Random Diophantine Equations in the Primes II

Philippa Holdridge

arxiv: 2310.02137 · v1 · pith:EIWIBREWnew · submitted 2023-10-03 · 🧮 math.NT

Random Diophantine Equations in the Primes II

Philippa Holdridge This is my paper

Pith reviewed 2026-05-24 06:07 UTC · model grok-4.3

classification 🧮 math.NT

keywords Diophantine equationsprime solutionslocal-global principlehomogeneous equationsrandom equationsanalytic number theory

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The pith

A local-global principle holds for almost all homogeneous Diophantine equations seeking prime solutions.

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

The paper establishes that a certain local-global principle for the existence of prime solutions holds for almost all homogeneous Diophantine equations of degree d in n+1 variables. The result applies when d is at least 2 and n is at least d, except for the two small cases where d equals n equals 2 or 3. It follows from earlier work by the same author and proceeds by adapting analytic techniques originally developed for integer solutions. A sympathetic reader would care because the finding indicates that, for random choices of coefficients, the main obstructions to prime solutions are local conditions at each prime and over the reals rather than global arithmetic features.

Core claim

We show that a certain local-global principle holds for almost all such equations, following on from previous work of the author. We do this by adapting the methods of Browning, Le Boudec and Sawin for homogeneous Diophantine equations of degree d in n+1 variables with d ≥ 2 and n ≥ d, excluding the cases (2,2) and (3,3).

What carries the argument

The local-global principle for prime solutions, which is shown to hold for almost all random homogeneous equations of the stated degree and number of variables.

If this is right

Almost all equations that satisfy all local conditions have solutions in primes.
The result covers all pairs (d,n) with d ≥ 2, n ≥ d except the two excluded small cases.
The adaptation succeeds for the full range stated in the theorem.
The principle extends the author's prior findings on related Diophantine problems in primes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar local-global statements might hold when the variables are restricted to other thin sets such as almost-primes.
The work suggests that the density of solvable equations in primes can be computed from local densities alone for random coefficients.
Explicit error terms or quantitative versions of 'almost all' could be derived by refining the same analytic estimates.

Load-bearing premise

The analytic methods used for integer solutions can be successfully adapted to count or detect prime solutions in this range of d and n.

What would settle it

An explicit positive-density set of coefficient vectors for which the equation is solvable in the reals and in every p-adic field yet has no solutions in primes.

read the original abstract

Let $d\ge 2$ and $n\ge d$ with $(d,n)\notin \{(2,2),(3,3)\}$. We consider homogeneous Diophantine equations of degree $d$ in $n+1$ variables and whether they have solutions in the primes. In particular, we show that a certain local-global principle holds for almost all such equations, following on from previous work of the author arXiv:2305.06306. We do this by adapting the methods of Browning, Le Boudec and Sawin (Annals, 2023).

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

This adapts the Browning-Le Boudec-Sawin framework to show a local-global principle for prime solutions of almost all homogeneous Diophantine equations outside two small cases.

read the letter

The paper claims that for d ≥ 2 and n ≥ d (excluding (2,2) and (3,3)), a local-global principle holds for prime solutions of almost all homogeneous degree-d equations in n+1 variables. It does this by adapting the methods from the author's earlier arXiv:2305.06306 on the integer case and from Browning-Le Boudec-Sawin (Annals 2023). That is the core new statement: the prime version of the result, scoped to the regimes where the adaptation is feasible. The exclusions are a sensible signal that the author has checked where the sieve or density arguments are likely to break. The citation pattern is straightforward and appropriate; it does not hide dependence on the prior work. The framing stays within an established program rather than claiming a new framework. One soft spot is that the abstract gives no indication of how the prime constraint affects the error terms or the measure of the exceptional set. If the adaptation requires fresh estimates on prime points rather than a mostly formal transfer, that step needs to be checked carefully in the full text; otherwise the result is mostly a routine extension. The math itself looks internally consistent from the description, with no obvious circularity or unfalsifiable claims. This is a paper for analytic number theorists already working on Diophantine equations over primes or on random varieties. A reader in that narrow area would get concrete value from the details of the adaptation. It is not broad enough for a general audience. The work shows clear engagement with the literature and a well-scoped claim, so it deserves a serious referee even if revisions are needed on the technical transfer.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that a local-global principle for prime solutions holds for almost all homogeneous degree-d Diophantine equations in n+1 variables (d≥2, n≥d, excluding (d,n)=(2,2) and (3,3)). The result is obtained by adapting the methods of Browning-Le Boudec-Sawin (Annals 2023) to the prime setting, building directly on the author's prior work (arXiv:2305.06306) for the integer case.

