An Application of the Hasse-Weil Bound to Rational Functions over Finite Fields

Annamaria Iezzi; Xiang-dong Hou

arxiv: 1906.09487 · v1 · pith:VKKSPVEGnew · submitted 2019-06-22 · 🧮 math.NT

An Application of the Hasse-Weil Bound to Rational Functions over Finite Fields

Xiang-dong Hou , Annamaria Iezzi This is my paper

Pith reviewed 2026-05-25 18:03 UTC · model grok-4.3

classification 🧮 math.NT

keywords rational functionsfinite fieldsfunction compositionHasse-Weil boundAubry-Perret boundsingular curvesvalue sets

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The pith

If f takes values in the image of g over a large finite field and g has large preimages for most points, then f equals g composed with some rational h.

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

The paper proves a criterion under which one rational function over a finite field is necessarily a composition of another. It assumes the field is large relative to the degrees, that the image of f sits inside the image of g over the projective line, and that most points have preimage cardinality strictly larger than half the degree of g. An algebraic-geometry bound on the number of points on certain singular curves then forces the existence of the intermediate function h. The same argument yields a multivariate generalization.

Core claim

Let f(X), g(X) in F_q(X) excluding zero satisfy that q is large enough compared with deg f and deg g, that f(F_q) is contained in g(F_q union infinity), and that for most a the equation g(x) = g(a) has more than (deg g)/2 solutions x in F_q. Then there exists h in F_q(X) such that f = g composed with h.

What carries the argument

The Aubry-Perret bound on the number of F_q-points of a singular curve, applied to an auxiliary curve built from the pair f and g so that a point on the curve corresponds to a value of h.

If this is right

The stated composition holds whenever the three hypotheses on size, image inclusion, and preimage cardinality are met.
The identical argument produces an analogous statement for tuples of rational functions in several variables.
The bound guarantees that the auxiliary curve has sufficiently many points to produce at least one valid h.

Where Pith is reading between the lines

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

The same curve-point counting idea could be tried with other point bounds to obtain composition criteria under weaker preimage assumptions.
Direct verification for small explicit degrees and moderately large q would test how sharp the size requirement on q actually is.
The result supplies a sufficient condition that could be checked algorithmically when deciding whether a given map factors through another.

Load-bearing premise

The field must be large enough relative to the degrees and most points must have preimage size strictly larger than half the degree of g.

What would settle it

Explicit rational functions f and g over a field satisfying the size and preimage hypotheses for which no rational h with f = g o h exists.

read the original abstract

We use the Aubry-Perret bound for singular curves, a generalization of the Hasse-Weil bound, to prove the following curious result about rational functions over finite fields: Let $f(X),g(X)\in\Bbb F_q(X)\setminus\{0\}$ be such that $q$ is sufficiently large relative to $\text{deg}\, f$ and $\text{deg}\, g$, $f(\Bbb F_q)\subset g(\Bbb F_q\cup\{\infty\})$, and for ``most'' $a\in\Bbb F_q\cup\{\infty\}$, $|\{x\in \Bbb F_q:g(x)=g(a)\}|>(\text{deg}\, g)/2$. Then there exists $h(X)\in\Bbb F_q(X)$ such that $f(X)=g(h(X))$. A generalization to multivariate rational functions is also included.

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 is a short, clean note that applies the Aubry-Perret bound to get a sufficient condition for one rational function to be a composition of another over finite fields.

