A Complete Classification of a Reciprocal Degree-Five Quadrinomial Family over F_{q^2}

Brian M. Woody

Home/Archive/arXiv

arXiv · 2607.01267

A Complete Classification of a Reciprocal Degree-Five Quadrinomial Family over F_{q^2}

Brian M. Woody

View original source ↗Read source document ↗Export metadata ↓

Abstract

We classify a reciprocal degree-five quadrinomial family over the quadratic extension F_{q^2}, where q is an odd prime power. The family has four terms, coefficients in F_q, and a coefficient constraint that makes the induced rational function on the unit circle highly structured. The classification has two sharply different branches. When q is congruent to 1 modulo 4, infinite families occur and are governed by two quadratic-character conditions on the parameter b. When q is congruent to 3 modulo 4, a square-class obstruction converts the problem into a character-sum problem on a conic. A Weil-bound argument eliminates all large fields in this branch, and finite verification leaves only the sporadic fields q = 7, 19, 23. The result is a complete classification of the nondegenerate members of the family for all odd prime powers q.

Explore related subjects

math.GM

Keep this discovery

BibTeX RIS

Brian M. Woody. 2026-06-29. A Complete Classification of a Reciprocal Degree-Five Quadrinomial Family over F_{q^2}. https://arxiv.org/abs/2607.01267

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Average Chord Lengths in a Triangle

Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.

math.GM↗

arXiv2026

The decrease in sectional curvature of the Fisher--Rao manifold of beta distributions

In the paper, the author develops a general convexity lemma and applies it to demonstrate the monotonic decrease of a function involving polygamma functions, arising from the sectional curvature of the Fisher--Rao manifold of beta distributions.

math.GM↗

arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that \[\sum_{\mathrm{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \geq 2+\left(\frac{2r}{R}\right)^k,~~k>1, \] with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.

math.GM↗