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Towards the classification of scattered binomials

Daniele Bartoli, Francesco Ghiandoni, Alessandro Giannoni, Giuseppe Marino

TL;DR

This work advances the classification of scattered ${\mathbb F}_q$-linearized binomials by translating the scatteredness condition into the geometry of associated algebraic varieties and applying Lang-Weil-type estimates. The authors establish that, for large base field size $q$ and prime $n$, a binomial $f(x)=x^{q^I}+\alpha x^{q^J}$ is scattered on ${\mathbb F}_{q^n}$ if and only if $I+J=n$ and $\text{N}_{q^n/q}(\alpha)\neq 1$ (LP-type), with analogous CMPZ-type conditions in the known special cases $n=6$ and $n=8$. They provide an explicit asymptotic classification for $3\le n\le 8$, showing that LP-type binomials dominate the scattered ones as $q$ grows, and they analyze the variety $\mathcal{Z}$ attached to known CMPZ examples to connect irreducibility properties with scattering criteria. The results support conjectures that LP-type binomials are the only scattered binomials for infinitely many $(n,q)$ and offer concrete criteria to decide scattering in low dimensions via algebraic-geometry tools. This provides a rigorous bridge between finite-field scattered linear sets and projective-geometry through explicit polynomial- and variety-based obstructions.

Abstract

Let \( q \) be a prime power and \( n \) an integer. An \( \mathbb{F}_q \)-linearized polynomial \( f \) is said to be scattered if it satisfies the condition that for all \( x, y \in \mathbb{F}_q^n \setminus \{ 0 \} \), whenever \( \frac{f(x)}{x} = \frac{f(y)}{y} \), it follows that \( \frac{x}{y} \in \mathbb{F}_q \). In this paper, we focus on scattered binomials. Two families of scattered binomials are currently known: the one from Lunardon and Polverino (LP), given by $f(x) = δx^{q^s} + x^{q^{n-s}},$ and the one from Csajbók, Marino, Polverino, and Zanella (CMPZ), given by $f(x) = δx^{q^s} + x^{q^{s + n/2}},$ where \( n = 6 \) or \( n = 8 \). Using algebraic varieties as a tool, we prove some necessary conditions for a binomial to be scattered. As a corollary, we obtain that when \( q \) is sufficiently large and \( n \) is prime, a binomial is scattered if and only if it is of the form (LP). Moreover we obtain a complete classification of scattered binomial in $\Fn$ when $n\leq8$ and $q$ is large enough.

Towards the classification of scattered binomials

TL;DR

This work advances the classification of scattered -linearized binomials by translating the scatteredness condition into the geometry of associated algebraic varieties and applying Lang-Weil-type estimates. The authors establish that, for large base field size and prime , a binomial is scattered on if and only if and (LP-type), with analogous CMPZ-type conditions in the known special cases and . They provide an explicit asymptotic classification for , showing that LP-type binomials dominate the scattered ones as grows, and they analyze the variety attached to known CMPZ examples to connect irreducibility properties with scattering criteria. The results support conjectures that LP-type binomials are the only scattered binomials for infinitely many and offer concrete criteria to decide scattering in low dimensions via algebraic-geometry tools. This provides a rigorous bridge between finite-field scattered linear sets and projective-geometry through explicit polynomial- and variety-based obstructions.

Abstract

Let be a prime power and an integer. An -linearized polynomial is said to be scattered if it satisfies the condition that for all , whenever \( \frac{f(x)}{x} = \frac{f(y)}{y} \), it follows that . In this paper, we focus on scattered binomials. Two families of scattered binomials are currently known: the one from Lunardon and Polverino (LP), given by and the one from Csajbók, Marino, Polverino, and Zanella (CMPZ), given by where or . Using algebraic varieties as a tool, we prove some necessary conditions for a binomial to be scattered. As a corollary, we obtain that when is sufficiently large and is prime, a binomial is scattered if and only if it is of the form (LP). Moreover we obtain a complete classification of scattered binomial in when and is large enough.

Paper Structure

This paper contains 17 sections, 25 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Scattered binomials
  4. On the equivalence of scattered binomials
  5. Algebraic varieties
  6. Link with algebraic varieties
  7. Structure of $\mathcal{W}$ in the case $\gcd(n,K)=1$
  8. Absolute irreducibility of $P(X)$
  9. Main result
  10. Classification of scattered binomials for $n \leq 8$
  11. Case $n=6,$ $\gcd(n,J-I)=2$
  12. Case $n=8,$ $\gcd(n,J-I)=2$
  13. Case $q$ even, $n=8,$ $\gcd(n,J-I)=4$
  14. Main result for scattered binomials when $n \leq 8$
  15. The variety $\mathcal{Z}$ for the known scattered binomials
  16. ...and 2 more sections

Key Result

Theorem 2.1

BartoliCsajbokMontanucci2021,PolverinoZullo2020rootsof... The ${\mathbb F}_{q}$-linear set with $\textnormal{N}_{q^6 / q^3}(\delta)=\delta^{1+q^3}\neq 1$, is maximum scattered if and only if the equation with $A=\frac{\delta^{1+q^3}}{\delta^{1+q^3}-1}$, admits two roots over ${\mathbb F}_{q}$. In particular, such a $\delta$ always exists for any $q > 2$. Additionally, there are exactly equivale

Theorems & Definitions (45)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Remark 2.5
  • Theorem 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Proposition 3.1
  • Remark 3.2
  • ...and 35 more