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The Hasse principle for random homogeneous polynomials in thin sets

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Pith's one-line read For coefficient vectors on a non-singular hypersurface P(a)=0, almost all associated homogeneous forms f_a(x)=0 satisfy the Hasse principle when n>24d and d≥17, and the share with rational solutions tends to the positive local-solubility…

desk verdict Worth a referee: a real threshold improvement with a new lattice count, but two fixable typos in the proof (big-O vs big-Omega and an invalid binomial inequality) need correcting. read the letter →

arxiv 2506.01291 v3 pith:AJN4OQZT submitted 2025-06-02 math.NT math.AG

Daniel Flores , Kiseok Yeon This is my paper
classification math.NTmath.AG MSC 11E7614G12
keywords HasseprinciplerandomDiophantineequationsthinsetsVeroneseembeddinglatticepointcountingadditiveformsgeometryofnumbershomogeneouspolynomials
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 studies random Diophantine equations whose coefficient vectors are constrained to a thin set: the integer points on a non-singular hypersurface P(a)=0. It proves that, once the number of variables n exceeds 24d and the degree d is at least 17, the proportion of such coefficient vectors for which the equation f_a(x)=0 satisfies the Hasse principle tends to 1 as the coefficient box grows. Moreover, when the local solubility density is positive, the proportion of coefficient vectors giving an actual rational solution tends to a positive constant c_P. The improvement over earlier work comes from a sharper bound on a lattice counting problem that controls the variance of the counting function.

What carries the argument

The engine of the paper is a new lattice counting problem, Lemma 3.1. It bounds the number $N(A,X)$ of integer solutions to the pair of additive equations $\sum_{1\le i\le s} a_i(x_i^d-z_i^d)=\sum_{1\le i\le s} a_i(y_i^d-w_i^d)=0$ with $|a_i|\le A$ and $|x_i|,|y_i|,|z_i|,|w_i|\le X$. The bound is $A^sX^{2s}+A^{s-1}X^{3s-d+\varepsilon}+A^{s-2}X^{4s-2d+\varepsilon}(1+X^{2d-s+3}+X^{4d-2s+4})$. The proof separates solutions where the two vectors of differences are linearly dependent from those where they are independent. In the independent case it uses the geometry of numbers: for fixed $x,y,z,w$, the solutions $a$ form a lattice whose determinant is controlled by the gcd of the $2\times2$ minors $\Delta_{i,j}$, and the count is reduced to the arithmetic sum $\Xi_d(X)=\sum 1/\Delta_{1,2}$. The decisive estimate $\Xi_d(X)\ll X^{8-2d+\varepsilon}$ rests on the gap between distinct $d$-th powers in $[X/2,X]$ being of order $X^{d-1}$. This sharper count, fed into the minor-arc estimate of Lemma 4.1, is what lowers the variable requirement from $n\ge 32d+17$ to $n>24d$.

What would settle it

Evaluate the arithmetic sum $\Xi_d(X)=\sum_{\Delta_{1,2}\ge1} 1/\Delta_{1,2}$ from (3.32) for moderate d and increasing X; the proof's bound is $\Xi_d(X)\ll X^{8-2d+\varepsilon}$, so any observed growth exponent larger than $8-2d$ (for example the trivial $10-2d$) would falsify Lemma 3.1 and with it the variance estimate driving the theorem.

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Extended reading notes

Core claim

The central claim is that the Hasse principle holds for almost all members of the thin family: with P a non-singular integer form of degree k≤d in N variables, the set $\mathfrak{A}(A;P)=\{\boldsymbol{a}\in\mathbb{Z}^N: P(\boldsymbol{a})=0,\ \|\boldsymbol{a}\|_\infty\leq A\}$ has the property that the proportion of $\boldsymbol{a}$ for which $f_{\boldsymbol{a}}(\boldsymbol{x})=0$ satisfies the Hasse principle converges to 1 as $A\to\infty$, provided $n>24d$ and $d\geq17$. If additionally the local solubility density $c_P$ is positive, the same proportion with an actual solution in $\mathbb{Q}^n$ converges to $c_P$. The proof runs through a mean-square estimate for the counting function $I_{\boldsymbol{a}}(X)$ against its expected value $S^*_{\boldsymbol{a}}J^*_{\boldsymbol{a}}$ (Theorem 1.2), a bound on exceptional locally soluble coefficient vectors (Theorem 1.3), and a statement that the proportion with $I_{\boldsymbol{a}}(X)$ abnormally small tends to 0 (Theorem 1.4).

