A standard form for scattered linearized polynomials and properties of the related translation planes

• Published in: Journal of algebraic combinatorics Vol. 59; no. 4; pp. 917 - 937
• Main Authors: Longobardi, Giovanni, Zanella, Corrado
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2024; Springer Nature B.V
• Subjects: Codes; Combinatorics; Computer Science; Convex and Discrete Geometry; Group Theory and Generalizations; Homology; Lattices; Linearization; Mathematics; Mathematics and Statistics; Order; Ordered Algebraic Structures; Polynomials; Vector space; Rank distance code; Partial spread; 51A40; Linear set; Projective line; 15B33; MRD code; Translation plane; 51E22; 51E14; André plane; Linearized polynomial
• ISSN: 0925-9899; 1572-9192; 1572-9192
• DOI: 10.1007/s10801-024-01317-y

In this paper, we present results concerning the stabilizer G f in GL ( 2 , q n ) of the subspace U f = { ( x , f ( x ) ) : x ∈ F q n } , f ( x ) a scattered linearized polynomial in F q n [ x ] . Each G f contains the q - 1 maps ( x , y ) ↦ ( a x , a y ) , a ∈ F q ∗ . By virtue of the results of Beard (Duke Math J, 39:313–321, 1972) and Willett (Duke Math J 40(3):701–704, 1973), the matrices in G f are simultaneously diagonalizable. This has several consequences: ( i ) the polynomials such that | G f | > q - 1 have a standard form of type ∑ j = 0 n / t - 1 a j x q s + j t 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 A f associated with f ( x ) admits nontrivial affine homologies if and only if | G f | > q - 1 , and in that case those with axis through the origin form two groups of cardinality ( q t - 1 ) / ( q - 1 ) that exchange axes and coaxes; ( iv ) no plane of type A f , f ( x ) a scattered polynomial not of pseudoregulus type, is a generalized André plane.