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

In this paper, we present results concerning the stabilizer Gf in GL(2,qn) of the subspace Uf={(x,f(x)):x∈Fqn}, f(x) a scattered linearized polynomial in Fqn[x]. Each Gf contains the q-1 maps (x,y)↦(ax,ay), a∈Fq∗. 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 are simultaneously diagonalizable. This has several consequences: (i) the polynomials such that |Gf|>q-1 have a standard form of type ∑j=0n/t-1ajxqs+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 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 Af, f(x) a scattered polynomial not of pseudoregulus type, is a generalized André plane.