skorobogatov,The Brauer-Grothendieck group.Ergebnisse der Math- ematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 71. Springer, Cham, 2021

2021

[4] [4]

org/pdf/2409.18116(2024)

K.Destagnol, E.Sofos, Averagesoffunctionsoverpolynomialsinmanyvariables.https://arxiv. org/pdf/2409.18116(2024)

arXiv 2024

[5] [5]

Destagnol, J

K. Destagnol, J. Lyczak, E. Sofos, Local solubility in generalised Châtelet surfaces.https:// arxiv.org/pdf/2504.11388(2025)

arXiv 2025

[6] [6]

Friedlander, H

J. Friedlander, H. Iwaniec, Ternary quadratic forms with rational zeros,J. Théor. Nombres Bor- deaux,22(2010) no. 1, 97-113

2010

[7] [7]

Gille and T

P. Gille and T. Szamuely,Central simple algebras and Galois cohomology.Cambridge Studies in Advanced Mathematics101(2006)

2006

[8] [8]

C. R. Guo, On solvability of ternary quadratic forms.Proc. London Math. Soc.(3)70(1995), no. 2, 241–263

1995