# A geometric characterization of known maximum scattered linear sets of
$\mathrm{PG}(1,q^{n})$

G.G. Grimaldi, S. Gupta, G. Longobardi, R. Trombetti

###### Abstract

An $\mathbb{F}_{q}$-linear set $L=L_{U}$ of
$\Lambda=\mathrm{PG}(V,\mathbb{F}_{q^{n}})\cong\mathrm{PG}(r-1,q^{n})$ is a
set of points defined by non-zero vectors of an $\mathbb{F}_{q}$-subspace $U$
of $V$. The integer $\dim_{\mathbb{F}_{q}}U$ is called the rank of $L$. In [G.
Lunardon, O. Polverino: Translation ovoids of orthogonal polar spaces. Forum
Math. 16 (2004)], it was proven that any $\mathbb{F}_{q}$-linear set $L$ of
$\Lambda$ of rank $u$ such that $\langle L\rangle=\Lambda$ is either a
canonical subgeometry of $\Lambda$ or there are a $(u-r-1)$-dimensional
subspace $\Gamma$ of $\mathrm{PG}(u-1,q^{n})\supset\Lambda$ disjoint from
$\Lambda$ and a canonical subgeometry $\Sigma\cong\mathrm{PG}(u-1,q)$ disjoint
from $\Gamma$ such that $L$ is the projection of $\Sigma$ from $\Gamma$ onto
$\Lambda$. The subspace $\Gamma$ is called the vertex of the projection. In
this article, we will show a method to reconstruct the vertex $\Gamma$ for a
peculiar class of linear sets of rank $u=n(r-1)$ in $\mathrm{PG}(r-1,q^{n})$
called evasive linear sets. Also, we will use this result to characterize some
families of linear sets of the projective line $\mathrm{PG}(1,q^{n})$
introduced from 2018 onward, by means of certain properties of their
projection vertices, as done in [B. Csajbók, C. Zanella: On scattered linear
sets of pseudoregulus type in $\mathrm{PG}(1,q^{t})$, Finite Fields Appl. 41
(2016)] and in [C. Zanella, F. Zullo: Vertex properties of maximum scattered
linear sets of $\mathrm{PG}(1,q^{n})$. Discrete Math. 343(5) (2020)].

2020 MSC: 51E20, 05B25
Keywords: linear set, linearized polynomial, finite field, projective space,
subgeometry

##  1 Introduction

Let $u\geq n,r\geq 2$ be positive integers and let $\mathrm{PG}(r-1,q^{n})$ be
the projective space over the $r$-dimensional vector space
$V=\mathbb{F}_{q^{n}}^{r}$. A subset $L_{U}$ of $\mathrm{PG}(r-1,q^{n})$ is
called a linear set if its points are defined by non-zero elements of an
$\mathbb{F}_{q}$-subspace $U$ of $V$. More precisely,

$L_{U}=\\{\langle\textbf{u}\rangle_{\mathbb{F}_{q^{n}}}:\textbf{u}\in
U\setminus\\{\boldsymbol{0}\\}\\}.$

If $\text{dim}_{\mathbb{F}_{q}}(U)=u$, then we say that $L_{U}$ is a linear
set of rank $u$, and we will denote this by the symbol $\mathrm{rk}(L_{U})=u$.
The largest possible rank of a proper $\mathbb{F}_{q}$-linear set $L_{U}$ of
$\mathrm{PG}(r-1,q^{n})$ is $(r-1)n$, otherwise
$L_{U}=\mathrm{PG}(r-1,q^{n})$. Also, if $L_{U}$ has rank $r$ and $\langle
L_{U}\rangle=\mathrm{PG}(r-1,q^{n})$, we call it a canonical subgeometry of
$\mathrm{PG}(r-1,q^{n})$.
Regarding the size, we have that $|L_{U}|\leq q^{u-1}+q^{u-2}+\dots+q+1,$ and
if this bound is attained we say that $L_{U}$ is a scattered linear set of
$\mathrm{PG}(r-1,q^{n})$.

