Estimating the tail of the singular product in the multivariate Bateman-Horn conjecture

Victor Volfson

arxiv: 2604.25969 · v1 · submitted 2026-04-28 · 🧮 math.NT

Estimating the tail of the singular product in the multivariate Bateman-Horn conjecture

Victor Volfson This is my paper

Pith reviewed 2026-05-07 15:14 UTC · model grok-4.3

classification 🧮 math.NT

keywords Bateman-Horn conjecturesingular seriesmultivariate polynomialsprime valuesexponential sumsdiagonal polynomialstail estimatesalgebraic geometry

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The pith

The tail of the singular product can be bounded to control relative error when approximating the singular series for multivariate polynomials.

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

This paper derives upper bounds on the error incurred by truncating the infinite product that defines the singular series in the multivariate Bateman-Horn conjecture. The bounds are obtained by combining probabilistic heuristics for the expected count of prime values with estimates on multidimensional exponential sums. For polynomials satisfying suitable smoothness conditions a general bound holds, while diagonal polynomials admit a stronger result that exploits factorization of the sums together with Katz's formula. The estimates make it possible to compute the singular series to prescribed accuracy in concrete cases.

Core claim

The number of times a multivariate integer polynomial takes prime values is expected, under the multidimensional Bateman-Horn conjecture, to be governed by a singular series given by an Euler product over primes. The paper supplies explicit upper bounds for the tail of this product. Under smoothness assumptions on the polynomial the relative error after truncation is controlled; in the diagonal case both the decay exponent and the implied constant are improved by factoring the relevant exponential sums and invoking Katz's theorem.

What carries the argument

The tail of the singular product, bounded through estimates on exponential sums and results from algebraic geometry including Katz's formula.

Load-bearing premise

The polynomials must satisfy the smoothness conditions that make the exponential-sum bounds applicable, and the multidimensional Bateman-Horn heuristic must correctly describe the main-term order of growth.

What would settle it

An explicit smooth or diagonal polynomial for which the measured tail of the singular product exceeds the stated upper bound on the relative error would falsify the estimate.

read the original abstract

This paper investigates the asymptotics of the number of prime values taken by a polynomial in several variables with integer coefficients. Based on probabilistic heuristics and the multidimensional Bateman Horn conjecture, the expected order of growth of this number is derived. The main focus is on the accuracy of computing the singular series. Using methods of algebraic geometry and analytic number theory, estimates for the tail of the singular product are obtained. For the general case of polynomials satisfying smoothness conditions, an upper bound for the relative error is established. For diagonal polynomials, due to the factorization of exponential sums and formula of Katz, an improvement is achieved both in the decay exponent and in the constant. Comparative tables are provided demonstrating the effectiveness of the diagonal case. The results make it possible to control the accuracy of the singular series approximation in practical computations and can be used for further progress towards the proof of the Bateman Horn conjecture for multivariate polynomials.

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

Volfson supplies explicit tail bounds for the singular product in the multivariate Bateman-Horn setting, with a claimed improvement for diagonal polynomials via Katz, but the abstract leaves the derivations and geometric hypotheses unshown.

read the letter

The main takeaway is that this paper works out explicit upper bounds on the relative error when truncating the singular product for polynomials in several variables that meet smoothness conditions. For the general case it gives a usable error control, and for diagonal polynomials it claims a better decay exponent and constant by factoring exponential sums and invoking Katz's formula, with tables showing the practical difference in approximation quality. That focus on making the infinite product computable with controlled error is the concrete advance, and it directly addresses a bottleneck that comes up when you want to test or apply the conjecture numerically in higher dimensions. The framing around the expected growth order from the heuristic is standard and the motivation for error control is clear, so the work sits in a useful spot for people doing effective analytic number theory on prime values of polynomials. The comparative tables are a plus for anyone who needs to see how much the diagonal improvement buys in practice. On the soft spots, the abstract states the bounds exist but shows none of the derivations or explicit constants, which makes it impossible to judge sharpness or tightness without the full text. The stress-test concern about Katz's formula is real: those results require non-degeneracy or smoothness of the associated projective variety that goes beyond generic C^1 or C^infty assumptions, and nothing in the summary confirms that the paper's hypotheses deliver exactly that input. If the check is missing or the conditions fall short, the sharper diagonal bound collapses back to the general estimate. The reliance on the Bateman-Horn heuristic itself for the growth order is fine as background but keeps the results conditional. This is aimed at specialists working on multivariate prime-producing polynomials and effective versions of the conjecture. A reader who needs tail estimates for singular-series computations would get direct value from the stated bounds and tables. It has enough relevance and technical substance to deserve a serious referee rather than a desk reject, though any review should focus on verifying the geometric hypotheses for Katz and seeing the actual proofs.

