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Counting rational points on transcendental curves in valued fields

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A transcendental definable curve in a 1-h-minimal valued field of mixed characteristic has at most $c_\varepsilon H^\varepsilon$ rational points of height at most H.

desk verdict A genuinely new scaling-map trick gives a Pila-Wilkie bound for curves in arbitrary 1-h-minimal mixed-characteristic fields, but the proof has a load-bearing gap in the ball-counting step that needs fixing. read the letter →

arxiv 2506.19411 v2 pith:ADEWXWOV submitted 2025-06-24 math.NT math.LO

Floris Vermeulen This is my paper · ORCID
classification math.NTmath.LO
keywords rationalpointstranscendentalcurvesvaluedfields1-h-minimalityheightboundsdeterminantmethodparametrizationsmixedcharacteristic
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 proves a counting theorem for rational points on transcendental curves in arbitrary valued fields of mixed characteristic that carry a 1-h-minimal structure: for every $\varepsilon>0$, the number of rational points of height at most $H$ on such a curve is bounded by $O_\varepsilon(H^\varepsilon)$. The theorem extends to this general setting a bound previously known for $p$-adic fields with analytic structure, and it brings the non-archimedean world into line with the classical o-minimal counting result. The proof covers the rational points on the curve by few algebraic curves of large degree, combining the determinant method with parametrizations by scaling maps instead of $r$-th power maps.

What carries the argument

The central object is the $r$-parametrizing map: a definable function $f:U \subset \mathcal{O}_K \to \mathcal{O}_K$ whose derivatives satisfy $|f^{(i)}(x)| \le |M|^{-i}|x-c|^{-i}$ on balls prepared by a centre $c$ and integer $M$. Lemma 3.3 asserts that every definable curve is a finite union of graphs of such maps, and Lemma 3.5 shows that composing an $r$-parametrizing map with the scaling map $s_a(x)=(a-c)(1+p^M x)+c$ turns it into a $T^r$ map, meaning uniform Taylor approximation and derivative bounds. This scaling trick replaces the $r$-th power substitutions used in earlier $p$-adic work and feeds the determinant estimate of Lemma 4.3, which bounds a determinant of values of $T^r$ maps by the ball radius to the power $e=\binom{r}{2}$. That estimate is what forces the rational points to lie on few algebraic curves.

What would settle it

Find a 1-h-minimal valued field of mixed characteristic and a definable transcendental curve whose rational-point count grows faster than every $H^\varepsilon$, for example roughly $H^c$ with $c>0$; such a curve would refute Theorem 1.2. At the lemma level, a single pair of $T^r$ maps whose product or scaling fails to be $T^r$ would break the determinant step.

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Extended reading notes

Core claim

Let $K$ be a valued field of mixed characteristic $(0,p)$ equipped with a 1-h-minimal structure, and let $C \subset K^n$ be a definable curve whose intersection with every algebraic curve is finite; call such a curve transcendental. Theorem 1.2 states that $\#C(\mathbb{Q},H) \le c_\varepsilon H^\varepsilon$ for every $\varepsilon>0$, where $C(\mathbb{Q},H)$ is the set of rational points on $C$ with numerator and denominator bounded by $H$. The paper proves this by first projecting $C$ to a planar curve that is non-algebraic up to a high degree, then decomposing it into graphs of $r$-parametrizing functions, and then applying the determinant method to confine all rational points of bounded height to few algebraic curves. Since each algebraic curve meets a transcendental curve in only finitely many points, the sub-polynomial bound follows.

Load-bearing premise

The proof depends on the full tame-geometry package on the valued field: definable curves decompose into finitely many graphs with controlled derivatives and Taylor approximations, and definable choice is available; if any part of that package fails, the counting argument stops.

Editorial extensions

If this is right

  • If Theorem 1.2 is correct, every transcendental definable curve in a mixed-characteristic 1-h-minimal valued field has sub-polynomial rational point counts, matching the classical o-minimal bound in a new setting.
  • The form of the constant $c = M p^\alpha$, with $M$ and $\alpha$ independent of the field, gives a uniform-in-$p$ version and an extension to large positive characteristic once the characteristic is large enough for the curve.
  • A compactness argument turns the single-curve bound into a bound for definable families of transcendental curves with a constant independent of the parameter.
  • At the level of a single parametrizing map, the rational points of height at most $H$ lie on at most $O(H^\varepsilon)$ algebraic curves of degree $d$, and each such algebraic curve contributes only finitely many points.