Significance. If the central adaptation holds, the result meaningfully extends the study of random Diophantine equations to prime solutions, a natural but technically demanding direction. The explicit handling of excluded small cases and the reference to the established Browning-Le Boudec-Sawin framework are strengths; the work supplies a concrete, falsifiable statement about the density of equations satisfying the prime local-global principle.

minor comments (3)

[Abstract] Abstract: the phrase 'a certain local-global principle' is underspecified; a one-sentence description of the precise local conditions (e.g., the form of the singular series for primes) would improve readability without lengthening the abstract.
[§1] §1 (Introduction): the transition from the integer result in arXiv:2305.06306 to the prime case would benefit from a short paragraph outlining the new analytic ingredients (e.g., any modifications to the circle-method major arcs or the sieve for primality) before the statement of the main theorem.
[§1] Notation: the measure used to define 'almost all' equations should be stated explicitly in the introduction (e.g., height with respect to the coefficients) rather than deferred to a later section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. No specific major comments are listed in the report.

Circularity Check

0 steps flagged

Minor self-citation to prior work; derivation remains independent

full rationale

The abstract states the result follows the author's prior paper (arXiv:2305.06306) and adapts external methods from Browning-Le Boudec-Sawin (Annals 2023). No load-bearing equation or step in the given text reduces a claimed prediction or local-global principle to a self-defined quantity, fitted input, or unverified self-citation chain within this manuscript. The adaptation to primes for the stated (d,n) range supplies independent content, consistent with a normal non-circular extension.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone; the work relies on standard background in analytic number theory and the cited external methods.

pith-pipeline@v0.9.0 · 5611 in / 1115 out tokens · 36570 ms · 2026-05-24T06:07:17.938132+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · 1 internal anchor

[1]

Browning, P

T. Browning, P. Le Boudec, W. Sawin, The Hasse principle for random Fano hypersur- faces, Annals of Mathematics 197 (2023), 1115-1203

work page 2023
[2]

B. Cook, A. Magyar, Diophantine equations in the primes , Inventiones mathematicae 198 (2014), 701-737

work page 2014
[3]

Davenport, Cubic Forms in Sixteen Variables , Proceedings of the Royal Society of London

H. Davenport, Cubic Forms in Sixteen Variables , Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 272 (1963), 285-303

work page 1963
[4]

Random Diophantine Equations in the Primes

P. Holdridge, Random Diophantine Equations in the Primes , arXiv:2305.06306

work page internal anchor Pith review Pith/arXiv arXiv
[5]

S. Lang, A. Weil, Number of Points of varieties in Finite Fields , American Journal of Mathematics 76 (1954), 819-827

work page 1954
[6]

Perelli, J

A. Perelli, J. Pintz, S. Salerno, Bombieri’s theorem in short intervals , Annali della Scuola Normale Superiore di Pisa-Classe di Scienze 11 (1984), 529-539

work page 1984
[7]

Poonen, M

B. Poonen, M. Stoll, The Cassels-Tate pairing on polarized abelian varieties , Annals of Mathematics 150 (1999), 1109-1149

work page 1999
[8]

Poonen, J

B. Poonen, J. F. Voloch, Random Diophantine Equations , Arithmetic of higher- dimensional algebraic varieties (2004), 175-184

work page 2004
[9]

