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

Abstract In this paper, we present results concerning the stabilizer Gf G f in {\, {GL\, }} (2, qⁿ) GL (2, q n) of the subspace Uf=\ (x, f (x) ): x {Fₐ䂞\} U f = (x, f (x) ): x ∈ F q n, f (x) a scattered linearized polynomial in Fₐ䂞x F q n x. Each Gf G f contains the q-1 q - 1 maps (x, y) (ax, ay) (x, y) ↦ (a x, a y), a Fₐ^* 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 Gf G f are simultaneously diagonalizable. This has several consequences: (i) the polynomials such that |Gf|>q-1 | G f | > q - 1 have a standard form of type ₉=₀^n/t-1aⱼx^q^{s+jt} ∑ j = 0 n / t - 1 a j x q s + j t for some s and t such that (s, t) =1 (s, t) = 1, t>1 t > 1 a divisor of n ; (ii) this standard form is essentially unique; (iii) for n>2 n > 2 and q>3 q > 3, the translation plane Af A f associated with f (x) admits nontrivial affine homologies if and only if |Gf|>q-1 | G f | > q - 1, and in that case those with axis through the origin form two groups of cardinality (qᵗ-1) / (q-1) (q t - 1) / (q - 1) that exchange axes and coaxes; (iv) no plane of type Af A f, f (x) a scattered polynomial not of pseudoregulus type, is a generalized André plane.