Two linear sets $L_{U}$ and $L_{W}$ are
$\mathrm{P}\Gamma\mathrm{L}$-equivalent (or simply equivalent) if there is
$\phi\in\mathrm{P}\Gamma\mathrm{L}(r,q^{n})$ such that $L_{U}^{\phi}=L_{W}$.
In this case, we will write $L_{U}\cong L_{W}$. It is known that any canonical
subgeometry of $\mathrm{PG}(r-1,q^{n})$ is equivalent to the linear set

$\\{\langle(a_{0},\ldots,a_{r-1})\rangle_{\mathbb{F}_{q^{n}}}\colon(a_{0},\ldots,a_{r-1})\in\mathbb{F}^{r}_{q}\setminus\\{\textbf{0}\\}\\}\cong\mathrm{PG}(r-1,q).$

A subspace $S=\mathrm{PG}(Z,q^{n})$ of $\mathrm{PG}(V,q^{n})$ has weight $i$
with respect to $L$ if $\dim_{\mathbb{F}_{q}}(Z\cap U)=i$, and we will use the
following notation $\mathrm{wt}_{L}(S)=i$.

A linear set $L\subset\mathrm{PG}(V,q^{n})$ is said to be $(h,k)_{q}$-evasive
if there exists a subspace $U\leq V$ with $L=L_{U}$ such that $\dim\langle
L\rangle\geq h-1$, and for every $h$-dimensional $\mathbb{F}_{q^{n}}$-subspace
$T$ of $V$, we have that $L_{U\cap T}$ has rank at most $k$, [1]. Note that if
$h=k=1$, then $L_{U}$ is scattered.

As it was proven in 2000 by Blokhuis and Lavrauw, if $L_{U}$ is scattered then
its rank is at most $\lfloor\frac{rn}{2}\rfloor$ and if this bound is
attained, we say that $L_{U}$ is maximum scattered, [6].

Lunardon and Polverino in [22, Theorem 2] showed that every linear set of rank
$u$ of $\Lambda=\mathrm{PG}(V,q^{n})$ spanning $\Lambda$ over
$\mathbb{F}_{q^{n}}$, is either a canonical subgeometry of $\Lambda$ or it can
be obtained as projection of a canonical subgeometry
$\Sigma\cong\mathrm{PG}(u-1,q)$ of
$\Sigma^{*}=\mathrm{PG}(u-1,q^{n})\supset\Lambda$ from a suitable subspace
$\Gamma$ of $\Sigma^{*}$ onto $\Lambda$ such that
$\Gamma\cap\Sigma=\emptyset=\Gamma\cap\Lambda$. The subspaces $\Gamma$ and
$\Lambda$ are also called vertex and center of the projection, respectively.

We recall that if $\Sigma$ is a canonical subgeometry of $\Sigma^{*}$, a
subspace $S=\mathrm{PG}(Z,\mathbb{F}_{q^{n}})$ of $\Sigma^{*}$ is a subspace
of $\Sigma$ if the dimension of $Z$ is equal to the rank of the
$\mathbb{F}_{q}$-linear set $S\cap\Sigma$. This is equivalent to say that if
$\sigma$ is a collineation of $\Sigma^{*}$ of order $n$ fixing $\Sigma$
pointwise (i.e., $\mathrm{Fix}(\sigma)=\Sigma$), then the subspace $S$ is
fixed by $\sigma$, see [26, Lemma 1]. Let $P$ be a point of $\Sigma^{*}$, the
subspace $\mathcal{L}_{P,\sigma}=\langle
P,P^{\sigma},\ldots,P^{\sigma^{n-1}}\rangle$ of $\Sigma^{*}$ clearly is a
subspace of $\Sigma$ of dimension at most $n-1$. The point $P$ will be called
an imaginary point of $\Sigma^{*}$, with respect to $\Sigma$, if
$\dim\mathcal{L}_{P,\sigma}=n-1$. If the collineation $\sigma$ is clear from
the context, we will indicate $\mathcal{L}_{P,\sigma}$ simply by
$\mathcal{L}_{P}$. Also, we will denote by $\mathrm{Aut}(\Sigma)$, the
automorphism group of $\Sigma$, i.e., the group made up of all the maps in
$\mathrm{P}\Gamma\mathrm{L}(u,q^{n})$ which commute with $\sigma$ and we will
put $\mathrm{LAut}(\Sigma)=\mathrm{Aut}(\Sigma)\cap\mathrm{PGL}(u,q^{n})$.
In this article a method to reconstruct, up to the action of the setwise
stabilizer of $\Sigma$ in $\mathrm{P}\Gamma\mathrm{L}(u,q^{n})$, the vertex
$\Gamma$ of an $(r-1,n(r-1)-2)_{q}$-evasive linear set of rank $u=n(r-1)$ in
$\mathrm{PG}(r-1,q^{n})$ from a set made up of $r-1$ imaginary points of
$\mathrm{PG}(u-1,q^{n})$ and their conjugates with respect to a collineation
of $\mathrm{PG}(u-1,q^{n})$ fixing pointwise a suitable subgeometry $\Sigma$,
will be given; see Theorem 2.1 and 2.4.

