Research manuscript by Brian M. Woody. This page presents a finite field theory research manuscript on Dickson trace curves, reciprocal quadrinomials, permutation polynomials, and asymptotic sparsity over the quadratic finite field extension \(\mathbb{F}_{q^2}\).

The paper develops a general framework for studying a coefficient-linked reciprocal family of sparse polynomials over finite fields. It builds on the complete degree-five classification in A Complete Classification of a Reciprocal Degree-Five Quadrinomial Family over \(\mathbb{F}_{q^2}\) and extends the collision method to a general odd-degree Dickson trace curve setting.

This research is part of Brian Woody’s broader work in finite field research, finite field theory, Galois fields, cryptography, and abstract algebra. It also connects naturally to abstract algebra, field extensions, Galois Theory, quotient groups, the Galois group of \(x^3-2\), linear algebra, real analysis, character sums, algebraic curves over finite fields, and advanced university-level mathematics.

Dickson Trace Curves and Reciprocal Quadrinomials over Finite Fields

The manuscript studies the coefficient-linked reciprocal family

\[
F_{n,a,b}(x)=x^n+a x^{q+n-1}+b x^{(n-1)q+1}+\frac ab x^{nq},
\]

over \(\mathbb{F}_{q^2}\), where \(q\) is an odd prime power. This family can be written in the root-of-unity reduction form

\[
F_{n,a,b}(x)=x^n h_{n,a,b}(x^{q-1}),
\]

where

\[
h_{n,a,b}(z)=1+az+bz^{n-1}+\frac ab z^n.
\]

The reciprocal structure creates a universal collision identity. Outside the natural degenerate cases \(b=0,\pm1\) and \(a=\pm b\), all off-diagonal collisions on the unit circle \(\mu_{q+1}\) are independent of the parameter \(a\). This means the main obstruction to permutation behavior can be studied through the parameter \(b\) and a trace curve over \(\mathbb{F}_q\).

Abstract

We study a coefficient-linked reciprocal family of sparse polynomials over \(\mathbb{F}_{q^2}\). The reciprocal structure gives a universal collision identity: outside the natural degenerate cases \(b=0,\pm1\) and \(a=\pm b\), all off-diagonal collisions on the unit circle are independent of the parameter \(a\).

For odd degrees \(n=2m+1\), the collision equation reduces to a Dickson trace curve. This produces an effective exceptional polynomial \(E_{n,q}(b)\). When \(\operatorname{char}\mathbb{F}_q\nmid m\) and \(E_{n,q}(b)\neq0\), the paper proves an explicit lower bound for the number of strict lifted collisions.

Consequently, for each fixed odd \(n\), all sufficiently large-field permutation examples in this family must lie over the exceptional locus \(E_{n,q}(b)=0\). In particular, the possible permutation members among the nondegenerate parameter pairs have density \(O_n(1/q)\).

As a first explicit cubic trace-curve case, the paper analyzes \(n=7\), computes a degree-thirteen exceptional polynomial \(E_7(b)\), and proves a generic non-permutation theorem for sufficiently large \(q\) in characteristic not \(3\).

Main Results

The manuscript proves three central results for reciprocal quadrinomials over finite fields:

  • Universal collision identity: for the reciprocal quadrinomial family, off-diagonal collision behavior on the unit circle is independent of \(a\) outside the natural degenerate cases.
  • Effective generic collision theorem: for odd \(n=2m+1\), if \(E_{n,q}(b)\neq0\), then a quantitative lower bound forces collisions for sufficiently large \(q\).
  • Asymptotic sparsity: for each fixed odd \(n\), possible permutation members in the nondegenerate parameter space have density \(O_n(1/q)\).

Dickson Trace Curve Framework

For odd \(n=2m+1\), the collision equation can be expressed using Dickson polynomials. If \(D_j\) is defined by

\[
D_j(t+t^{-1})=t^j+t^{-j},
\]

and

\[
S_m(X)=1+\sum_{j=1}^{m}D_j(X),
\]

then the trace collision curve is

\[
\mathcal C_{n,b}:\quad bD_m(W)+S_m(V)-b^2S_{m-1}(V)=0.
\]

This Dickson trace curve controls strict lifted collisions on the unit circle \(\mu_{q+1}\). The first odd-degree cases have a clean geometric progression:

  • \(n=3\): a line.
  • \(n=5\): a conic.
  • \(n=7\): a cubic.
  • \(n=9\): a quartic.

This is the structural bridge between sparse permutation polynomials and algebraic curves over finite fields.

Asymptotic Sparsity over Finite Fields

The main sparsity theorem says that, for fixed odd \(n\), sufficiently large finite fields force possible permutation examples into a small exceptional \(b\)-locus. Outside that exceptional algebraic set, the trace curve produces strict lifted collisions, so the polynomial cannot permute \(\mathbb{F}_{q^2}\).

This gives a strong structural conclusion: possible permutation members are asymptotically sparse in the nondegenerate parameter space. As \(q\) grows, the proportion of possible permutation examples tends to zero at the rate \(O_n(1/q)\).

The Generic Degree-Seven Case

The paper gives an explicit worked case for \(n=7\). The trace curve becomes the cubic

\[
\mathcal E_b:\quad
b(W^3-3W)+V^3+(1-b^2)V^2-(b^2+2)V+(b^2-1)=0.
\]

The exceptional polynomial is

\[
E_7(b)=b(b^2-1)(25b^2-49)Q_+(b)Q_-(b),
\]

where

\[
Q_+(b)=5b^4+8b^3+18b^2+28b+49,
\]

and

\[
Q_-(b)=5b^4-8b^3+18b^2-28b+49.
\]

The resulting theorem proves that if \(\operatorname{char}\mathbb{F}_q\neq3\), \(E_7(b)\neq0\), and \(q>3000\), then the degree-seven family is not a permutation polynomial over \(\mathbb{F}_{q^2}\) for any nondegenerate value of \(a\).

Relation to the Degree-Five Classification

The degree-five case \(n=5\) corresponds to a conic trace curve and has been completely classified in A Complete Classification of a Reciprocal Degree-Five Quadrinomial Family over \(\mathbb{F}_{q^2}\). The present manuscript explains the general Dickson trace mechanism behind that classification and extends the framework to higher odd degrees.

For students, the degree-five guide is the best entry point before reading the broader Dickson trace framework:
Finite Field Permutation Polynomial Research Guide.

Download the Paper

Download the full research manuscript PDF

Suggested Citation

Brian M. Woody, Dickson Trace Curves and Asymptotic Sparsity for Reciprocal Quadrinomials over \(\mathbb{F}_{q^2}\), 2026.

Research Area

This paper belongs to finite field theory and the study of permutation polynomials. It is connected to Brian M. Woody’s research hub, finite fields and Galois fields, abstract algebra, field extensions, Galois Theory, the Galois group of \(x^3-2\), Dickson polynomials, reciprocal quadrinomials, algebraic curves over finite fields, character sums, sparse polynomial maps, and asymptotic counting arguments.

Related Mathematical Topics

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About Brian Woody

Brian M. Woody is a mathematician, Private Professor, and founder of Woody Calculus. His work includes university-level mathematics instruction, finite field theory, abstract algebra, differential equations, advanced calculus, and mathematical research connected to permutation polynomials and sparse finite-field maps.

For advanced mathematics instruction or private study support, visit BrianWoody.com, explore the Woody Calculus Mastery Lab, or learn more about private mathematics instruction.