A Carlitz type result for linearized polynomials

For an arbitrary $q$-polynomial $f$ over $\mathbb{F}_{q^n}$ we study the problem of finding those $q$-polynomials $g$ over $\mathbb{F}_{q^n}$ for which the image sets of $f(x)/x$ and $g(x)/x$ coincide. For $n\leq 5$ we provide sufficient and necessary conditions and then apply our result to study maximum scattered linear sets of $\mathrm{PG}(1,q^5)$

A Carlitz type result for linearized polynomials

For an arbitrary q-polynomial f over GF(q^n) we study the problem of finding those q-polynomials g over GF(q^n) for which the image sets of f(x)/x and g(x)/x coincide. For n < 6 we provide sufficient and necessary conditions and then apply our result to study maximum scattered linear sets of PG(1,q^5)

Vertex properties of maximum scattered linear sets of PG(1,qn)\mathrm{PG}(1,q^n)

In this paper we investigate the geometric properties of the configuration consisting of a $k$-subspace $\Gamma$ and a canonical subgeometry $\Sigma$ in $\mathrm{PG}(n-1,q^n)$, with $\Gamma\cap\Sigma=\emptyset$. The idea motivating is that such properties are reflected in the algebraic structure of the linear set which is projection of $\Sigma$ from the vertex $\Gamma$. In particular we deal with the maximum scattered linear sets of the line $\mathrm{PG}(1,q^n)$ found by Lunardon and Polverino and recently generalized by Sheekey. Our aim is to characterize this family by means of the properties of the vertex of the projection as done by Csajb\'ok and the first author of this paper for linear sets of pseudoregulus type. With reference to such properties, we construct new examples of scattered linear sets in $\mathrm{PG}(1,q^6)$, yielding also to new examples of MRD-codes in $\mathbb F_q^{6\times 6}$ with left idealiser isomorphic to $\mathbb F_{q^6}$