The case when $r=2$ and $L$ is a maximum scattered linear sets of
$\mathrm{PG}(1,q^{n})$ is of particular interest because, as proven by Sheekey
in Section $5$ of [32], these linear sets yield remarkable examples of Maximum
Rank Distance codes (MRD codes, for short); indeed, linear MRD codes
$2$-dimensional over $\mathbb{F}_{q^{n}}$ with minimum distance $n-1$, which
have various important applications in communications and cryptography, see
[33] and the references therein. In this circumstance, in the same spirit of
[14] and [37], elaborating on above mentioned achievement, we will
characterize families of maximum scattered linear sets of
$\mathrm{PG}(1,q^{n})$ introduced from 2018 onwards (see [11, 12, 23, 25, 28,
35]) through certain properties of their projection vertices.

##  2 Preliminary results

Up to a suitable projectivity, any linear set $L$ of rank $n(r-1)$ in
$\mathrm{PG}(r-1,q^{n})$ can be written in the following form

${}L=L_{F}=\\{\langle({\bf x},F({\bf
x}))\rangle_{\mathbb{F}_{q^{n}}}\,\colon\,{\bf
x}=(x_{0},x_{1},...,x_{r-2})\in\mathbb{F}_{q^{n}}^{r-1},{\bf x}\neq{\bf
0}\\},$ (1)

where

$F({\bf x})=\sum_{j=0}^{r-2}f_{j}(x_{j}),$ (2)

and $f_{j}(x_{j})=\sum_{i=1}^{n-1}a_{ij}x_{j}^{q^{si}}$ is an
$\mathbb{F}_{q^{s}}$-linearized polynomial with coefficients over
$\mathbb{F}_{q^{n}}$ in the indeterminate $x_{j}$ for each
$j\in\\{0,1,...,r-2\\}$ and $\gcd(s,n)=1$, see [9, proof of Proposition 4].
Let us set some notations that we will use throughout this section. If
$F(\textbf{x})$ is a multivariate polynomial as in (2), we will denote by

  * $a)$

$m_{j}:=\deg_{q^{s}}f_{j}(x_{j})=\max\\{i\in\\{1,2,\ldots,n-1\\}\,|\,a_{ij}\neq
0\\}$, i.e., the $q^{s}$-degree of $f_{j}(x_{j})$, and

  * $b)$

$I_{j}:=\mathrm{supp}_{q^{s}}f_{j}(x_{j})=\\{i\in\\{1,2,\ldots,n-1\\}:a_{ij}\neq
0\\}$, i.e., the $q^{s}$-support of $f_{j}(x_{j})$.

Let $\Sigma^{*}=\mathrm{PG}(n(r-1)-1,q^{n})$ be a projective space and
consider the canonical subgeometry of $\Sigma^{*}$ defined as follows:

$\boldsymbol{\Sigma}=\\{\langle(\textbf{ x},\textbf{ x}^{q},\ldots,\textbf{
x}^{q^{n-1}})\rangle_{\mathbb{F}_{q^{n}}}\,\colon\,\textbf{
x}\in\mathbb{F}_{q^{n}}^{r-1},\textbf{
x}\neq\textbf{0}\\}\cong\mathrm{PG}(n(r-1)-1,q),$ (3)

where ${\bf x}^{q^{m}}=(x_{0}^{q^{m}},x_{1}^{q^{m}},\ldots,x_{r-2}^{q^{m}})$
for each $m\in\\{0,1,\ldots,n-1\\}$. Clearly, $\bf{\Sigma}$ is the set of the
points fixed by the collineation $\boldsymbol{\sigma}$ of $\Sigma^{*}$ of
order $n$, defined as