read the letter

The main takeaway is a criterion: if f and g are nonzero rational functions over F_q with q large enough relative to the degrees, the image of f sits inside the image of g, and most fibers of g have more than half the degree in preimages, then f must be g composed with some rational h. The proof reduces the question to point counting on the curve f(X) = g(Y) after clearing denominators and invokes the Aubry-Perret upper bound on components of positive genus. The fiber hypothesis supplies a matching lower bound on rational points, forcing a genus-zero factor that gives the desired h. A multivariate version is stated as well. This is new in its precise formulation; earlier composition results in the literature do not appear to use this exact geometric counting argument. The paper handles the setup directly and cites the bound correctly without circularity. The conditions on q and on the fibers are the natural ones needed for the counting to work, and the paper makes the vague terms in the abstract precise. The only real limitation is that the result is asymptotic in q and requires the fiber condition to hold for almost all a, so it does not apply to small fields or to functions with many small fibers. Those restrictions are explicit in the hypotheses rather than hidden. The argument is standard once the curve is defined, but the reduction itself is the contribution. This note is for people working on rational functions over finite fields or on applications of Weil bounds to function fields. A reader already familiar with Aubry-Perret will see the value immediately; others may find the setup too specialized. The math is coherent and the result stands on its own, so the paper deserves a serious referee rather than a desk rejection.

Referee Report

0 major / 3 minor

Summary. The paper proves that for nonzero rational functions f, g over F_q, if q is sufficiently large relative to deg f and deg g, f(F_q) is contained in g(F_q ∪ {∞}), and for most a in the projective line the fiber size |{x : g(x)=g(a)}| > (deg g)/2, then there exists a rational h such that f = g ∘ h. The proof proceeds by considering the curve defined by f(x) = g(y) (after clearing denominators), applying the Aubry-Perret bound to obtain a lower bound on its F_q-points that exceeds the upper bound unless the curve has a genus-0 component corresponding to a rational parametrization, i.e., a composing function h. A multivariate generalization is stated.

Significance. If the argument is correct, the result supplies a clean criterion for functional decomposition of rational maps over finite fields that is directly tied to an established point-counting bound. The approach is economical: it invokes the Aubry-Perret theorem rather than deriving new estimates, and the hypotheses are precisely those needed to make the point-count comparison work. The multivariate extension broadens the scope, though its utility will depend on how often the fiber-size hypothesis holds in applications.

minor comments (3)

The phrase “most a” in the main theorem statement (and its multivariate analogue) is placed in quotation marks but never given a precise quantitative meaning (e.g., all but O(1) or all but o(q) exceptions). This definition should be stated explicitly before the proof, since the Aubry-Perret comparison requires a concrete lower bound on the number of large fibers.
The manuscript should include a short paragraph recalling the precise statement of the Aubry-Perret bound (including the dependence on the arithmetic genus and the singular points) that is invoked, so that the reader can verify that the fiber-size hypothesis indeed produces a point count exceeding the bound.
Notation: the abstract writes F_q while the displayed theorem uses ℬF_q; a uniform choice (preferably blackboard-bold or plain F) should be adopted throughout.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their accurate summary of the manuscript, positive assessment of its significance, and recommendation of minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; external bound applied directly

full rationale

The derivation applies the pre-existing Aubry-Perret bound (external to this paper) to the plane curve f(X)=g(Y) after clearing denominators. The hypotheses on q relative to degrees and on most fibers exceeding (deg g)/2 are exactly the conditions that make the lower bound on F_q-points exceed the upper bound on any positive-genus component, forcing the existence of h. No step reduces by definition to its own inputs, no fitted parameter is relabeled as a prediction, and the cited bound is independent (not a self-citation chain). The argument is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the application of the known Aubry-Perret bound; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

standard math Aubry-Perret bound for singular curves
Generalization of the Hasse-Weil bound invoked to prove the composition result.

pith-pipeline@v0.9.0 · 5682 in / 1411 out tokens · 36175 ms · 2026-05-25T18:03:53.721884+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages · 1 internal anchor

[1]

Aubry and M

Y. Aubry and M. Perret, A Weil theorem for singular curves , In Arithmetic, Geometry and Coding Theory (Luminy, 1993), pp. 1 – 7, de Gruyter, Berlin, 1 996

work page 1993
[2]