Reading between the lines

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

  • A concrete next test is the dimension-two version of Lemma 4.2: the paper identifies bounding the number of twisted boxes that contain rational points as the main obstacle, and a counterexample or a proof for restricted derivative growth would show whether the method extends to definable surfaces.
  • Since the proof avoids $r$-th power maps and is insensitive to the residue field and value group, it suggests that the sub-polynomial bound is a feature of the definable structure itself; testing a valued field where power maps are not definable would delimit exactly how general the mechanism is.
  • The reliance on algebraic Skolem functions could be probed directly: if a 1-h-minimal structure without definable choice still satisfies Lemma 3.3, the theorem would transfer unchanged, and finding such a structure would separate the choice assumption from the counting phenomenon.
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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

3 major / 4 minor

Summary. The paper proves Theorem 1.2: for any 1-h-minimal valued field K of mixed characteristic (0,p), every transcendental definable curve C⊂K^n has at most O_{C,\varepsilon}(H^\varepsilon) rational points of height at most H. The proof strategy is to project C to a planar curve (Lemma 3.2), decompose it into finitely many graphs of functions satisfying strong derivative and Taylor bounds (Lemma 3.3), use scaling maps to produce T^r maps (Lemma 3.5), apply a Bombieri–Pila type determinant method to cover the rational points by few algebraic curves (Lemmas 4.3–4.4), and finally control the number of relevant balls with Lemma 4.2. The claimed bound is uniform in the residue field and value group, and applies for example to C_p with an analytic structure.

Significance. If the missing technical steps are supplied, this is a significant result: it extends the Pila–Wilkie counting philosophy and the Cluckers–Comte–Loeser p-adic results to arbitrary 1-h-minimal fields of mixed characteristic, and it is the first bound of this kind in such general valued fields. The proof strategy is attractive because it avoids r-th power maps and instead uses scaling maps, following the spirit of Binyamini–Novikov–Zak. The paper is not fully self-contained, relying on the 1-h-minimal machinery from [6,7] and on a projection lemma from [3], but this is made explicit and is acceptable for a research note. The explicit dependency of the constants on p and the curve is a useful additional feature.

major comments (3)
  1. [Section 4, Lemma 4.1] The proof of Lemma 4.1 is not valid as written. The assumption gives that the xi-c are distinct modulo p^{N+1}δ, and from this the proof concludes that the rational numbers xi-x1 are distinct modulo p^{N+1}δ and hence that m≤#(Z/p^{N+1}Z)=p^{N+1}. This does not imply the claimed bound m≤p^N+1, and more importantly the counting of distinct classes by Z/p^{N+1}Z is unjustified for arbitrary K of mixed characteristic: if the residue field is infinite, the quotient O_K/(p^{N+1}δ O_K) is infinite, and the classes of rational numbers inside a ball around c need not be parametrized by Z/p^{N+1}Z. What is needed is a packing argument showing that the rational points in B(c,δ) lie in a single ball over Q_p and that their separation condition leaves at most O(p^N) of them; no such argument is supplied. Since Lemma 4.2 and hence the H^ε ball count in Lemma 4.4 depend on this lemma, the main proof has a load-bearing gap at this point. The gap appears repairable, but it must be fixed explicitly.
  2. [Section 4, paragraph before Lemma 4.4] The assertion that 'Hence also s_a^i g^j is T^r for any i,j∈N' is made without proof and is not automatic. The class T^r as defined in Definition 2.3 is not closed under products with the stated constant 1: for a product of two T^r maps, the l-th derivative can involve l+1 terms, and the bound |(fg)^{(l)}|≤|l!| is not generally preserved. This matters because Lemma 4.3 uses the T^r bound |β_{i,j,k}|≤1 to obtain the determinant estimate, and Lemma 4.4 then applies the estimate to the functions s_a^i g^j. The author should either prove that these specific products satisfy the T^r bounds, or prove a variant of Lemma 4.3 with an extra constant factor and check that such a constant is harmless in the final count.
  3. [Section 4, Lemma 4.2 and Lemma 4.4] The ball-counting constants are stated inconsistently: Lemma 4.1 gives m≤p^N+1, Lemma 4.2 states a bound with a factor that is printed as 2Mp^N+1, and the proof of Lemma 4.4 later refers to 2Mp^{N+2}(1+2log_p H). These notational discrepancies should be harmonized. More substantively, Lemma 4.2 also asserts without proof that the number of possible values |c-x| for rationals x of height at most H is at most 2+4log_p(H); this estimate should be justified explicitly, since it is part of the logarithmic factor used in the final count.
minor comments (4)
  1. [Section 3, Lemma 3.5] The proof that f∘s_a inherits Taylor approximation of order r-1 is stated in one sentence; a short verification using the chain rule and the definition of an r-parametrizing map would make the argument easier to check.
  2. [Proof of Theorem 1.2] The claim that π restricted to C is finite-to-one 'otherwise C would contain a subset {a}×B' uses the 1-h-minimal fact that every infinite definable subset of O_K contains a ball; this fact should be cited or stated.
  3. [Section 2] The reduction to algebraic Skolem functions is justified by reference to [7, Prop. 3.2.3], but the sentence 'This is no loss in generality' would benefit from an explanation of why the rational-point count is preserved under the relevant extension.
  4. [Section 4, Lemma 4.3] In the expansion of the determinant, the case ℓ_j=0 should be addressed explicitly: then column j equals the first column, so the corresponding determinant vanishes; this is needed for the conclusion that non-zero terms must have sum of ℓ_j at least e.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main counting theorem is an independent determinant-method argument; self-citations supply framework theorems rather than the target result.