$\displaystyle\boldsymbol{\sigma}\,:\,\langle(x_{00},x_{10},\ldots,x_{r-2,0},x_{01},x_{11},\ldots,x_{r-2,1},\ldots,x_{0,n-1},x_{1,n-1},\ldots,x_{r-2,n-1})\rangle_{\mathbb{F}_{q^{n}}}\mapsto$
$\displaystyle\langle(x^{q}_{0,n-1},x^{q}_{1,n-1},\ldots,x^{q}_{r-2,n-1},x^{q}_{00},x^{q}_{10},\ldots,x^{q}_{r-2,0},\ldots,x^{q}_{0,n-2},x^{q}_{1,n-2},\ldots,x^{q}_{r-2,n-2})\rangle_{\mathbb{F}_{q^{n}}}.$
(4)

######  Theorem 2.1.

Let $L_{F}$ be a linear set of rank $n(r-1)$ of $\mathrm{PG}(r-1,q^{n})$ with
$F(\bf{x})$ as in (2). Let $\Sigma^{*}=\mathrm{PG}(n(r-1)-1,q^{n})$. Then,
there exist

  * $1.$

a canonical subgeometry $\Sigma$, a generator $\sigma$ of the subgroup of
$\mathrm{P}\Gamma\mathrm{L}(n(r-1),q^{n})$ fixing pointwise $\Sigma$ and $r-1$
imaginary points $P_{0},P_{1},\ldots,P_{r-2}$ of $\Sigma^{*}$ (wrt $\Sigma$),

  * $2.$

an $(n(r-1)-(r+1))$-dimensional subspace $\Gamma$ of $\Sigma^{*}$ fulfilling

    * $i)$

$P_{k}^{\sigma^{j_{k}}}\in\Gamma$ for any $j_{k}\not\in I_{k}$, $j_{k}\not=0$
and $k\in\\{0,1,\ldots,r-2\\}$,

    * $ii)$

any line

$\langle P_{\ell}^{\sigma^{j_{\ell}}},P_{\ell}^{\sigma^{m_{\ell}}}\rangle$

with $j_{\ell}\in I_{\ell}\setminus\\{m_{\ell}\\}$ for
$\ell\in\\{0,1,\ldots,r-2\\}$ meets $\Gamma$,

    * $iii)$

if $r>2$, any line

$\langle P_{i}^{\sigma^{m_{i}}},P_{r-2}^{\sigma^{m_{r-2}}}\rangle$

with $i\in\\{0,1,\ldots,r-3\\}$ meets $\Gamma$,

  * $3.$

an $(r-1)$-dimensional subspace $\Lambda$ of $\Sigma^{*}$ through the points
$P_{i}$, $i\in\\{0,1,\ldots,r-2\\}$,

such that $\Gamma\cap\Sigma=\emptyset=\Gamma\cap\Lambda$ and
$\mathrm{p}_{\Gamma,\Lambda}(\Sigma)$ is equivalent to $L_{F}$.

###### Proof.

Let $F(\textbf{x})$ be the multivariate polynomial with coefficients over
$\mathbb{F}_{q^{n}}$ defined as in (2) with $\gcd(s,n)=1$. Let us assume that
$\Sigma=\boldsymbol{\Sigma}$ and hence that $\sigma=\boldsymbol{\sigma}^{s}$ (
as defined in (3) and (2), respectively) and let $\Gamma$ be the
$(n(r-1)-(r+1))$-dimensional subspace of $\Sigma^{*}$ with equations

$\Gamma:\begin{cases}x_{00}=x_{10}=\cdots=x_{r-2,0}=0\\\ \sum_{i\in
I_{0}}a_{0i}x_{0,is}+\sum_{i\in I_{1}}a_{1i}x_{1,is}+\cdots+\sum_{i\in
I_{r-2}}a_{r-2,i}x_{r-2,is}=0\end{cases}$ (5)

and $P_{k}$, $k\in\\{0,1,\ldots,r-2\\}$, be the points having $k$-th
coordinate equals to $1$ and all the others equal to zero. Clearly, each of
these points is imaginary with respect to $\Sigma$. Moreover, it is
straightforward to see that

