2022/05/30 by Giovanni Longobardi, Corrado Zanella, Longobardi, Giovanni +1 · 1 citation
Mathematics · #15B33 #51A40 #51E14 #51E22 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research
paper · pdf · doi:10.48550/arxiv.2205.15429
openalex publication_date 2022/05/30 · openalex created_date 2023/02/14 · openalex updated_date 2026/07/28
In this paper we present results concerning the stabilizer Gf in GL(2,qn) of the subspace Uf=\(x,f(x))\colon x∈\mathbb Fqn[x]\, f(x) a scattered linearized polynomial in \mathbb Fqn[x]. Each Gf contains the q-1 maps (x,y)↦(ax,ay), a∈\mathbb Fq^*. By virtue of the results of Beard (1972) and Willett (1973), the matrices in Gf are simultaneously diagonalizable. This has several consequences: (i) the polynomials such that |Gf|>q-1 have a standard form of type ∑j=0n/t-1ajx^qs+jt for some s and t such that (s,t)=1, t>1 a divisor of n; (ii) this standard form is essentially unique; (iii) for n>2 and q>3, the translation plane \cal Af associated with f(x) admits nontrivial affine homologies if and only if |Gf|>q-1, and in that case those with axis through the origin form two groups of cardinality (qt-1)/(q-1) that exchange axes and coaxes; (iv) no plane of type \cal Af, f(x) a scattered polynomial not of pseudoregulus type, is a generalized André plane.