Investigating the exceptionality of scattered polynomials

Abstract

Scattered polynomials over a finite field ${\mathbb{F}}_{q^n}$ have been introduced by Sheekey in 2016, and a central open problem regards the classification of those that are exceptional. So far, only two families of exceptional scattered polynomials are known. Very recently, Longobardi and Zanella weakened the property of being scattered by introducing the notion of L-$q^t$-partially scattered and R-$q^t$-partially scattered polynomials, for t a divisor of n. Indeed, a polynomial is scattered if and only if it is both L-$q^t$-partially scattered and R-$q^t$-partially scattered. In this paper, by using techniques from algebraic geometry over finite fields and function fields theory, we show that the property which is the hardest to be preserved is the L-$q^t$-partially scattered one. We investigate a large family $\mathcal{F}$ of R-$q^t$-partially scattered polynomials, containing examples of exceptional R-$q^t$-partially scattered polynomials, which turn out to be connected with linear sets of so-called pseudoregulus type. We introduce two different notions of equivalence preserving the property of being R-$q^t$-partially scattered. Many polynomials in $\mathcal{F}$ are inequivalent and geometric arguments are used to determine their equivalence classes under the action of $\mathrm{\Gamma }\mathrm{L}(2n/t,q^t)$.

Introduction

Let q be a prime power, n be a positive integer, and $f(x)={\sum }_{i=0}^k{a}_i{x}^{q^i}\in {\mathbb{F}}_{q^n}[x]$ be an ${\mathbb{F}}_q$-linearized polynomial over the finite field ${\mathbb{F}}_{q^n}$. We also assume that the q-degree k of $f(x)$ is smaller than n, so that the identification with the map $x\mapsto f(x)$ defines a one-to-one correspondence between such polynomials and ${\mathbb{F}}_q$-linear maps over ${\mathbb{F}}_{q^n}$.

An ${\mathbb{F}}_q$-linearized polynomial $f(x)\in {\mathbb{F}}_{q^n}[x]$ is said to be scattered of index $\ell \in \{0,\dots ,n-1\}$ over ${\mathbb{F}}_{q^n}$ if, for any $y,z\in {\mathbb{F}}_{q^n}^{⁎}$,

$$\frac{f(y)}{y^{q^{\ell }}}=\frac{f(z)}{z^{q^{\ell }}}⟹\frac{y}{z}\in {\mathbb{F}}_q;$$

see [8]. Scattered polynomials $f(x)\in {\mathbb{F}}_{q^n}[x]$ yield scattered subspaces $U_f$ (w.r.t. a Desarguesian spread) in ${\mathbb{F}}_{q^n}\times {\mathbb{F}}_{q^n}$ by defining

$$U_f=\{(x^{q^{\ell }},f(x)):x\in {\mathbb{F}}_{q^n}\}.$$

Scattered subspaces of maximum dimension have many applications, such as translation hyperovals [17], translation caps in affine spaces [4], two-intersection sets [10], blocking sets [1], translation spreads of the Cayley generalized hexagon [31], finite semifields [22], coding theory [39], [45], and graph theory [11].

Starting from [8], a much stronger property regarding scattered polynomials, namely their exceptionality, has been defined and deeply investigated. An ${\mathbb{F}}_q$-linearized polynomial $f(x)\in {\mathbb{F}}_{q^n}[x]$ is said to be exceptional scattered of index $\ell \in \{0,\dots ,n-1\}$ if there exist infinitely many $m\in \mathbb{N}$ such that, for any $y,z\in {\mathbb{F}}_{q^{nm}}$, Condition (1) holds.

While several families of scattered polynomials have been constructed in recent years, only two families of exceptional scattered polynomials are known:

• $f(x)=x^{q^s}$ of index 0, with $\gcd (s,n)=1$ (polynomials of so-called pseudoregulus type);

• $f(x)=x+\delta x^{q^{2s}}$ of index s, with $\gcd (s,n)=1$ and ${\mathrm{N}}_{q^n/q}(\delta )\ne 1$ (so-called LP-polynomials).

Several tools have already been proposed in the study of exceptional scattered polynomials, related to certain algebraic curves or Galois extensions of function fields; see [2], [6], [8], [16]. However, their classification is still unknown when the index is greater than 1. In this paper we investigate the exceptional scatteredness of a polynomial by considering separately the exceptionality of two weaker properties defined in [26], namely the L-$q^t$-partial scatteredness and the R-$q^t$-partial scatteredness.