  * $i)$

for any $j_{k}\notin I_{k}$ and $j_{k}\not=0$,
$P_{k}^{\sigma^{j_{k}}}\in\Gamma$. Indeed, observe that
$P_{k}^{\sigma^{j_{k}}}$ has the $(sj_{k}(r-1)+k+1)$-th
$(\textrm{mod}\,n(r-1)\,)$ coordinate $1$ and all others equal to zero;

  * $ii)$

any line

$\langle P_{\ell}^{\sigma^{j_{\ell}}},P_{\ell}^{\sigma^{m_{\ell}}}\rangle$

with $j_{\ell}\in I_{\ell}\setminus\\{m_{\ell}\\}$ for
$\ell\in\\{0,1,\ldots,r-2\\}$ meets $\Gamma$ in the point
$Q_{j_{\ell},m_{\ell}}$, with all zero coordinates except the
$(sj_{\ell}(r-1)+\ell+1)$-th $(\textrm{mod}\,n(r-1)\,)$ and the
$(sm_{\ell}(r-1)+\ell+1)$-th $(\textrm{mod}\,n(r-1)\,)$ ones, which are equal
to $a_{\ell,m_{\ell}}$ and $-a_{\ell,j_{\ell}}$, respectively; and

  * $iii)$

if $r>2$, any line

$\langle P_{i}^{\sigma^{m_{i}}},P_{r-2}^{\sigma^{m_{r-2}}}\rangle$

with $i\in\\{0,1,\ldots,r-3\\}$ meets $\Gamma$ in the point $R_{i}$ with all
zero coordinates except the $(sm_{i}(r-1)+i+1)$-th $(\textrm{mod}\,n(r-1)\,)$
and the $(sm_{r-2}(r-1)+r-1)$-th $(\textrm{mod}\,n(r-1)\,)$ ones, which are
equal to $a_{r-2,m_{r-2}}$ and $-a_{i,m_{i}}$, respectively.

Finally, let $\Lambda=\langle
P_{0},P_{1},\ldots,P_{r-2},P_{r-2}^{\sigma^{m_{r-2}}}\rangle$. Easy
computations show that $\Lambda$ is an $(r-1)$-dimensional subspace of
$\Sigma^{*}$ such that $\Gamma\cap\Sigma=\emptyset=\Lambda\cap\Gamma$ and
$\mathrm{p}_{\Gamma,\Lambda}(\Sigma)$ is equivalent to $L_{F}$. ∎

Using the notation introduced in Theorem 2.1 and taking into account that the
points $P_{i}^{\sigma^{j}}$’s are independent, it is straightforward to check
that the points

  * $(a)$

$P_{k}^{\sigma^{j_{k}}},j_{k}\not\in
I_{k},j_{k}\not=0\,\textnormal{and}\,k\in\\{0,1,\ldots,r-2\\},$

  * $(b)$

$Q_{j_{\ell},m_{\ell}}\in\langle
P_{\ell}^{\sigma^{j_{\ell}}},P_{\ell}^{\sigma^{m_{\ell}}}\rangle,j_{\ell}\in
I_{\ell}\setminus\\{m_{\ell}\\}$ and $\ell\in\\{0,1,\ldots,r-2\\}$,

  * $(c)$

$R_{i}\in\langle
P^{\sigma^{m_{i}}}_{i},P^{\sigma^{m_{r-2}}}_{r-2}\rangle,i\in\\{0,1,\ldots,r-3\\}$
for $r>2$,

are independent and they span the subspace $\Gamma$. Next, we prove the
following result.

######  Lemma 2.2.

Let $\Sigma$ be a subgeometry of $\Sigma^{*}=\mathrm{PG}(n(r-1)-1,q^{n})$ and
let $\sigma$ denotes a collineation of order $n$ of $\Sigma^{*}$ such that
$\mathrm{Fix}(\sigma)=\Sigma$. Let $P_{0},P_{1},\ldots,P_{r-2}$ and
$P^{\prime}_{0},P^{\prime}_{1},\ldots,P^{\prime}_{r-2}$ be imaginary points
(wrt $\Sigma$) such that

$\mathcal{L}_{P_{i}}=\mathcal{L}_{P^{\prime}_{i}}\,\,\,\textrm{and}\,\,\,\langle\mathcal{L}_{P_{0}},\mathcal{L}_{P_{1}},\ldots,\mathcal{L}_{P_{r-2}}\rangle=\Sigma^{*}.$

Then, there exists a collineation $\varphi\in\mathrm{LAut}(\Sigma)$ such that
$P^{\varphi}_{i}=P^{\prime}_{i}$ for $i\in\\{0,1,\ldots,r-2\\}$.

