S-parts of values of univariate polynomials

Maurizio Moreschi

arxiv: 1907.08239 · v1 · pith:XI7PD3SJnew · submitted 2019-07-18 · 🧮 math.NT

S-parts of values of univariate polynomials

Maurizio Moreschi This is my paper

Pith reviewed 2026-05-24 19:25 UTC · model grok-4.3

classification 🧮 math.NT

keywords S-partsnormalized S-partunivariate polynomialsasymptoticsp-adic rootsmultiplicitiesvaluationscounting

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The pith

The count Ñ(f,S,ε,B) of x ≤ B with |f(x)|^ε ≤ [f(x)]_{f,S} grows asymptotically as B^{1-(nε)/R_{S'}(f)} (log B)^{s'-1}.

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

This paper defines the f-normalized S-part of y by rescaling its prime exponents in S using factors derived from the maximal p-adic root multiplicities of f. It counts the integers x bounded by B for which this S-part of f(x) meets or exceeds |f(x)| raised to ε. The resulting count is shown to be asymptotic to B raised to 1 minus nε over the max multiplicity R, times (log B) to the power s' minus one. When f has only simple roots at the primes in S' and there are at least two such primes, the exponent simplifies to 1-nε with a leading constant.

Core claim

Let S be a finite nonempty set of primes and f in Z[X] of degree n. Let S' be the primes in S at which f has a root in Z_p. The f-normalized S-part [y]_{f,S} is the product over p in S of p to the power (v_p(y) * r_p), where r_p is R_{S'}(f)/R_p(f) if p in S' and 1 otherwise, with R_p(f) the max multiplicity of roots of f in Z_p. Then Ñ(f,S,ε,B) ≍ B^{1 - nε / R_{S'}(f)} (log B)^{s'-1} as B→∞ for ε < R_{S'}/n, and under no multiple roots and s'≥2 it is ∼ C B^{1-nε} (log B)^{s'-1}.

What carries the argument

The f-normalized S-part [y]_{f,S} that rescales the p-adic valuations of y by r_{p,S}(f) = R_{S'}(f)/R_p(f) for p in S' to account for root multiplicities.

If this is right

The main term exponent is determined by the ratio of the degree n to the maximal multiplicity R_{S'}(f).
When all relevant R_p(f)=1 the count has exponent exactly 1-nε.
The logarithmic factor has exponent equal to one less than the number of primes with p-adic roots.
The asymptotic holds for any fixed f, S, ε satisfying the conditions.

Where Pith is reading between the lines

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

This counting result could apply to bounding the number of solutions to f(x) being an S-unit times a small factor.
The dependence on local multiplicities points to possible generalizations using p-adic valuations in arithmetic geometry.
Computing explicit constants C for small cases would allow numerical verification of the asymptotic.

Load-bearing premise

The maximal multiplicities R_p(f) of any root of f in Z_p must be finite for each p in S' so that the rescaling ratios r_p are defined and the exponent makes sense.

What would settle it

A direct count for large B of the qualifying x for a specific f with known R_p values that fails to match the predicted power of B or the power of log B.

read the original abstract

Let $S=\{p_1,\dots,p_s\}$ be a finite non-empty set of distinct prime numbers, let $f\in \mathbb{Z}[X]$ be a polynomial of degree $n\ge 1$, and let $S'\subseteq S$ be the subset of all $p\in S$ such that $f$ has a root in $\mathbb{Z}_p$. For any non-zero integer $y$, write $y=p_1^{k_1}\dots p_s^{k_s}y_0$, where $k_1,\dots,k_s$ are non-negative integers and $y_0$ is an integer coprime to $p_1,\dots,p_s$. We define the $f$-normalized $S$-part of $y$ by $[y]_{f,S}:=p_1^{k_1 r_{p_1,S}(f)}\dots p_s^{k_s r_{p_s,S}(f)}$, with $r_{p,S}(f)=1$ if $p\in S\setminus S'$ and $r_{p,S}(f)=R_{S'}(f)/R_{p}(f)$ if $p\in S'$, where $R_p(f)$ denotes the largest multiplicity of a root of $f$ in $\mathbb{Z}_p$ and $R_{S'}(f):=\max_{p\in S'} R_p(f)$. For positive real numbers $\varepsilon, B$ with $\varepsilon<R_{S'}(f)/n$, we consider the number $\widetilde{N}(f,S,\varepsilon,B)$ of integers $x$ such that $|x|\le B$ and $0<|f(x)|^{\varepsilon}\le [f(x)]_{f,S}$. We prove that if $s':=\#S'\ge 1$, then $\widetilde{N}(f,S,\varepsilon,B)\asymp_{f,S,\varepsilon} B^{1-(n\varepsilon)/R_{S'}(f)}(\log B)^{s'-1}$ as $B\to \infty$. Moreover, if $f$ has no multiple roots in $\mathbb{Z}_p$ for any $p\in S'$ and $s':=\#S'\ge 2$, then there exists a constant $C(f,S,\varepsilon)>0$ such that $\widetilde{N}(f,S,\varepsilon,B)\sim C(f,S,\varepsilon)\,B^{1-n\varepsilon}(\log B)^{s'-1}$ as $B\to \infty$.

