1 Introduction

A point P of the projective space \(\mathrm {PG}(d-1,q^n)\) is a one-dimensional subspace of the vector space \({\mathbb {F}}_{q^n}^d\); that is, \(P=\langle v\rangle _{{\mathbb {F}}_{q^n}}=\{cv:c\in {\mathbb {F}}_{q^n}\}\) for some nonzero \(v\in {\mathbb {F}}_{q^n}^d\).

Let U be an r-dimensional \({\mathbb {F}}_q\)-subspace of \({\mathbb {F}}_{q^n}^d\). Then

$$\begin{aligned} L_U:=\{\langle v\rangle _{{\mathbb {F}}_{q^n}}:v\in U,v\ne 0\} \end{aligned}$$

is an \({\mathbb {F}}_q\)-linear set (or just linear set) of rank r in \(\mathrm {PG}(d-1,q^n)\). Let \(u,v\in U\). If \(u=cv\), \(c\in {\mathbb {F}}_q\), then clearly \(\langle u\rangle _{{\mathbb {F}}_{q^n}}=\langle v\rangle _{{\mathbb {F}}_{q^n}}\). If this is the only case in which two vectors of U determine the same point of \(\mathrm {PG}(d-1,q)\), that is, \(\langle v\rangle _{{\mathbb {F}}_{q^n}}=\langle u\rangle _{{\mathbb {F}}_{q^n}}\) if and only if \(\langle v\rangle _{{\mathbb {F}}_q}=\langle u\rangle _{{\mathbb {F}}_q}\), then \(L_U\) is called a scattered linear set. Equivalently, \(L_U\) is scattered if and only if it has maximum size \((q^r-1)/(q-1)\) with respect to r. The linear sets are related to combinatorial objects, such as blocking sets, two-intersection sets, finite semifields, rank-distance codes, and many others. The interested reader is referred to the survey by Polverino [18] and to [20], where J. Sheekey builds a bridge with the rank-distance codes.

Assume that in particular U is an \({\mathbb {F}}_q\)-subspace of \({\mathbb {F}}_{q^n}^2\), \(\dim _{{\mathbb {F}}_q}U=n\). In this case \(L_U:=\{\langle v\rangle _{{\mathbb {F}}_{q^n}}:v\in U,v\ne 0\}\subseteq \mathrm {PG}(1,q^n)\), is called a maximum linear set of \(\mathrm {PG}(1,q^n)\), since by the dimension formula any linear set of rank greater than n equals \(\mathrm {PG}(1,q^n)\). Up to projectivities of \(\mathrm {PG}(1,q^n)\) it may be assumed that \(\langle (0,1)\rangle _{{\mathbb {F}}_{q^n}}\not \in L_U\). Hence

$$\begin{aligned} L_U={L_f}=\{\langle (x,f(x))\rangle _{{\mathbb {F}}_{q^n}}:x\in {\mathbb {F}}_{q^n}^*\} \end{aligned}$$

where f(x) is a suitable \({\mathbb {F}}_q\)-linear map, that is a linearized polynomial:

$$\begin{aligned} f(x)=\sum _{i=0}^{n-1}a_ix^{q^i},\quad a_i\in {\mathbb {F}}_{q^n},\quad i=0,1,\ldots ,n-1. \end{aligned}$$
(1)

If \(L_f\) is scattered, then f(x) is called a scattered linearized polynomial, or scattered q-polynomial with respect to n. A property characterizing the scattered q-polynomials is that for any \(x,y\in {\mathbb {F}}_{q^n}^*\), \({f(x)}/x={f(y)}/y\) if and only if \(\langle x\rangle _{{\mathbb {F}}_q}=\langle y\rangle _{{\mathbb {F}}_q}\).

A first example of scattered q-polynomial is \(f(x)=x^q\) [3], with respect to any n. Indeed, for any \(x,y\in {\mathbb {F}}_{q^n}^*\), \({f(x)}/x={f(y)}/y\), is equivalent to \(x^{q-1}=y^{q-1}\), hence to \(x/y\in {\mathbb {F}}_q^*\). A derived example is \(f(x)=x^{q^s}\), \(\gcd (n,s)=1\). Indeed \((x/y)^{q^s-1}=1\) implies \(x/y\in {\mathbb {F}}_{q^s}\cap {\mathbb {F}}_{q^n}^*={\mathbb {F}}_q^*\). In both cases above, \(L_f=\{\langle (x,f(x))\rangle _{{\mathbb {F}}_{q^n}}:x\in {\mathbb {F}}_{q^n}^*\}= \{\langle (1,z)\rangle _{{\mathbb {F}}_{q^n}}:z\in {\mathbb {F}}_{q^n},\,{{\,\mathrm{N}\,}}_{q^n/q}(z)=1\}\), where \({{\,\mathrm{N}\,}}_{q^n/q}(z)=z^{(q^n-1)/(q-1)}\) denotes the norm over \({\mathbb {F}}_q\) of \(z\in {\mathbb {F}}_{q^n}\). The related linear set is called a linear set of pseudoregulus type.