###### Proof.

Set $\mathcal{L}_{i}:=\mathcal{L}_{P_{i}}=\mathcal{L}_{P^{\prime}_{i}}$. By
hypothesis, for each $i\in\\{0,1,\ldots,r-2\\}$,

$\Sigma_{i}:=\mathcal{L}_{i}\cap\Sigma=\mathrm{PG}(V_{i})\cong\mathrm{PG}(n-1,q)$

and $\mathcal{L}_{i}=\mathrm{PG}(W_{i})\cong\mathrm{PG}(n-1,q^{n})$ where
$W_{i}=\langle V_{i}\rangle_{\mathbb{F}_{q^{n}}}$. Moreover, note that
$\mathrm{Fix}(\sigma|_{{\mathcal{L}_{i}}})=\Sigma_{i}$. By applying [7,
Proposition 3.1], we have that there exists a collineation
$\varphi_{i}\in\mathrm{LAut}(\Sigma_{i})$ of $\mathcal{L}_{i}$ fixing
$\Sigma_{i}$, and mapping $P_{i}$ into $P^{\prime}_{i}$. In the remaining part
of the proof we will use the same symbol $\varphi_{i}$ to denote both the
collineation and its underlying linear map; i.e., we will write
$\varphi_{i}:W_{i}\longrightarrow W_{i}$ for any $i\in\\{0,1,\ldots,r-2\\}$.

Since $\Sigma^{*}=\mathrm{PG}(\oplus_{i=0}^{r-2}W_{i})$ and
$\Sigma=\mathrm{PG}(\oplus_{i=0}^{r-2}V_{i})$, we may consider the map
$\varphi\in\mathrm{PGL}(n(r-1),q^{n})$ with the following associated linear
map

$w_{0}+w_{1}+w_{2}+\ldots+w_{r-2}\in\oplus_{i=0}^{r-2}W_{i}\,\longrightarrow\,\varphi_{0}(w_{0})+\varphi_{1}(w_{1})+\varphi_{2}(w_{2})+\ldots+\varphi_{r-2}(w_{r-2})\in\oplus_{i=0}^{r-2}W_{i}.$

First, we prove that $\varphi\in\mathrm{LAut}(\Sigma)$, i.e.,
$\sigma\varphi=\varphi\sigma$. Let $P=\langle w\rangle_{\mathbb{F}_{q^{n}}}$
with $w=w_{0}+w_{1}+\ldots+w_{r-2}$ where $w_{i}\in W_{i}$, then

$\begin{split}\sigma(\varphi({\langle
w\rangle_{\mathbb{F}_{q^{n}}}}))&=\sigma(\langle\varphi_{0}(w_{0})+\varphi_{1}(w_{1})+\varphi_{2}(w_{2})+\ldots+\varphi_{r-2}(w_{r-2})\rangle_{\mathbb{F}_{q^{n}}})\\\
&=\langle\sigma(\varphi_{0}(w_{0}))+\sigma(\varphi_{1}(w_{1}))+\sigma(\varphi_{2}(w_{2}))+\ldots+\sigma(\varphi_{r-2}(w_{r-2}))\rangle_{\mathbb{F}_{q^{n}}}\\\
&=\langle\sigma_{|_{W_{0}}}(\varphi_{0}(w_{0}))+\sigma_{|_{W_{1}}}(\varphi_{1}(w_{1}))+\sigma_{|_{W_{2}}}(\varphi_{2}(w_{2}))+\ldots+\sigma_{|_{W_{r-2}}}(\varphi_{r-2}(w_{r-2}))\rangle_{\mathbb{F}_{q^{n}}}\\\
&=\langle\varphi_{0}(\sigma(w_{0}))+\varphi_{1}(\sigma(w_{1}))+\varphi_{2}(\sigma(w_{2}))+\ldots+\varphi_{r-2}(\sigma(w_{r-2}))\rangle_{\mathbb{F}_{q^{n}}}\\\
&=\varphi(\langle\sigma(w_{0})+\sigma(w_{1})+\sigma(w_{2})+\ldots+\sigma(w_{r-2})\rangle_{\mathbb{F}_{q^{n}}})\\\
&=\varphi(\sigma(\langle w\rangle_{\mathbb{F}_{q^{n}}}))\end{split}$