Load-bearing premise

The whole conclusion depends on Lemma 3.1's bound for the number of solutions to the two additive equations; that bound in turn needs the smallest gap between distinct d-th powers in [X/2,X] to be at least a fixed multiple of $X^{{d-1}}$, and the proof writes this as an O-bound rather than the required lower bound.

Editorial extensions

If this is right

  • For every non-singular P of degree k≤d, the proportion of thin-set coefficients producing a form satisfying the Hasse principle is 1 in the limit.
  • If local solubility has positive density c_P, then the proportion with a genuine rational solution is asymptotically c_P, not just close to 1.
  • The mean-square estimate Theorem 1.2 gives a polynomial rate: the variance of I_a(X) around S_a^*J_a^* is O(A^{N-4}X^{2n-2d}(\log A)^{-\delta}).
  • The condition n>24d with d≥17 is a concrete improvement over the earlier n≥32d+17 with d≥14.
  • The argument allows d to be as small as 17, expanding the range of degrees for which the thin-set Hasse principle is known.

Reading between the lines

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

  • A similar lattice-counting treatment could handle coefficient sets defined by several polynomial equations, as long as the corresponding mean values of the exponential sums can be bounded by the same geometry-of-numbers method.
  • The gap estimate behind (3.32) suggests that what matters is not the size of the coefficient box but the separation of pure powers; one might expect an analogous theorem for other additive structures, such as forms with fewer but larger-degree terms.
  • If the exponent 4s-2d in the N_2 bound could be improved by a different treatment of the determinant sum, the condition n>24d could likely be pushed further toward the unconstrained threshold n≥d+1.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies the Hasse principle for random homogeneous polynomials f_a(x) of degree d in n variables, where the coefficient vector a is restricted to the thin set A(A;P) = {a ∈ Z^N : P(a)=0, ||a||_∞ ≤ A} defined by a non-singular form P of degree k ≤ d in N = binom(n+d-1,d) variables. The main claim is that, under the numerical conditions n > 24d and d ≥ 17, the proportion of a ∈ A(A;P) for which f_a(x)=0 satisfies the Hasse principle tends to 1, and if the local solubility density is positive, the proportion with an actual rational point tends to the positive constant c_P. The central new ingredient is Lemma 3.1, a lattice-counting estimate for the number of solutions of two additive equations (3.1); this estimate is fed into a mean-value bound in §4 and then into the minor-arc second-moment estimate behind Theorem 1.2. Theorems 1.3 and 1.4 are then used to convert the variance estimate into the proportion-1 Hasse-principle conclusion.

Significance. If the stated results are correct, the paper improves the second author's earlier condition n ≥ 32d+17 to n > 24d, and the new lattice-counting lemma is a genuine methodological contribution in the style of Brüdern-Dietmann. The paper is clearly written and the structure is standard: geometry-of-numbers input (Lemma 3.1), minor-arc and major-arc bounds (Lemmas 4.1, 5.1, 6.1, 6.2), and a variance estimate. The reliance on the published papers [13] and [9] is explicit and disclosed. The two technical problems identified below are local and appear fixable, but they affect load-bearing points: the proof of Lemma 3.1's key bound as printed is not valid, and the proof of Corollary 1.5 contains a false inequality chain. These prevent the paper from being accepted in its current form.