The next example has been given by Lunardon and Polverino [12] and generalized in [11, 20]:

$$\begin{aligned} f(x)=x^{q^s}+\delta x^{q^{n-s}},\quad n\ge 4,\quad \gcd (n,s)=1,\quad {{\,\mathrm{N}\,}}_{q^n/q}(\delta )\ne 1. \end{aligned}$$

In particular cases, the condition \({{\,\mathrm{N}\,}}_{q^n/q}(\delta )\ne 1\) has been proved to be necessary for f(x) to be scattered [2, 10, 11, 22]. In Sect. 3 it will proved that actually it is necessary for any n and s. Further examples of scattered q-polynomials are given in [5, 6, 14, 22]. All of them are with respect to \(n=6\) or \(n=8\). Bartoli et al. [1] proved that if \({\hat{f}}(x)\) is the adjoint of f(x) with respect to the bilinear form \(\langle x,y\rangle =\hbox {Tr}_{q^n/q}(xy)\) in \({\mathbb {F}}_{q^n}^2\), where \(\hbox {Tr}_{q^n/q}(z)=\sum _{i=0}^{n-1}z^{q^i}\) denotes the trace over \({\mathbb {F}}_q\) of \(z\in {\mathbb {F}}_{q^n}\), then \(L_f=L_{{\hat{f}}}\). This implies that if the polynomial f(x) in (1) is scattered, then also \({\hat{f}}(x)=\sum _{i=0}^{n-1}a_i^{q^{n-i}}x^{q^{n-i}}\) is. Up to the knowledge of the author of this paper, no more examples of scattered q-polynomials are known. So, it would seem that scattered q-polynomials are rare. Bartoli and Zhou [2] formalized such an idea of scarcity by proving that the pseudoregulus and Lunardon–Polverino polynomials are, roughly speaking, the only q-polynomials of a certain type which are scattered for infinitely many n.

Recently, a great deal of effort has been put in finding conditions for q-polynomials to be scattered [4, 7, 15, 19]. Some of them are based on the Dickson matrix associated with the q-polynomial in (1), that is, the \(n\times n\) matrix

$$\begin{aligned} M_{q,f}= \begin{pmatrix} a_0&{}\quad a_1&{}\quad a_2&{}\quad \cdots &{}\quad a_{n-1}\\ a_{n-1}^q&{}\quad a_0^q&{}\quad a_1^q&{}\quad \cdots &{}\quad a_{n-2}^q\\ a_{n-2}^{q^2}&{}\quad a_{n-1}^{q^2}&{}\quad a_0^{q^2}&{}\quad \cdots &{}\quad a_{n-3}^{q^2}\\ \vdots &{}\quad &{}\quad &{}\quad &{}\quad \vdots \\ a_{1}^{q^{n-1}}&{}\quad a_{2}^{q^{n-1}}&{}\quad a_3^{q^{n-1}}&{}\quad \cdots &{}\quad a_{0} ^{q^{n-1}}\end{pmatrix}. \end{aligned}$$

It is well-known that the rank of \(M_{q,f}\) equals the rank of f(x), see for example [21, Proposition 4.4]. This rank can be computed by applying the following result by B. Csajbók:

Theorem 1.1

([4, Theorem 3.4]) Let \(M_{q,f}\) be the Dickson matrix associated with the q-polynomial in (1). Denote by \(M_{q,f}^{(r)}\) the \(r\times r\) submatrix of \(M_{q,f}\) obtained by considering the last r columns and the first r rows of \(M_{q,f}\). Then the rank of f(x) is t if and only if \(\det (M_{q,f}^{(n)}) =\det (M_{q,f}^{(n-1)}) =\cdots = \det (M_{q,f}^{(t+1)}) =0\), and \(\det (M_{q,f}^{(t)}) \ne 0\).

