2025/02/17 by Bartoli, Daniele, Ghiandoni, Francesco, Giannoni, Alessandro +1
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2502.11666
Let \( q \) be a prime power and \( n \) an integer. An \( \mathbbFq \)-linearized polynomial \( f \) is said to be scattered if it satisfies the condition that for all \( x, y ∈ \mathbbFqn ∖ \ 0 \ \), whenever \( (f(x))/(x) = (f(y))/(y) \), it follows that \( (x)/(y) ∈ \mathbbFq \). 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) = δxqs + x^qn-s, and the one from Csajbók, Marino, Polverino, and Zanella (CMPZ), given by f(x) = δxqs + x^qs + 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≤8 and q is large enough.