major comments (2)
  1. [§3, Eqs. (3.33)–(3.34)] The proof of the bound Ξ_d(X) ≪ X^{8-2d+ε} requires a lower bound on the gaps between distinct values of x_2^d - \tilde x_2^d. The text writes min |x_2^d - \tilde x_2^d| ≥ B with B = O(X^{d-1}), but then uses B^{-2} as if B were a lower bound of size X^{d-1}. If B is only subject to B = O(X^{d-1}), it could be arbitrarily small and the chain ∑_{n,m≤X} 1/(1+nmB^2) ≪ ∑_{n≤X^2} 1/(1+nB^2) ≪ B^{-2} does not follow. This invalidates (3.32) as written and hence the N_2 bound (3.34), which is the new lattice-counting input used in (4.8) to obtain the mean-value estimate and ultimately the variance estimate in Theorem 1.2. The defect appears typographical: the correct statement is B ≍ X^{d-1}, and with that replacement the displayed estimates propagate correctly.
  2. [Corollary 1.5] The displayed inequality chain contains the step 1000·8^d·(n+d-1)^2 ≤ binom(n+d-1,d)^d ≤ binom(n+d-1,d) = N_{d,n}. For d ≥ 2 and binom(n+d-1,d) > 1, the inequality x^d ≤ x is false, so the proof that d ≥ 17, k ≤ d and n > 24d imply the hypotheses of Theorems 1.2–1.4 is invalid. This is load-bearing because these numerical conditions are exactly the advertised range of the main theorem. The chain appears to be a typographical/intended-comparison error, but it must be corrected, for example by replacing the final two terms with a direct lower bound on binom(n+d-1,d), or by a different valid comparison.
minor comments (5)
  1. [§1, paragraph after the abstract] The text says 'Let P be a non-singular form in n variables of degree k ≥ 2' and then defines A(A;P) as a subset of Z^N. Since P is evaluated at coefficient vectors a ∈ Z^N, P should be a form in N variables, as stated in the abstract and in Theorem 1.2. Please make this consistent.
  2. [§6, application of Lemma 6.1] In the proof of Theorem 1.2, Lemma 6.1 is applied 'with k=2'. Lemma 6.1's hypotheses and bound depend on the degree k of P, and a non-singular form of degree k cannot be treated by setting k=2. Since the displayed bound in (6.8) correctly retains A^{N-k}, this is presumably a typo and should read 'with k equal to the degree of P'.
  3. [Theorem 1.3 and Lemma 6.2] The proofs of Theorem 1.3 and Lemma 6.2 say that the choice of ζ = w^{-4-1/(8d)} 'does not harm' the arguments in [13, Proposition 5.12] and [13, Lemma 4.2]. Because ζ is a weight parameter that also appears in the final bounds, the authors should either spell out the verification or give precise pointers to the displayed estimates in [13] that are preserved under this change.
  4. [§1, Notation] The sentence introducing x^{(1)}, x^{(2)}, ..., x^{(l)} and the notation X_x!^s appears garbled in the typeset text and should be rephrased for clarity.
  5. [§3, around (3.33)] In (3.33), the symbol \tilde x_2 is used without an explicit definition; it denotes a second value of the variable x_2 in the same range, and this should be stated.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the new lattice-counting lemma is an independent input; self-citations are to published theorems and do not smuggle in the target conclusion.

full rationale

The derivation chain is not circular. The Hasse-principle conclusion is reached through Theorem 1.2 (variance estimate) and Theorem 1.4, whose proof is quoted from [13, Theorem 1.3] under prior, stricter hypotheses; the improvement in variable range is supplied by the new Lemma 3.1, a genuinely new lattice count proved via the geometry of numbers. Lemma 3.1 is not a restatement of the Hasse principle, local solubility, or any fitted constant; it feeds into Lemma 4.1 through the mean value estimate (4.8) and is an original auxiliary estimate. The paper's self-citations [9] and [13] (both involving the second author) are to published, externally checkable theorems and are used as ingredients, not as the conclusion itself; no uniqueness theorem or ansatz is imported from the authors' prior work to forbid alternatives. The quantities I_a(X), S*_a, and J*_a are defined independently, and the variance bound is an estimate rather than a definitional identity. The only questionable passage found, (3.33) writing B=O(X^{d-1}) where the subsequent denominator appears to require a lower bound B ≫ X^{d-1}, is a potential correctness defect in the proof of Lemma 3.1, not a circular step, since it does not assume the target theorem. Hence no circularity is present.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central claim involves no fitted constants from data; the only hand-chosen auxiliary parameter is ζ. Several cited lemmas from [13] and [9] are not rederived, but they are published results rather than assumptions built to force the conclusion. The geometry-of-numbers gap assumption in (3.33) is the least stable ingredient.