A q-polynomial \(f(x)\in {\mathbb {F}}_{q^n}[x]\) is scattered if and only if for any \(m\in {\mathbb {F}}_{q^n}\) the dimension of the kernel of \(f_m(x)=mx+f(x)\) is at most one. So, by Theorem 1.1 a necessary and sufficient condition for f(x) to be scattered is that the system of two equations

$$\begin{aligned} \det (M_{q,f_m}^{(n)})=\det (M_{q,f_m}^{(n-1)})=0 \end{aligned}$$

has no solution in the variable \(m\in {\mathbb {F}}_{q^n}\).

In this paper a condition consisting of one equation (Proposition 2.2) is proved, and applied to two binomials. It would seem that one equation is better than two in order to prove that a given q-polynomial f(x) is not scattered, while two equations will usually be more helpful in the proof that f(x) is. As a matter of fact, here the condition \({{\,\mathrm{N}\,}}_{q^n/q}(\delta )\ne 1\) is proved to be necessary for the Lunardon–Polverino binomial to be scattered (cf. Theorem 3.4). Furthermore, two necessary and sufficient conditions for \(x^{q^s}+bx^{q^{2s}}\) (where \(\gcd (s,n)=1\)) to be scattered are stated in Propositions 3.5 and 3.10 . This leads to the fact that the polynomial \(x^q+bx^{q^2}\), \(b\ne 0\), is never scattered if \(n\ge 5\) (cf. Proposition 3.8 and Remark 3.11).

2 A condition for scattered linearized polynomials

In this paper s, n, q and \(\sigma \) will always denote natural numbers such that \(n\ge 3\), \(\gcd (s,n)=1\), q is the power of a prime and \(\sigma =q^s\). Any \({\mathbb {F}}_q\)-linear endomorphism of \({\mathbb {F}}_{q^n}\) can be represented in the form

$$\begin{aligned} f(x)=a_0x+a_1x^\sigma +a_2x^{\sigma ^2}+\cdots +a_{n-1}x^{\sigma ^{n-1}}\in {\mathbb {F}}_{q^n}[x]. \end{aligned}$$
(2)

As a matter of fact, if \(\tau \) is the permutation \(i\mapsto is\) of \({\mathbb {Z}}/(n)\), then f(x) is the same function of \(\tilde{f}(x)=\sum _{i=0}^{n-1}a_{\tau ^{-1}(i)}x^{q^i}\). Generalizing the notion of Dickson matrix given in the previous section, the \(\sigma \)-matrix of Dickson associated with the linearized polynomial \(g(t)=\sum _{i=0}^{n-1}a_it^{\sigma ^i}\) is

$$\begin{aligned} M_{\sigma ,g}=\begin{pmatrix} a_0&{}\quad a_1&{}\quad a_2&{}\quad \cdots &{}\quad a_{n-1}\\ a_{n-1}^\sigma &{}\quad a_0^\sigma &{}\quad a_1^\sigma &{}\quad \cdots &{}\quad a_{n-2}^\sigma \\ a_{n-2}^{\sigma ^2}&{}\quad a_{n-1}^{\sigma ^2}&{}\quad a_0^{\sigma ^2}&{}\quad \cdots &{}\quad a_{n-3}^{\sigma ^2}\\ \vdots &{}\quad &{}\quad &{}\quad &{}\quad \vdots \\ a_{1}^{\sigma ^{n-1}}&{}\quad a_{2}^{\sigma ^{n-1}}&{}\quad a_3^{\sigma ^{n-1}}&{}\quad \cdots &{}\quad a_{0} ^{\sigma ^{n-1}}\end{pmatrix}. \end{aligned}$$

This is just the Dickson matrix \(M_{q,{\tilde{g}}}\) associated with \({\tilde{g}}(t)\) after a permutation of the row and columns. Indeed, the element in row r and column c of \(M_{q,{\tilde{g}}}\), \(r,c\in \{0,1,\ldots ,n-1\}\), is \(m_{rc}=a_{\tau ^{-1}(c-r)}^{q^r}=a_{\tau ^{-1}(c)-\tau ^{-1}(r)}^{q^r}\). By applying \(\tau \) to both the row and column index, \(m_{\tau (r)\tau (c)}=a_{c-r}^{\sigma ^r}\) follows. Therefore, the rank of \(M_{\sigma ,g}\) equals the rank of g(t).