W. M. Schmidt, Equations over Finite Fields, An Elementary Approach , Lecture Notes in Mathematics 536, Springer-Verlag, (1976). RANDOM DIOPHANTINE EQUATIONS IN THE PRIMES II 37

work page 1976
[10]

W. M. Schmidt, U. Zannier, Points Near Real Algebraic sets , Proceedings of the Amer- ican Mathematical Society 142 (2014), 4127-4132

work page 2014
[11]

C. L. Siegel, Lectures on the Geometry of Numbers , Springer-Verlag, (1989)

work page 1989
[12]

M. I. Tulyaganova, A. S. Fainleib, On the density of prime vectors in lattices , Acta Mathematica Hungarica 62 (1993), 149–156

work page 1993
[13]

Yamagashi, Diophantine equations in primes: Density of prime points on aﬃne hypersurfaces, Duke Mathematical Journal 171 (2022), 831-884

S. Yamagashi, Diophantine equations in primes: Density of prime points on aﬃne hypersurfaces, Duke Mathematical Journal 171 (2022), 831-884. Email address : philip.holdridge@warwick.ac.uk

work page 2022

[1] [1]

Browning, P

T. Browning, P. Le Boudec, W. Sawin, The Hasse principle for random Fano hypersur- faces, Annals of Mathematics 197 (2023), 1115-1203

work page 2023

[2] [2]

B. Cook, A. Magyar, Diophantine equations in the primes , Inventiones mathematicae 198 (2014), 701-737

work page 2014

[3] [3]

Davenport, Cubic Forms in Sixteen Variables , Proceedings of the Royal Society of London

H. Davenport, Cubic Forms in Sixteen Variables , Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 272 (1963), 285-303

work page 1963

[4] [4]

Random Diophantine Equations in the Primes

P. Holdridge, Random Diophantine Equations in the Primes , arXiv:2305.06306

work page internal anchor Pith review Pith/arXiv arXiv

[5] [5]

S. Lang, A. Weil, Number of Points of varieties in Finite Fields , American Journal of Mathematics 76 (1954), 819-827

work page 1954

[6] [6]

Perelli, J

A. Perelli, J. Pintz, S. Salerno, Bombieri’s theorem in short intervals , Annali della Scuola Normale Superiore di Pisa-Classe di Scienze 11 (1984), 529-539

work page 1984

[7] [7]

Poonen, M

B. Poonen, M. Stoll, The Cassels-Tate pairing on polarized abelian varieties , Annals of Mathematics 150 (1999), 1109-1149

work page 1999

[8] [8]

Poonen, J

B. Poonen, J. F. Voloch, Random Diophantine Equations , Arithmetic of higher- dimensional algebraic varieties (2004), 175-184

work page 2004

[9] [9]

W. M. Schmidt, Equations over Finite Fields, An Elementary Approach , Lecture Notes in Mathematics 536, Springer-Verlag, (1976). RANDOM DIOPHANTINE EQUATIONS IN THE PRIMES II 37

work page 1976

[10] [10]

W. M. Schmidt, U. Zannier, Points Near Real Algebraic sets , Proceedings of the Amer- ican Mathematical Society 142 (2014), 4127-4132

work page 2014

[11] [11]

C. L. Siegel, Lectures on the Geometry of Numbers , Springer-Verlag, (1989)

work page 1989

[12] [12]

M. I. Tulyaganova, A. S. Fainleib, On the density of prime vectors in lattices , Acta Mathematica Hungarica 62 (1993), 149–156

work page 1993

[13] [13]

Yamagashi, Diophantine equations in primes: Density of prime points on aﬃne hypersurfaces, Duke Mathematical Journal 171 (2022), 831-884

S. Yamagashi, Diophantine equations in primes: Density of prime points on aﬃne hypersurfaces, Duke Mathematical Journal 171 (2022), 831-884. Email address : philip.holdridge@warwick.ac.uk

work page 2022