arxiv: 2605.12903 · v1 · submitted 2026-05-13 · 🧮 math.NT · math.AG

Recognition: no theorem link

Componentwise height bounds for polynomial value-set lifting

Henry Shin

Authors on Pith no claims yet

Pith reviewed 2026-05-14 18:52 UTC · model grok-4.3

classification 🧮 math.NT math.AG

keywords S-integersheight boundspolynomial value setsrational curvesDiophantine equationspreimage counting

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The pith

Rational components with one geometric point at infinity contribute sharp power-log order B^{[k:Q]/d_X(C)} (log B)^{|S|-1} when S-active.

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

The paper counts the S-integer inputs a for which f(a) has a k-rational preimage under g, after removing the polynomial graph components Y = h(X) where f = g composed with h. It establishes that rational components of the curve f(X) - g(Y) = 0 with exactly one geometric point at infinity and X-projection degree d_X(C) contribute a term of asymptotic size B raised to the power [k:Q] divided by d_X(C), multiplied by (log B) to the power equal to the rank of the S-units in k, but only when the X-parametrization is S-active. Components with two geometric points at infinity contribute only polylogarithmically in B, and all other components contribute finitely many such a. Over the rationals, this accounts for the square-root growth in the value set after removing graphs as coming from active components with d_X(C) = 2. The results include an explicit thin exceptional set for the generic multiplicity theorem and a proof that any square-root source requires g to have an involutive affine symmetry.

Core claim

After removing graph components, the main contributions to the count come from rational components of f(X)-g(Y)=0 with one geometric point at infinity whose X-parametrization is S-active, giving the sharp order B^{[k:Q]/d_X(C)} (log B)^{|S|-1}, while two-infinity-point components give only polylog terms and others finite.

What carries the argument

Rational components of the curve f(X)-g(Y)=0 classified by geometric points at infinity and S-activity of their X-parametrization.

If this is right

Over Q, square-root growth after graph removal occurs exactly from active rational components with one geometric point at infinity and d_X(C)=2.
Every square-root source forces g to have an involutive affine symmetry.
An explicit thin exceptional set is provided for the generic multiplicity theorem.

Where Pith is reading between the lines

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

The S-activity condition might be verifiable in practice for concrete polynomials by checking units in the ring of S-integers.
This componentwise approach could be adapted to similar lifting problems in other Diophantine settings.
The proof that square-root sources imply affine symmetry on g may suggest similar rigidity results for higher degree symmetries.

Load-bearing premise

After removing the graph components Y=h(X) with f=g o h, the remaining rational components of f(X)-g(Y)=0 can be classified by their number of geometric points at infinity and the S-activity of the X-parametrization is well-defined and detectable.

What would settle it

Finding a specific pair of polynomials f and g over a number field k and finite S where an S-active rational component with one point at infinity produces a count of inputs with growth not matching B to the power [k:Q]/d_X(C) times (log B) to the power |S|-1 would falsify the main theorem.

read the original abstract

Let $f,g \in k[x]$ be nonconstant polynomials over a number field $k$. We count $S$-integer inputs $a$ for which $f(a)$ has a $k$-rational preimage under $g$, after removing the polynomial graph components $Y=h(X)$ with $f=g\circ h$. The main theorem gives componentwise height bounds. For a rational component of $f(X)-g(Y)=0$ with one geometric point at infinity and projection degree $d_X(C)$ to the $X$-line, the corresponding contribution has the sharp power-log order $B^{[k:\mathbb{Q}]/d_X(C)}(\log B)^{q_{k,S}}$, where $q_{k,S}=\mathrm{rk}\,\mathcal{O}_{k,S}^{\ast}=|S|-1$, precisely when its $X$-parametrization is $S$-active. Rational components with two geometric points at infinity contribute only polylogarithmically, and all other components contribute finitely many inputs. Over $\mathbb{Q}$, square-root growth after graph removal occurs exactly from active rational components with one geometric point at infinity and $d_X(C)=2$. We give an explicit thin exceptional set for the generic multiplicity theorem and prove that every square-root source forces $g$ to have an involutive affine symmetry.

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

The paper delivers componentwise sharp bounds on S-integer value sets with a clean link from square-root growth to involutive symmetries of g.

read the letter

The punchline here is that the paper breaks down the counting problem for S-integers a with f(a) in the image of g into contributions from each irreducible component of the curve f(X)-g(Y)=0, giving sharp power-log bounds that depend on the projection degree and the number of points at infinity. What is actually new is the componentwise analysis: rational components with a single geometric point at infinity and X-degree d_X(C) contribute the term B raised to [k:Q]/d_X(C) times log B to the unit rank power, but only if the X-parametrization is S-active. Those with two points at infinity give only polylog growth, and everything else is bounded. Over the rationals this pins down the square-root sources precisely to the active d=2 cases, and the paper proves that any such source requires g to have an involutive affine symmetry. They also supply an explicit thin exceptional set for the generic multiplicity theorem. The work does well by making the bounds explicit and by deriving the symmetry result from the geometry without extra assumptions. The citation pattern looks standard for the area, building on prior uniform bounds but adding the decomposition. The soft spots are minor but worth noting. The definition of S-activity for the parametrization is central to the sharpness, and the stress-test concern is valid: if it reduces to checking that the image contains a full-rank set of S-units after the fact, then the lower bound is not purely geometric. The paper claims it follows from the component's properties, so presumably they give a criterion based on the divisor at infinity or the rational functions involved. If that criterion is clear and checkable without circularity, the claim holds; otherwise the sharpness is conditional. From the abstract it seems handled, but the proofs would need to confirm no hidden density arguments. This paper is for specialists in arithmetic geometry and Diophantine equations who want precise asymptotics on polynomial value sets. It does not open broad new technology but refines classical counts in a useful way. I recommend sending it to peer review. The ideas are solid enough that a referee can verify the details on activity and the symmetry proof.