Remark 2.1

Each row of an n-order \(\sigma \)-matrix of Dickson is obtained from the previous one (cyclically) by the map

$$\begin{aligned} \phi :(X_0,X_1,\ldots ,X_{n-1})\mapsto (X_{n-1},X_0,\ldots ,X_{n-2})^\sigma \end{aligned}$$

which is an invertible semilinear map of \({{\mathbb {F}}}_{q^n}^n\) into itself.

The polynomial (2) is scattered if and only if \(f_1(x)=\sum _{i=1}^{n-1}a_ix^{\sigma ^i}\) is. Hence in the following \(a_0\) will always be zero.

Proposition 2.2

Let \(f(x)=\sum _{i=1}^{n-1}a_ix^{\sigma ^i}\) be a linearized polynomial over \({{\mathbb {F}}}_{q^n}\), and

$$\begin{aligned} g(t)=g_x(t)=-f(x)t+\sum _{i=1}^{n-1}a_ix^{\sigma ^i}t^{\sigma ^i}=-f(x)t+f(xt). \end{aligned}$$

Then the following conditions are equivalent:

  1. (i)

    the polynomial f(x) is scattered;

  2. (ii)

    for any \(x\in {{\mathbb {F}}}_{q^n}^*\), a nonsingular \((n-1)\)-order minor of \(M_{\sigma ,g}\) exists;

  3. (iii)

    for any \(x\in {{\mathbb {F}}}_{q^n}^*\), all \((n-1)\)-order minors of \(M_{\sigma ,g}\) are nonsingular.

Proof

The polynomial f(x) is scattered if and only if for any \(x\in {{\mathbb {F}}}_{q^n}^*\) the rank of \(h(t)=xf(t)-tf(x)\) is \(n-1\), that is, the rank of

$$\begin{aligned} M_{\sigma ,h}=\begin{pmatrix}-f(x)&{}\quad a_1x&{}\quad a_2x&{}\quad \ldots &{}\quad a_{n-1}x\\ a_{n-1}^\sigma x^\sigma &{}\quad -f(x)^\sigma &{}\quad a_1^\sigma x^\sigma &{}\quad \ldots &{}\quad a_{n-2}^\sigma x^\sigma \\ a_{n-2}^{\sigma ^2} x^{\sigma ^2}&{}\quad a_{n-1}^{\sigma ^2}x^{\sigma ^2}&{}\quad -f(x)^{\sigma ^2}&{}\quad \ldots &{}\quad a_{n-3}^{\sigma ^2} x^{\sigma ^2}\\ \vdots &{}\quad &{}\quad &{}\quad &{}\quad \vdots \\ a_1^{\sigma ^{n-1}}x^{\sigma ^{n-1}}&{}\quad a_2^{\sigma ^{n-1}}x^{\sigma ^{n-1}}&{}\quad a_3^{\sigma ^{n-1}} x^{\sigma ^{n-1}}&{}\quad \ldots &{}\quad -f(x)^{\sigma ^{n-1}}\end{pmatrix} \end{aligned}$$

is always \(n-1\). By dividing the rows of \(M_{\sigma ,h}\) by \(x{\hbox {'}}s\), \(x^\sigma \), \(x^{\sigma ^2}\), \(\ldots \), \(x^{\sigma ^{n-1}}\), respectively, and then multiplying the columns for that same elements, one obtains

$$\begin{aligned} \begin{pmatrix} -f(x)&{}\quad a_1x^\sigma &{}\quad a_2x^{\sigma ^{2}}&{}\quad \ldots &{}\quad a_{n-1}x^{\sigma ^{n-1}}\\ a_{n-1}^\sigma x&{}\quad -f(x)^\sigma &{}\quad a_1^{\sigma } x^{\sigma ^2}&{}\quad \ldots &{}\quad a_{n-2}^\sigma x^{\sigma ^{n-1}}\\ a_{n-2}^{\sigma ^2} x&{}\quad a_{n-1}^{\sigma ^2}x^{\sigma }&{}\quad -f(x)^{\sigma ^2}&{}\quad \ldots &{}\quad a_{n-3}^{\sigma ^2} x^{\sigma ^{n-1}}\\ \vdots &{}\quad &{}\quad &{}\quad &{}\quad \vdots \\ a_1^{\sigma ^{n-1}}x&{}\quad a_2^{\sigma ^{n-1}}x^{\sigma }&{}\quad a_3^{\sigma ^{n-1}} x^{\sigma ^{2}}&{}\quad \ldots &{}\quad -f(x)^{\sigma ^{n-1}}\end{pmatrix}, \end{aligned}$$