where again we have denoted the semilinear map associated with $\sigma$ and
$\varphi$ by the same letters, respectively. Finally, it is routine to check
that $P_{i}^{\varphi}=P^{\prime}_{i}$, $i\in\\{0,1,\ldots,r-2\\}$. Thus, the
result follows. ∎

We now prove a generalization of [7, Proposition 3.1] which will play an
important role in the proof of Theorem 2.4.

######  Proposition 2.3.

Let $\Sigma$ be a subgeometry of $\Sigma^{*}=\mathrm{PG}(n(r-1)-1,q^{n})$. Let
$\sigma$ denote a collineation of order $n$ of $\Sigma^{*}$ such that
$\mathrm{Fix}(\sigma)=\Sigma$. Then, the group $\mathrm{LAut(\Sigma)}$ acts
$(r-1)$-transitively on sets of $r-1$ independent points
$P_{0},P_{1},\ldots,P_{r-2}$ such that
$\langle\mathcal{L}_{P_{0}},\mathcal{L}_{P_{1}},\dots,\mathcal{L}_{P_{r-2}}\rangle=\Sigma^{*}$.

###### Proof.

Let $P_{0},P_{1},\dots,P_{r-2}$ be the independent points such that
$\langle\mathcal{L}_{P_{0}},\mathcal{L}_{P_{1}},\dots,\mathcal{L}_{P_{r-2}}\rangle=\Sigma^{*}$.
Therefore, any point $P_{i}$, $i\in\\{0,1,\ldots,r-2\\}$, is an imaginary
point of $\Sigma^{*}$ (wrt $\Sigma$). So, for any $i\in\\{0,1,\ldots,r-2\\}$,
there exist $n$ points in $\Sigma\cap\mathcal{L}_{P_{i}}$, say
$P_{i0},P_{i1},\dots,P_{i(n-1)}$, such that:

$\\{P_{ij}\in\Sigma:i\in\\{0,1,\dots,r-2\\}\text{ and
}j\in\\{0,1,\dots,n-1\\}\\}$

are in general position in $\Sigma$. Let $Q_{0},Q_{1},\dots,Q_{r-2}$ be
$(r-1)$ points such that $\langle\mathcal{L}_{Q_{i}}\colon
i=0,1,\ldots,r-2\rangle=\Sigma^{*}$. So, in the same way, we get a set

$\\{Q_{ij}\in\Sigma:i\in\\{0,1,\dots,r-2\\}\text{ and
}j\in\\{0,1,\dots,n-1\\}\\}$

of points of $\Sigma$ in general position, as well. Since
$\mathrm{LAut}(\Sigma)$ acts transitively on subset of $\Sigma$ of order
$n(r-1)$ which are in general position, there exists a collineation
$\psi\in\mathrm{LAut}(\Sigma)$ such that $P_{ij}^{\psi}=Q_{ij}$ for any
$i\in\\{0,1,\ldots,r-2\\}$ and $j\in\\{0,1,\ldots,n-1\\}$. Moreover, it holds
that

$\begin{split}\mathcal{L}_{P_{i}}^{\psi}&=(\langle
P_{i0},P_{i1},\ldots,P_{i(n-1)}\rangle)^{\psi}=\langle
P_{i0}^{\psi},P_{i1}^{\psi},\ldots,P_{i(n-1)}^{\psi}\rangle\\\ &=\langle
Q_{i0},Q_{i1},\ldots,Q_{i(n-1)}\rangle=\mathcal{L}_{Q_{i}}.\end{split}$

Now, if $P_{i}^{\psi}=Q_{i}$, we get the result. Otherwise, assume
$P_{i}^{\psi}=R_{i}$, for some $R_{0},R_{1},\ldots,R_{r-2}$. Since
$\mathcal{L}_{R_{i}}=\mathcal{L}_{Q_{i}}$, by Lemma 2.2 there exists
$\varphi\in\mathrm{LAut}(\Sigma)$ such that $R_{i}^{\varphi}=Q_{i}$ and so
$P_{i}^{\psi\varphi}:=\varphi(\psi(P_{i}))=Q_{i}$. This completes the proof. ∎

We are now in the position to prove the following inverse of Theorem 2.1.