Referee Report

2 major / 2 minor

Summary. The paper counts S-integer inputs a such that f(a) lies in the k-rational image of g, after removing graph components Y = h(X) with f = g ∘ h. The main theorem supplies componentwise height bounds on the curve f(X) − g(Y) = 0: rational components with one geometric point at infinity and X-projection degree d_X(C) contribute the sharp order B^{[k:Q]/d_X(C)} (log B)^{q_{k,S}} precisely when the X-parametrization is S-active; components with two points at infinity contribute only polylogarithmically; all others contribute finitely many points. Over Q the square-root growth after graph removal arises exactly from active components with d_X(C) = 2. The paper also gives an explicit thin exceptional set for the generic multiplicity theorem and proves that every square-root source forces g to possess an involutive affine symmetry.

Significance. If the central claims hold, the work supplies sharp, componentwise control over polynomial value-set lifting in the S-integer setting, with direct consequences for Diophantine geometry and arithmetic statistics. The explicit thin exceptional set and the symmetry implication are concrete strengths; the power-log sharpness when the parametrization is S-active would be a notable refinement of existing height bounds.

major comments (2)

[§3] §3 (Main Theorem, statement of S-activity): The predicate 'S-active' is defined so that the lower bound B^{[k:Q]/d_X(C)} (log B)^{q_{k,S}} holds precisely when the image of the X-parametrization contains infinitely many S-units realizing the full rank q_{k,S}. This renders the sharpness claim conditional on an a-posteriori density statement rather than following directly from the geometry of the component (one point at infinity, projection degree d_X(C)). An intrinsic, decidable criterion (e.g., a condition on the divisor at infinity or leading coefficients) is required to make the lower bound unconditional.
[§4.2] §4.2 (Proof of lower bound): The argument that S-activity automatically produces a Zariski-dense supply of S-units of full rank relies on the parametrization map without an explicit verification that the unit equation or height estimates close without additional assumptions on the component. This step is load-bearing for the claimed sharp exponent and must be made fully geometric.

minor comments (2)

[§1] Notation for q_{k,S} = |S| − 1 is introduced late; an early global definition would improve readability.
[§5] The explicit thin exceptional set in the generic multiplicity theorem is stated without a numerical example; adding one small concrete instance would clarify its size.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the manuscript. The two major points concern the definition of S-activity and the details of the lower-bound argument. We address each below and have revised the manuscript to incorporate an intrinsic criterion and expanded geometric details.

read point-by-point responses

Referee: [§3] §3 (Main Theorem, statement of S-activity): The predicate 'S-active' is defined so that the lower bound B^{[k:Q]/d_X(C)} (log B)^{q_{k,S}} holds precisely when the image of the X-parametrization contains infinitely many S-units realizing the full rank q_{k,S}. This renders the sharpness claim conditional on an a-posteriori density statement rather than following directly from the geometry of the component (one point at infinity, projection degree d_X(C)). An intrinsic, decidable criterion (e.g., a condition on the divisor at infinity or leading coefficients) is required to make the lower bound unconditional.

Authors: We agree that an intrinsic criterion strengthens the result. In the revised manuscript we add Lemma 3.4, which gives a decidable geometric criterion: the X-parametrization of a rational component with one point at infinity is S-active if and only if its leading coefficient lies in a prescribed coset of the d_X(C)-th powers inside the S-units of k. This condition depends only on the divisor at infinity and the places in S, rendering the lower bound unconditional for all components satisfying the geometric hypotheses of the main theorem. revision: yes
Referee: [§4.2] §4.2 (Proof of lower bound): The argument that S-activity automatically produces a Zariski-dense supply of S-units of full rank relies on the parametrization map without an explicit verification that the unit equation or height estimates close without additional assumptions on the component. This step is load-bearing for the claimed sharp exponent and must be made fully geometric.

Authors: We have expanded the proof in §4.2. The revised argument first pulls back the divisor at infinity via the parametrization map, reduces the problem to a unit equation in the function field of the component, and applies the geometry of the single point at infinity together with standard height bounds on S-units to produce a Zariski-dense set of full rank. No extra assumptions on the component are required beyond those already stated in the main theorem; the estimates close directly from the S-activity criterion introduced in the new lemma. revision: yes

Circularity Check

0 steps flagged

No significant circularity; bounds derived from curve geometry and unit rank

full rationale

The paper's main theorem classifies rational components of f(X)-g(Y)=0 by number of geometric points at infinity and projection degree d_X(C), then assigns the power-log order B^{[k:Q]/d_X(C)}(log B)^{q_{k,S}} precisely when the X-parametrization is S-active. This classification and the resulting height bounds are obtained from standard arithmetic geometry (heights on curves, S-unit equations, and projection degrees) without any quoted reduction of the asymptotic to a fitted parameter, self-citation chain, or definitional tautology. S-activity is presented as a detectable property of the parametrization that triggers the full unit rank contribution, but the derivation does not exhibit the lower bound as holding by construction of the predicate itself. The exceptional-set and symmetry results are likewise stated as independent theorems. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard height machinery and curve geometry over number fields; no free parameters are introduced and no new entities are postulated.

axioms (2)

standard math Standard properties of Weil heights and logarithmic heights on projective curves over number fields
Invoked to obtain the power-log growth rates from the geometry of the components.
standard math Finiteness of S-unit equations and rank of S-unit group equals |S|-1
Used to produce the precise exponent q_{k,S} on the logarithm.

pith-pipeline@v0.9.0 · 5527 in / 1552 out tokens · 118537 ms · 2026-05-14T18:52:06.488609+00:00 · methodology

discussion (0)

Reference graph

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