Referee Report

1 major / 2 minor

Summary. The manuscript investigates tail estimates for the singular product in the multivariate Bateman-Horn conjecture. It derives the expected asymptotic order for the count of prime values taken by multivariate integer polynomials under the conjecture's heuristics. The main results are upper bounds on the relative error of the truncated singular series: a general bound for polynomials satisfying smoothness conditions, obtained via algebraic geometry and analytic number theory, and sharper decay (both exponent and constant) for diagonal polynomials via factorization of exponential sums together with Katz's formula. Comparative tables are included to illustrate the practical improvement in the diagonal case. The work is framed as enabling accurate numerical control of the singular series and supporting further progress toward a proof of the conjecture.

Significance. If the bounds are valid, the estimates supply explicit error controls that are directly usable in computational checks of prime-producing multivariate polynomials, a setting where the singular series must be approximated to high precision. The separation into general and diagonal cases, with a concrete improvement for the latter, adds practical value. The reliance on Katz's theorems and exponential-sum factorization, when the geometric hypotheses are met, would constitute a technically substantive refinement of existing heuristic machinery.

major comments (1)

[section deriving improved bounds for diagonal polynomials] The claimed improvement in decay exponent and constant for diagonal polynomials rests on the factorization of exponential sums followed by an application of Katz's formula. Katz's theorems on exponential sums and monodromy require non-degeneracy (or smoothness) hypotheses on the projective hypersurface that are strictly stronger than the generic C^1 or C^∞ smoothness conditions stated for the general case. No explicit verification is supplied that the diagonal polynomials satisfy these geometric hypotheses; without it the sharper bound reduces to the general-case estimate.

minor comments (2)

The abstract and introduction refer to 'comparative tables' without section or table numbers; explicit cross-references would improve navigation.
Notation for the tail of the singular product and the relative error is introduced gradually; a single early definition or displayed equation would aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive criticism of our manuscript. The single major comment is addressed point by point below.

read point-by-point responses

Referee: [section deriving improved bounds for diagonal polynomials] The claimed improvement in decay exponent and constant for diagonal polynomials rests on the factorization of exponential sums followed by an application of Katz's formula. Katz's theorems on exponential sums and monodromy require non-degeneracy (or smoothness) hypotheses on the projective hypersurface that are strictly stronger than the generic C^1 or C^∞ smoothness conditions stated for the general case. No explicit verification is supplied that the diagonal polynomials satisfy these geometric hypotheses; without it the sharper bound reduces to the general-case estimate.

Authors: We agree with the referee that Katz's theorems require non-degeneracy conditions on the projective hypersurface that are stronger than the C^1 or C^∞ smoothness assumed in the general case, and that the manuscript does not contain an explicit verification for the diagonal polynomials. In the revised version we will insert a short subsection (immediately preceding the statement of the sharper bound) that verifies these conditions for the diagonal forms under consideration. The verification proceeds by direct computation of the partial derivatives and appeal to the fact that the diagonal hypersurface is a Fermat hypersurface of degree d in projective space, which is known to be smooth over the algebraic closure when the coefficients are nonzero and the characteristic does not divide d; the monodromy group is then the full symmetric group by Katz's criterion, justifying the improved exponent and constant. This addition will make the application of Katz's formula fully rigorous while leaving the general-case bound unchanged. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The paper derives upper bounds on the tail of the singular product using external tools from algebraic geometry and analytic number theory (factorization of exponential sums, Katz formula) applied to polynomials satisfying stated smoothness conditions. The multidimensional Bateman-Horn conjecture appears only as a heuristic motivating the expected order of growth, not as an input that the estimates are fitted to or defined in terms of. No equations or steps reduce by construction to self-citations, fitted parameters renamed as predictions, or ansatzes smuggled via prior work by the same author. The central claims are therefore independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the multidimensional Bateman-Horn conjecture as a heuristic and on standard smoothness assumptions for the polynomials; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)

domain assumption Multidimensional Bateman-Horn conjecture
Used to derive the expected order of growth of prime values
domain assumption Smoothness conditions on the polynomials
Required to obtain the upper bound on relative error

pith-pipeline@v0.9.0 · 5444 in / 1267 out tokens · 61825 ms · 2026-05-07T15:14:34.859689+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Prime Number Theorems for Polynomials from Homogeneous Dynamics

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work page arXiv 2023
[2]

& Sofos, E

Destagnol, K. & Sofos, E. (2019). The Bateman–Horn conjecture for polynomials in many variables. Mathematische Annalen, 374(1–2), 1–35

work page 2019
[3]