full rationale

The central claim (Theorem 1.2) is not reduced to its inputs by construction. The proof's new content is the scaling-map parametrization plus a determinant-method count (Section 4), and no parameter is fitted to the rational points being counted. The earlier results cited by the author ([3], [7], [8]) provide the 1-h-minimal framework—Jacobian property, Taylor approximation, cell decomposition, and the planar projection lemma—whose assumptions do not include Theorem 1.2 and whose content is independent of the point count. That is ordinary reliance on prior theorems, not circularity. The possible flaw in Lemma 4.1's packing bound is an internal correctness issue, not a circular dependence, and per the hard rules does not affect the circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper adds no fitted numbers or new ontological entities. Its load-bearing assumptions are the axioms of 1-h-minimality and the imported theorems from the author's prior work, most importantly the Jacobian property, Taylor approximation, algebraic Skolem functions, and the planar projection lemma.

assumptions (5)
  • domain assumption Th_L(K) is 1-h-minimal
    The whole paper works inside this framework; it gives dimension theory, cell decomposition, and uniform finiteness used throughout.
  • domain assumption Th_L(K) has algebraic Skolem functions
    Section 2 assumes this and cites [7, Prop. 3.2.3] for why it is no loss of generality; Lemma 3.3 uses it to write definable curves as unions of graphs.
  • domain assumption Jacobian property (Theorem 2.1) and Taylor approximation (Theorem 2.2)
    Imported from [7]; these give the derivative bounds and Taylor estimates that define r-parametrizing maps in Lemma 3.3.
  • domain assumption Lemma 3.2 (planar projection)
    Cited from [3, Lem. 4.2.2]; it lets the author reduce to planar curves non-algebraic up to degree d, a step essential for the determinant method.
  • domain assumption Closure of T^r maps under multiplication and powers
    Section 4 asserts that s_a^i g^j is T^r for the monomials used in the determinant; this is not proved in the paper.

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Pith. "Pith review of Counting rational points on transcendental curves in valued fields." pith.science (2026). https://pith.science/paper/ADEWXWOV

@misc{pith2026250619411,
  author       = {Pith},
  title        = {Pith review of: Counting rational points on transcendental curves in valued fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ADEWXWOV}},
  note         = {Machine review of arXiv:2506.19411}
}
abstract

We prove upper bounds on the number of rational points on transcendental curves in arbitrary $1$-h-minimal fields, similar to the Pila--Wilkie counting theorem in the o-minimal setting. These results extend results due to Cluckers--Comte--Loeser from $p$-adic fields to arbitrary valued fields of mixed characteristic. Our methods rely on parametrizations, where we avoid the usage of $r$-th power maps, combined with the determinant method.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Strong stratifications and uniform Yomdin-Gromov parametrizations in valued fields with analytic structure

    math.LO 2025-05 conditional novelty 6.0 of 10

    Definable sets in equicharacteristic zero valued fields with analytic structure admit uniform T^r Yomdin-Gromov parametrizations with bound s times b_r^m.

Reference graph

Works this paper leans on

11 extracted references · 10 canonical work pages · cited by 1 Pith paper

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