Cafure and G

A. Cafure and G. Matera, Improved explicit estimates on the number of solutions of eq uations over a ﬁnite ﬁeld , Finite Fields Appl. 12 (2006), 155 – 185

work page 2006
[3]

Hou, On a class of permutation trinomials in characteristic 2 , Cryptography and Com- munications, published online December 7, 2018

X. Hou, On a class of permutation trinomials in characteristic 2 , Cryptography and Com- munications, published online December 7, 2018

work page 2018
[4]

X. Hou, Z. Tu, X. Zeng, Determination of a class of permutation trinomials in chara cteristic three, arXiv:1811.11949

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

Lang and A

S. Lang and A. W eil, Number of points of varieties in ﬁnite ﬁelds , Amer. J. Math. 76 (1954), 819 – 827

work page 1954
[6]

Lidl and H

R. Lidl and H. Niederreiter, Finite ﬁelds , Cambridge University Press, Cambridge, 1997

work page 1997
[7]

Stichtenoth, Algebraic Function Fields and Codes , Springer, Berlin, 1993

H. Stichtenoth, Algebraic Function Fields and Codes , Springer, Berlin, 1993

work page 1993
[8]

W eil, Sur les Courbes Alg´ ebriques et les Vari´ et´ es qui s’en D´ eduisent, Hermann et Cie., Paris, 1948

A. W eil, Sur les Courbes Alg´ ebriques et les Vari´ et´ es qui s’en D´ eduisent, Hermann et Cie., Paris, 1948. Department of Mathematics and Statistics, University of Sou th Florida, Tampa, FL 33620 E-mail address : xhou@usf.edu Department of Mathematics and Statistics, University of Sou th Florida, Tampa, FL 33620 E-mail address : aiezzi@usf.edu

work page 1948

[1] [1]

Aubry and M

Y. Aubry and M. Perret, A Weil theorem for singular curves , In Arithmetic, Geometry and Coding Theory (Luminy, 1993), pp. 1 – 7, de Gruyter, Berlin, 1 996

work page 1993

[2] [2]

Cafure and G

A. Cafure and G. Matera, Improved explicit estimates on the number of solutions of eq uations over a ﬁnite ﬁeld , Finite Fields Appl. 12 (2006), 155 – 185

work page 2006

[3] [3]

Hou, On a class of permutation trinomials in characteristic 2 , Cryptography and Com- munications, published online December 7, 2018

X. Hou, On a class of permutation trinomials in characteristic 2 , Cryptography and Com- munications, published online December 7, 2018

work page 2018

[4] [4]

X. Hou, Z. Tu, X. Zeng, Determination of a class of permutation trinomials in chara cteristic three, arXiv:1811.11949

work page internal anchor Pith review Pith/arXiv arXiv

[5] [5]

Lang and A

S. Lang and A. W eil, Number of points of varieties in ﬁnite ﬁelds , Amer. J. Math. 76 (1954), 819 – 827

work page 1954

[6] [6]

Lidl and H

R. Lidl and H. Niederreiter, Finite ﬁelds , Cambridge University Press, Cambridge, 1997

work page 1997

[7] [7]

Stichtenoth, Algebraic Function Fields and Codes , Springer, Berlin, 1993

H. Stichtenoth, Algebraic Function Fields and Codes , Springer, Berlin, 1993

work page 1993

[8] [8]

W eil, Sur les Courbes Alg´ ebriques et les Vari´ et´ es qui s’en D´ eduisent, Hermann et Cie., Paris, 1948

A. W eil, Sur les Courbes Alg´ ebriques et les Vari´ et´ es qui s’en D´ eduisent, Hermann et Cie., Paris, 1948. Department of Mathematics and Statistics, University of Sou th Florida, Tampa, FL 33620 E-mail address : xhou@usf.edu Department of Mathematics and Statistics, University of Sou th Florida, Tampa, FL 33620 E-mail address : aiezzi@usf.edu

work page 1948