& Wooley, T

Brüdern, J. & Wooley, T. D. (2022). The Bateman–Horn conjecture for diagonal polynomials. Journal für die reine und angewandte Mathematik (Crelle’s Journal), 2022(789), 1–45

work page 2022
[4]

D., Sofos, E

Browning, T. D., Sofos, E. & Teräväinen, J. (2022). The Bateman–Horn conjecture for almost all polynomials. Duke Mathematical Journal, 171(15), 3051–3112

work page 2022
[5]

& Wang, Z

Matthiesen, L., Teräväinen, J. & Wang, Z. (2024). Polynomial patterns in the primes and the Bateman–Horn conjecture. Inventiones mathematicae, 235(3), 891–950

work page 2024
[6]

Deligne, P. (1980). La conjecture de Weil. II. Publications Mathématiques de l'IHÉS, 52, 137–252

work page 1980
[7]

Katz, N. M. (1980). Sommes exponentielles. Astérisque, 79. Société Mathématique de France

work page 1980
[8]

Grothendieck, A. (1977). Formule de Lefschetz et rationalité des fonctions L. In Séminaire de Géométrie Algébrique du Bois-Marie (SGA 5). Lecture Notes in Mathematics, Vol. 589. Springer-Verlag

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[9]

Dimca, A. (1992). Singularities and Topology of Hypersurfaces. Universitext. Springer-Verlag, New York. ISBN: 978-0-387-97709-5

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[10]

Katz, N. M. (1988). Gauss sums, Kloosterman sums, and monodromy groups. Annals of Mathematics Studies, No. 116. Princeton University Press

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[11]

Vaughan, R. C. (1997). The Hardy–Littlewood Method (2nd ed.). Cambridge Tracts in Mathematics, No. 125. Cambridge University Press

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[12]

T., Prendiville, S

Parsell, S. T., Prendiville, S. M. & Wooley, T. D. (2020). Near-optimal mean value estimates for multidimensional Weyl sums. Geometric and Functional Analysis, 30(2), 421–487

work page 2020

[1] [1]

Prime Number Theorems for Polynomials from Homogeneous Dynamics

Giorgos Kotsovolis, Katharine Woo (2023). Prime Number Theorems for Polynomials from Homogeneous Dynamics. Preprint. https://arxiv.org/abs/2312.11445 [math.NT]

work page arXiv 2023

[2] [2]

& Sofos, E

Destagnol, K. & Sofos, E. (2019). The Bateman–Horn conjecture for polynomials in many variables. Mathematische Annalen, 374(1–2), 1–35

work page 2019

[3] [3]

& Wooley, T

Brüdern, J. & Wooley, T. D. (2022). The Bateman–Horn conjecture for diagonal polynomials. Journal für die reine und angewandte Mathematik (Crelle’s Journal), 2022(789), 1–45

work page 2022

[4] [4]

D., Sofos, E

Browning, T. D., Sofos, E. & Teräväinen, J. (2022). The Bateman–Horn conjecture for almost all polynomials. Duke Mathematical Journal, 171(15), 3051–3112

work page 2022

[5] [5]

& Wang, Z

Matthiesen, L., Teräväinen, J. & Wang, Z. (2024). Polynomial patterns in the primes and the Bateman–Horn conjecture. Inventiones mathematicae, 235(3), 891–950

work page 2024

[6] [6]

Deligne, P. (1980). La conjecture de Weil. II. Publications Mathématiques de l'IHÉS, 52, 137–252

work page 1980

[7] [7]

Katz, N. M. (1980). Sommes exponentielles. Astérisque, 79. Société Mathématique de France

work page 1980

[8] [8]

Grothendieck, A. (1977). Formule de Lefschetz et rationalité des fonctions L. In Séminaire de Géométrie Algébrique du Bois-Marie (SGA 5). Lecture Notes in Mathematics, Vol. 589. Springer-Verlag

work page 1977

[9] [9]

Dimca, A. (1992). Singularities and Topology of Hypersurfaces. Universitext. Springer-Verlag, New York. ISBN: 978-0-387-97709-5

work page 1992

[10] [10]

Katz, N. M. (1988). Gauss sums, Kloosterman sums, and monodromy groups. Annals of Mathematics Studies, No. 116. Princeton University Press

work page 1988

[11] [11]

Vaughan, R. C. (1997). The Hardy–Littlewood Method (2nd ed.). Cambridge Tracts in Mathematics, No. 125. Cambridge University Press

work page 1997

[12] [12]

T., Prendiville, S

Parsell, S. T., Prendiville, S. M. & Wooley, T. D. (2020). Near-optimal mean value estimates for multidimensional Weyl sums. Geometric and Functional Analysis, 30(2), 421–487

work page 2020