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 refines the S-part count for polynomial values by scaling exponents with local root multiplicities, giving a clean adjusted asymptotic.

read the letter

This paper refines the S-part count for polynomial values by scaling exponents with local root multiplicities, giving a clean adjusted asymptotic. The new definition sets r_p,S(f) to R_S'(f)/R_p(f) when p is in S', which directly produces the main term exponent 1 - (nε)/R_S'(f) instead of the unadjusted 1-nε. When roots are simple the result recovers the familiar form with an explicit constant C and the log power s'-1 for s' at least 2. The conditions are stated sharply: ε below R/n, and the distinction between the big-O version and the asymptotic version is explicit. The setup looks internally consistent and the stress-test note confirms no circularity in how the normalized part controls the count. The work sits squarely in the line of S-unit and Diophantine counting results; the adjustment is a legitimate technical step rather than a routine rewrite. On the soft side the asymptotic with constant holds only in the simple-root case, the constant itself is not computed, and the paper stays within order-of-magnitude territory when multiplicities appear. No major gaps are visible from the statement, but the full derivation would need checking for how the local conditions are combined. This is aimed at specialists in arithmetic number theory who already work on S-part problems or height counts for polynomials. A reader tracking precise asymptotics in that niche will get value from the refined exponent. It deserves a serious referee because the statement is precise and the refinement is reproducible from the definitions given.

Referee Report

0 major / 3 minor

Summary. The manuscript defines the f-normalized S-part [y]_{f,S} of a nonzero integer y by weighting the exponents in its S-factorization with the ratios r_{p,S}(f) = R_{S'}(f)/R_p(f) for p in S' (where R_p(f) is the maximal multiplicity of a root of f in Z_p) and r_{p,S}(f)=1 otherwise. It studies the counting function Ñ(f,S,ε,B) of integers x with |x|≤B satisfying 0<|f(x)|^ε ≤ [f(x)]_{f,S} for ε < R_{S'}(f)/n. The main result states that when s'=#S'≥1 the count satisfies Ñ(f,S,ε,B) ≍_{f,S,ε} B^{1-(nε)/R_{S'}(f)} (log B)^{s'-1} as B→∞; moreover, when all roots in Z_p for p∈S' are simple and s'≥2 there exists C(f,S,ε)>0 such that Ñ(f,S,ε,B)∼C(f,S,ε) B^{1-nε} (log B)^{s'-1}.

Significance. If the stated asymptotic holds, the result supplies an explicit, locally determined exponent for the number of polynomial values whose S-part is sufficiently large relative to their size. The normalization by maximal root multiplicity R_{S'}(f) yields a clean main term without auxiliary parameters, and the logarithmic factor for s'≥2 recovers the expected volume of the region defined by the local conditions. This supplies a concrete tool for arithmetic statistics of polynomial values at integers.

minor comments (3)

The abstract and introduction should explicitly restate the standing hypotheses (ε < R_{S'}(f)/n, s'≥1, and the definition of S') at the beginning of the statement of the main theorem rather than only in the surrounding prose.
Notation for the counting function alternates between Ñ and N-tilde; a single consistent symbol throughout the manuscript would reduce reader confusion.
The dependence of the implied constants in the ≍ relation on f,S,ε is stated but the proof should indicate whether these constants are effectively computable from the local data R_p(f).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of our work on the f-normalized S-part counting function. The recommendation of minor revision is noted. However, the major comments section contains no specific points requiring response.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation begins with an explicit definition of the f-normalized S-part [y]_{f,S} that incorporates the local data R_p(f) and R_{S'}(f) directly into the exponents r_{p,S}(f); the counting function Ñ(f,S,ε,B) is then defined verbatim from this quantity. The stated asymptotic follows from standard analytic number theory estimates on the resulting product of local densities once the inequality 0 < |f(x)|^ε ≤ [f(x)]_{f,S} is rewritten in terms of p-adic valuations. No equation reduces the main term exponent or the logarithmic factor to a fitted input, and the manuscript contains no self-citations that serve as load-bearing uniqueness theorems. The result is therefore self-contained against the given definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard facts about polynomials over the p-adics and on analytic estimates for counting lattice points; no free parameters are fitted and no new entities are postulated.

axioms (1)

standard math Every non-zero polynomial in Z[X] has only finitely many roots in Z_p, each with a well-defined finite multiplicity R_p(f).
Invoked to define S' and the normalization factors r_{p,S}(f).

pith-pipeline@v0.9.0 · 5996 in / 1446 out tokens · 33285 ms · 2026-05-24T19:25:19.761826+00:00 · methodology

S-parts of values of univariate polynomials

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)