that is, the matrix \(M_{\sigma ,g}\). By Remark 2.1, if a \(\sigma \)-matrix of Dickson is singular, then any row is a linear combination of the remaining ones. Hence the rank of \(M_{\sigma ,g}\) equals the rank of any \((n-1)\times n\) matrix obtained from it by deleting a row. Furthermore, since the sum of the columns of \(M_{\sigma ,g}\) is zero, all \((n-1)\)-order minors have the same rank of \(M_{\sigma ,g}\). \(\square \)

3 Two linearized binomials

Definition 3.1

For any \(\delta \in {{\mathbb {F}}}_{q^n}\),

$$\begin{aligned} f_{\sigma ,\delta }(x)=x^\sigma +\delta x^{\sigma ^{n-1}} \end{aligned}$$

is the Lunardon–Polverino binomial.

If \({{\,\mathrm{N}\,}}_{q^n/q}(\delta )\ne 1\), then \(f_{\sigma ,\delta }\) is scattered [11,12,13, 20].

Proposition 3.2

The polynomial \(f_{\sigma ,\delta }(x)\) is scattered if only if there is no \(x\in {{\mathbb {F}}}_{q^n}^*\) such that

$$\begin{aligned} \sum _{i=0}^{n-1}z^{(\sigma ^i-1)/(\sigma -1)}=0, \end{aligned}$$
(3)

where \(z=\delta x^{\sigma ^{n-1}-\sigma }\).

Proof

The \((n-1)\)-th order North-West principal minor of the \(\sigma \)-matrix of Dickson associated with the polynomial

$$\begin{aligned} g(t)=-f_{\sigma ,\delta }(x)t+\sum _{i=1}^{n-1}a_ix^{\sigma ^i}t^{\sigma ^i}= -f_{\sigma ,\delta }(x)t+x^\sigma t^\sigma +\delta x^{\sigma ^{n-1}}t^{\sigma ^{n-1}}, \end{aligned}$$

further normalized row by row, is

$$\begin{aligned} B(z)=\begin{pmatrix} -(1+z)&{}\quad 1&{}\quad 0&{}\quad 0&{}\quad \cdots &{}\quad 0&{}\quad 0\\ z^\sigma &{}\quad -(1+z)^\sigma &{}\quad 1&{}\quad 0&{}\quad \cdots &{}\quad 0&{}\quad 0\\ 0&{}\quad z^{\sigma ^2}&{}\quad -(1+z)^{\sigma ^2}&{}\quad 1&{}\quad \cdots &{}\quad 0&{}\quad 0\\ \vdots &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \vdots \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad \cdots &{}\quad -(1+z)^{\sigma ^{n-3}}&{}\quad 1\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad \cdots &{}\quad z^{\sigma ^{n-2}}&{}\quad -(1+z)^{\sigma ^{n-2}} \end{pmatrix}.\nonumber \\ \end{aligned}$$
(4)

By Laplace expansion along the last column and induction on n, the determinant of B(z) can be computed as \((-1)^{n+1}\sum _{i=0}^{n-1}z^{(\sigma ^i-1)/(\sigma -1)}\). \(\square \)

The following can be useful in understanding the role of \(\delta \):

Proposition 3.3

Let \(z\in {\mathbb {F}}_{q^n}\). Then (3) holds if and only if there exists a \(y\in {\mathbb {F}}_{q^n}^*\) such that \(z=y^{\sigma -1}\) and \(\hbox {Tr}_{q^n/q}(y)=0\).

Proof

Any solution of (3) is nonzero. Raising \(\sum _{i=0}^{n-1}z^{(\sigma ^i-1)/(\sigma -1)}\) to the \(\sigma \), multiplying by z and then subtracting to the original equation yields \(1-{{\,\mathrm{N}\,}}_{q^n/q}(z)=0\). So, z is a solution of (3) if and only if \(z=y^{\sigma -1}\) for some \(y\in {\mathbb {F}}_{q^n}^*\), and \(\sum _{i=0}^{n-1}y^{\sigma ^i-1}=0\). The latter equation is equivalent to \(\hbox {Tr}_{q^n/q}(y)=0\). \(\square \)

