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

In this paper, we present results concerning the stabilizer \(G_f\) $G_f$ in \({{\,\mathrm{{GL}}\,}}(2,q^n)\) $\mathrm{GL}(2,q^n)$ of the subspace \(U_f=\{(x,f(x)):x\in \mathbb {F}_{q^n}\}\) $U_f=\{(x,f(x)):x\in {\mathbb{F}}_{q^n}\}$ , f(x) a scattered linearized polynomial in \(\mathbb {F}_{q^n}[x]\) ${\mathbb{F}}_{q^n}[x]$ . Each \(G_f\) $G_f$ contains the \(q-1\) $q-1$ maps \((x,y)\mapsto (ax,ay)\) $(x,y)\mapsto (ax,ay)$ , \(a\in \mathbb {F}_{q}^*\) $a\in {\mathbb{F}}_q^{\ast }$ . 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 \(G_f\) $G_f$ are simultaneously diagonalizable. This has several consequences: (i) the polynomials such that \(|G_f|>q-1\) $|G_f|>q-1$ have a standard form of type \(\sum _{j=0}^{n/t-1}a_jx^{q^{s+jt}}\) ${\sum }_{j=0}^{n/t-1}{a}_j{x}^{q^{s+jt}}$ 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 \(\mathcal {A}_f\) ${\mathcal{A}}_f$ associated with f(x) admits nontrivial affine homologies if and only if \(|G_f|>q-1\) $|G_f|>q-1$ , and in that case those with axis through the origin form two groups of cardinality \((q^t-1)/(q-1)\) $(q^t-1)/(q-1)$ that exchange axes and coaxes; (iv) no plane of type \(\mathcal {A}_f\) ${\mathcal{A}}_f$ , f(x) a scattered polynomial not of pseudoregulus type, is a generalized André plane.