Let $f(x)$ be an ${\mathbb{F}}_q$-linearized polynomial over ${\mathbb{F}}_{q^n}$, t a divisor of n, so that $n=t{t}^{\prime }$, and $\ell \in \{0,\dots ,n-1\}$. We say that $f(x)$ is L-$q^t$-partially scattered of index ℓ if for any $y,z\in {\mathbb{F}}_{q^n}^{⁎}$,

$$\frac{f(y)}{y^{q^{\ell }}}=\frac{f(z)}{z^{q^{\ell }}}⟹\frac{y}{z}\in {\mathbb{F}}_{q^t},$$

and that $f(x)$ is R-$q^t$-partially scattered of index ℓ if for any $y,z\in {\mathbb{F}}_{q^n}^{⁎}$,

$$\frac{f(y)}{y^{q^{\ell }}}=\frac{f(z)}{z^{q^{\ell }}}\text{ and }\frac{y}{z}\in {\mathbb{F}}_{q^t}⟹\frac{y}{z}\in {\mathbb{F}}_q.$$

From now on, whenever the index is not specified, we mean $\ell =0$.

We say that $f(x)$ is exceptional L-$q^t$-partially scattered of index ℓ (resp. exceptional R-$q^t$-partially scattered of index ℓ) if there exist infinitely many $m\in \mathbb{N}$ such that Condition (2) (resp. Condition (3)) holds for any $y,z\in {\mathbb{F}}_{q^{nm}}^{⁎}$. Clearly, for any t, $f(x)$ is (exceptional) scattered of index ℓ if and only if it is (exceptional) both L- and R-$q^t$-partially scattered of index ℓ.

Some results on, and characterizations of L- and R-$q^t$-partially scattered polynomials have been provided in [26]; see Section 2.

In this paper, we start the investigation of such properties by those families that contain the known examples of exceptional scattered polynomials, namely the monomials $x^{q^u}$ of index 0 and the LP-polynomials $x+\delta x^{q^{2s}}$ of index s. In this way we obtain exceptional L- and R-$q^t$-partially scattered monomials which are not scattered, and other results for L- and R-$q^t$-partially scattered polynomials of LP type; see Section 3.

Afterwards, we prove in Section 4 several necessary conditions for a polynomial to be exceptional L-$q^t$-partially scattered, classifying for instance those of index at most 1. This is done by means of tools already exploited in the literature in connection with the exceptional scatteredness, such as algebraic curves and function fields over finite fields, also exploiting a method due to G. Micheli in [33], [34]. Interestingly, such connections can be generalized to exceptional L-$q^t$-partial scatteredness.

Turning to the R-side of $q^t$-partial scatteredness, we investigate in Section 5 an explicit large family of R-$q^t$-partially scattered polynomials of the form

$$f(x)=\sum _{i=0}^{n/t-1}{a}_i{x}^{q^{it+s}}\in {\mathbb{F}}_{q^n}[x],$$

obtained through a criterion in [26], and whose exact number we are able to count. As a byproduct, we provide families of exceptional R-$q^t$-partially scattered binomials of the shape $x^{q^{kt+s}}+\alpha x^{q^s}$. Such a construction suggests that the existence of exceptional L-$q^t$-partially scattered polynomials is much harder to prove than that of exceptional R-$q^t$-partially scattered ones.

We further investigate the family (4) in Section 6 from a geometric point of view: indeed, this family can be alternatively constructed by considering linear sets of pseudoregulus type in the projective space $\mathrm{PG}(2n/t-1,q^t)$.

The search for new R-$q^t$-partially scattered polynomials naturally requires the investigation of the equivalence issue, in the sense of a suitable group action on the polynomials $f(x)\in {\mathbb{F}}_{q^{t{t}^{\prime }}}[x]$ preserving the desired property. To this aim, we analyse in Section 7 the equivalence defined by a natural action of the group $\mathrm{\Gamma }\mathrm{L}(2,q^{t{t}^{\prime }})$ on the elements $(x,f(x))$. In Section 8 we study a weaker equivalence defined by a natural action of the larger group $\mathrm{\Gamma }\mathrm{L}(2t^{\prime },q^t)$. We solve the weak equivalence issue for the family of R-$q^t$-partially scattered polynomials described in Section 5. Some open problems in different directions conclude the paper.