free parameters (1)
  • ζ = w^{-4-1/(8d)}
    Auxiliary smoothing parameter in the function w_ζ(β); it is chosen by hand to optimize the balance between major and minor arc estimates and differs from the choice in [13].
assumptions (6)
  • standard math Prime number theorem: log W ≤ 2w, so W ≤ X^2
    Used in Section 1, equation (1.3), to size the auxiliary modulus W.
  • domain assumption Classical major arc analysis from Birch [2, Lemma 5.1] converts the major arc integral into S_a(q)J_a(w) plus a small error
    Invoked in equation (6.2) in the proof of Theorem 1.2.
  • standard math Weil bound for complete exponential sums S(q,a) ≪ q^{1-1/d} from Vaughan [12, Theorem 4.2]
    Used in Section 5, around equation (5.6).
  • domain assumption Estimates from Yeon [13]: Lemma 2.10, Proposition 3.4, Lemmas 4.1 and 4.2, and the structure of Theorems 1.2 and 1.3
    Quoted repeatedly; Theorem 1.3 and Theorem 1.4 are proved only by reference to [13]. Since one author is also the author of [13], this reliance creates a verifiability burden without being circular.
  • domain assumption Local solubility density result from Lee-Lee-Yeon [9]: the proportion of locally soluble coefficient vectors in A(A;P) converges to c_P
    Used in the concluding paragraph of Section 1 to convert the proportion-1 Hasse principle statement into a positive proportion with rational solutions.
  • ad hoc to paper Distinct values x^d-z^d with X/2≤x,z≤X are separated by at least c X^{d-1}
    Needed in equation (3.33) to bound Ξ_d(X); the text writes B=O(X^{d-1}) instead of stating the required lower bound.

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Pith. "Pith review of The Hasse principle for random homogeneous polynomials in thin sets." pith.science (2026). https://pith.science/paper/AJN4OQZT

@misc{pith2026250601291,
  author       = {Pith},
  title        = {Pith review of: The Hasse principle for random homogeneous polynomials in thin sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AJN4OQZT}},
  note         = {Machine review of arXiv:2506.01291}
}
abstract

Let $d$ and $n$ be natural numbers. Let $\nu_{d,n}: \mathbb{R}^n\rightarrow \mathbb{R}^{N}$ denote the Veronese embedding with $N=N_{n,d}:=\binom{n+d-1}{d}$, defined by listing all the monomials of degree $d$ in $n$ variables using the lexicographical ordering. Let $\langle \boldsymbol{a}, \nu_{d,n}(\boldsymbol{x})\rangle\in \mathbb{Z}[\boldsymbol{x}]$ be a homogeneous polynomial in $n$ variables of degree $d$ with integer coefficients $\boldsymbol{a}$, where $\langle\cdot,\cdot\rangle$ denotes the inner product. For a non-singular form $P\in \mathbb{Z}[\boldsymbol{x}]$ of degree $k\ (\leq d)$ in $N$ variables, consider a set of integer vectors $\boldsymbol{a}\in \mathbb{Z}^N$, defined by $$\mathfrak{A}(A;P)=\{\boldsymbol{a}\in \mathbb{Z}^N:\ P(\boldsymbol{a})=0,\ \|\boldsymbol{a}\|_{\infty}\leq A\}.$$ By handling a new lattice problem via the geometry of numbers, we confirm that whenever $n> 24d$ and $d\geq 17,$ the proportion of integer coefficients $\boldsymbol{a}\in \mathfrak{A}(A;P)$, whose associated equation $f_{\boldsymbol{a}}(\boldsymbol{x})=0$ satisfies the Hasse principle, converges to $1$ as $A\rightarrow\infty$. This improves on the recent work of the second author.

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Works this paper leans on

13 extracted references · 13 canonical work pages

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