Propositions 3.2 and 3.3 together show that, if \(f_{\sigma ,\delta }\) is not scattered, then there is an \(x\in {\mathbb {F}}_{q^n}\) such that \({{\,\mathrm{N}\,}}_{q^n/q}(\delta x^{\sigma ^{n-1}-\sigma })={{\,\mathrm{N}\,}}_{q^n/q}(\delta ) {{\,\mathrm{N}\,}}_{q^n/q}(x^{\sigma ^{n-1}-\sigma })=1\). On the other hand, \(q-1\) divides \(\sigma ^{n-1}-\sigma \), hence \({{\,\mathrm{N}\,}}_{q^n/q}( x^{\sigma ^{n-1}-\sigma })=1\). Summarizing, if \({{\,\mathrm{N}\,}}_{q^n/q}(\delta )\ne 1\), then the Lunardon–Polverino binomial is scattered, as is known.

Theorem 3.4

If \({{\,\mathrm{N}\,}}_{q^n/q}(\delta )=1\), then the Lunardon–Polverino binomial \(f_{\sigma ,\delta }(x)\) is not scattered.

Proof

\(\underline{\hbox {Case odd } n.}\) Since

$$\begin{aligned} x^{\sigma ^{n-1}-\sigma }=(x^\sigma )^{\sigma ^{n-2}-1} \end{aligned}$$

and \(\gcd (s(n-2),n)=1\), the expression \(x^{\sigma ^{n-1}-\sigma }\) takes all values in \({{\mathbb {F}}}_{q^n}\) whose norm over \({\mathbb {F}}_q\) is equal to one. This allows the substitution \(\delta x^{\sigma ^{n-1}-\sigma }=w^{\sigma -1}\) into (3). So, \(f_{\sigma ,\delta }(x)\) is not scattered if and only if \(\hbox {Tr}_{q^n/q}(w)=0\) for some nonzero w and this is trivial.

\(\underline{\hbox {Case even }n.}\) Since \(\gcd (\sigma ^{n-2}-1,\sigma ^n-1)=\sigma ^2-1\), the set of all powers of elements in \({\mathbb {F}}_{q^n}\) with exponent \(\sigma ^{n-2}-1\) coincides with the set of all powers with exponent \(\sigma ^{2}-1\). Hence for any \(x\in {{\mathbb {F}}}_{q^n}\) there exists \(u\in {{\mathbb {F}}}_{q^n}\) such that \(x^{\sigma ^{n-1}-\sigma }=u^{\sigma ^2-1}\), and conversely. This allows the substitution \(z=\delta u^{\sigma ^2-1}\) in (3), meaning that if there is u such that

$$\begin{aligned} \sum _{i=0}^{n-1}\left( \delta u^{\sigma ^2-1}\right) ^{(\sigma ^i-1)/(\sigma -1)}=0, \end{aligned}$$
(5)

then \(f_{\sigma ,\delta }(x)\) is not scattered. So, taking \(\delta =d^{\sigma -1}\), (5) is equivalent to \(\hbox {Tr}_{q^n/q}(du^{\sigma +1})=0\), \(u\ne 0\). This is a quadratic form in u in a vector space over \({\mathbb {F}}_q\) of dimension greater than two which has at least one nontrivial zero.\(\square \)

The theorem above has been proved in the particular cases \(n=4\) in [10], \(s=1\) in [2], both n and q odd in [11], and odd n in [22].

Proposition 3.5

The polynomial \(f(x)=x^\sigma +bx^{\sigma ^2}\) is scattered if only if there is no \(x\in {{\mathbb {F}}}_{q^n}^*\) such that

$$\begin{aligned} \sum _{i=0}^{n-1}w^{(\sigma ^i-1)/(\sigma -1)}=0,\ \text{ where } w=-(1+b^{-1}x^{\sigma -\sigma ^2}). \end{aligned}$$
(6)

