A standard form for scattered linearized polynomials and properties of the related translation planes (Q6570594)

Scattered polynomials, a type of linearised polynomials, have been studied intensively over the past years. Recall that an \(\mathbb{F}_q\)-linearised polynomial is of the form \(f(x)=\sum_{i=0}^{n-1} a_i x^{q^i}\), where \(a_i\in \mathbb{F}_{q^n}\), and such a polynomial is called scattered if \(\frac{f(x)}{x}=\frac{f(y)}{y}\) implies that \(y=\lambda x\) for some \(x\in \mathbb{F}_q^*\). One of the main problems in the study of scattered polynomials (and their associated subspaces) is their equivalence.\N\NIn the present paper, the authors use algebraic tools to study the stabiliser in \(\mathrm{GL}(2,q^n)\) of the subspace \(U_f=\{(x,f(x))\mid x\in \mathbb{F}_{q^n}\}\). It is clear that this stabiliser always contains the \(q-1\) (diagonal) maps induced by multiplying with a non-zero element of \(\mathbb{F}_q\); the interest lies in those polynomials for which the stabiliser is strictly larger.\N\NThe authors first describe the stabilisers of the known scatttered polynomials before showing the main result. This says that, in the case that the associate stabiliser has indeed larger size than \(q-1\), the polynomials can be put in an essentially unique standard form. They then use this to derive results on the equivalence of such polynomials.\N\NThe final part of the paper deals with the application of their results to translation planes. Some interesting properties of the collineation groups of the planes associated with the polynomials are derived. The main result of this section generalises a known result for scattered polynomials of LP type: it is shown that if \(q>3\), a scattered polynomial that is not of pseudo-regulus type can't give rise to a generalised André plane.