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

In this paper, we present results concerning the stabilizer \(G_f\) in \({{\,\mathrm{{GL}}\,}}(2,q^n)\) of the subspace \(U_f=\{(x,f(x)):x\in \mathbb {F}_{q^n}\}\), f(x) a scattered linearized polynomial in \(\mathbb {F}_{q^n}[x]\). Each \(G_f\) contains the \(q-1\) maps \((x,y)\mapsto (ax,ay)\), \(a\in \mathbb {F}_{q}^*\). 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 \(\sum _{j=0}^{n/t-1}a_jx^{q^{s+jt}}\) 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 \(\mathcal {A}_f\) 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 \(\mathcal {A}_f\), f(x) a scattered polynomial not of pseudoregulus type, is a generalized André plane.