Proof

The \(\sigma \)-matrix of Dickson associated with the polynomial

$$\begin{aligned} g(t)=-f(x)t+x^\sigma t^\sigma +bx^{\sigma ^2}t^{\sigma ^2}, \end{aligned}$$

further normalized by dividing the rows by \(bx^{\sigma ^2}\), \(b^{\sigma }x^{\sigma ^3}\), \(\ldots \) is

$$\begin{aligned} A=\begin{pmatrix} w&{}\quad -(1+w)&{}\quad 1&{}\quad \cdots &{}\quad 0&{}\quad 0\\ 0&{}\quad w^\sigma &{}\quad -(1+w)^\sigma &{}\quad \cdots &{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad w^{\sigma ^2}&{}\quad \cdots &{}\quad 0&{}\quad 0\\ \vdots &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \vdots \\ 1&{}\quad 0&{}\quad 0&{}\quad \cdots &{}\quad w^{\sigma ^{n-2}}&{}\quad -(1+w)^{\sigma ^{n-2}}\\ -(1+w)^{\sigma ^{n-1}}&{}\quad 1&{}\quad 0&{}\quad \cdots &{}\quad 0&{}\quad w^{\sigma ^{n-1}} \end{pmatrix}. \end{aligned}$$

The matrix obtained by deleting the last row and first column is B(w) (cf. (4)). \(\square \)

Corollary 3.6

Assume \(b_1,b_2\in {{\mathbb {F}}}_{q^n}\) and \({{\,\mathrm{N}\,}}_{q^n/q}(b_1)={{\,\mathrm{N}\,}}_{q^n/q}(b_2)\). Then the polynomials \(f_i(x)=x^\sigma +b_ix^{\sigma ^2}\), \(i=1,2\), are either both scattered, or both non-scattered.

Proof

If the norm of \(b_1\) is zero then the statement is trivial, so assume that it is not. Define \(w_i(x)=-(1+b_i^{-1}x^{\sigma -\sigma ^2})\) for \(i=1,2\), and note that \(w_1(x)=w_2(y)\) is equivalent to \(b_1/b_2=((x/y)^\sigma )^{\sigma -1}\), that is, \(((x/y)^\sigma )^{\sigma -1}=c^{\sigma -1}\) for some \(c\in {\mathbb {F}}_{q^n}^*\). This equation can be always solved in both x and y, whence \(w_1(x)\) and \(w_2(y)\) take the same set of values. \(\square \)

Remark 3.7

Corollary 3.6 allows to look at only \(q-1\) linearized polynomials, given s, n, and q. This makes a computer search easier. Computations with GAPFootnote 1 show that there are no scattered linearized polynomials of the form \(l_b(x)=x^q+bx^{q^2}\), \(b\ne 0\), for any \(q<223\) if \(n=5\). In [17] it is proved that for \(n=5\) and \(q\ge 223\) the linearized polynomial \(l_b(x)\) is not scattered for any \(b\ne 0\). The next proposition summarizes this.

Proposition 3.8

If \(n=5\) and \(b\in {{\mathbb {F}}}_{q^5}^*\), then the q-polynomial \(l_b(x)=x^q+bx^{q^2}\in {{\mathbb {F}}}_{q^5}[x]\) is non-scattered.

Remark 3.9

For \(n=4\) there are scattered polynomials of type \(l_b(x)\), \(b\ne 0\). By the results in [9, 10], all the related linear sets are of Lunardon–Polverino type, up to collineations.

Proposition 3.10

Let \(b\in {\mathbb {F}}_{q^n}^*\). The polynomial \(x^\sigma +bx^{\sigma ^2}\in {\mathbb {F}}_{q^n}[x]\) is not scattered if and only if the algebraic curve \(b^{-1}X^{q-1}+Y^{\sigma -1}+1=0\) in \(\mathrm {AG}(2,q^n)\) has a point \((x_0,y_0)\) with coordinates in \({\mathbb {F}}_{q^n}^*\), such that \(\hbox {Tr}_{q^n/q}(y_0)=0\).

Proof

By Proposition 3.3, the first equation in (6) is equivalent to the existence of \(y\in {\mathbb {F}}_{q^n}^*\) such that \(w=y^{\sigma -1}\), \(\hbox {Tr}_{q^n/q}(y)=0\). The second equation \(y^{\sigma -1}+1+b^{-1}x^{q-q^2}=0\) has solutions with \(x\ne 0\) if and only if \(b^{-1}x^{q-1}+y^{\sigma -1}+1=0\) does. \(\square \)

Remark 3.11

Very recently, Montanucci [16] proved that if \(n>5\), then for any q the algebraic curve \(b^{-1}X^{q-1}+Y^{q-1}+1=0\) has a point with the properties above. Together with Propositions 3.8 and 3.10 this implies that for \(n\ge 5\) no q-polynomial of type \(l_b(x)=x^q+bx^{q^2}\), \(b\ne 0\), is scattered.