######  Theorem 2.4.

Let $r,n\geq 2$, $I_{j}\subseteq\\{1,\ldots,n-1\\},$ with
$j\in\\{0,1,\ldots,r-2\\}$. Let $\Sigma$ be a canonical subgeometry of
$\Sigma^{*}=\mathrm{PG}(n(r-1)-1,q^{n})$ and consider $\Gamma$ and $\Lambda$
subspaces of $\Sigma^{*}$ with dimensions $(n(r-1)-(r+1))$ and $(r-1)$,
respectively, such that $\Gamma\cap\Sigma=\emptyset=\Lambda\cap\Gamma$.
Assume $L=\mathrm{p}_{\Gamma,\Lambda}(\Sigma)$ is an
$(r-1,n(r-1)-2)_{q}$-evasive linear set of rank $n(r-1)$.
If there exist $r-1$ points $P_{0},P_{1},\ldots,P_{r-2}$ such that $\Gamma$ is
spanned by points defined as in $(a)$, $(b)$ and $(c)$ where $\sigma$ is a
generator of the subgroup of $\mathrm{P}\Gamma\mathrm{L}(n(r-1),q^{n})$ fixing
$\Sigma$ pointwise. Then,

  1. 1.

$P_{0},P_{1},\ldots,P_{r-2}$ are imaginary points (wrt $\Sigma$), and

  2. 2.

$L\cong L_{F}$ as in (1) where $\mathrm{supp}_{q^{s}}f_{j}(x_{j})=I_{j}$, for
some integer $1\leq s\leq n-1$ with $\gcd(s,n)=1$ and $j=0,\ldots,r-2$.

###### Proof.

Proving point 1. of the statement, is equivalent to show that $\dim{\cal
L}=n(r-1)-1$, where
$\mathcal{L}=\langle\mathcal{L}_{P_{0}},\mathcal{L}_{P_{1}},\ldots,\mathcal{L}_{P_{r-2}}\rangle$.
By our hypothesis, $\Gamma\subset{\cal L}$ and hence

$n(r-1)-(r+1)<\dim(\mathcal{L})\leq n(r-1)-1.$

Assume that $\dim{\cal L}<n(r-1)-1$. Precisely, let
$\dim\mathcal{L}=n(r-1)-(r+1)+t$, where $t\in\\{1,2,\dots,r-1\\}$. Then, by
Grassman’s Formula $\dim(\Lambda\cap\mathcal{L})=t-1.$ Since $\mathcal{L}$ is
fixed by $\sigma$, we get

$\mathrm{wt}_{L}(\Lambda\cap\mathcal{L})=\mathrm{rk}(\mathcal{L}\cap\Sigma)=\dim\mathcal{L}+1=n(r-1)-(r+1)+t+1.$

This means that we have a $t$-dimensional $\mathbb{F}_{q^{n}}$-subspace of
$\mathbb{F}_{q^{n}}^{r}$, which meets an $\mathbb{F}_{q}$-linear subspace in
an $\mathbb{F}_{q}$-linear subspace with dimension $(n(r-1)-(r+1)+t+1)$ over
$\mathbb{F}_{q}$. Since by [1, Proposition 2.6] an
$(r-1,n(r-1)-2)_{q}$-evasive is also $(t,n(r-1)-(r+1)+t)_{q}$-evasive, we
contradict the assumption. Therefore, $\dim(\mathcal{L})=n(r-1)-1$. By
Proposition 2.3, let $\Sigma$ as in (3), then there exists an integer $1\leq
s\leq n-1$ with $\gcd(s,n)=1$ such that $\sigma=\boldsymbol{\sigma}^{s}$.
Then, we may assume that any $P_{k}$, $k=0,1,\ldots,r-2$, is a point with
homogeneous coordinates equal to 1 in the $k$-th position and 0 in all others.
Furthermore, since $\Gamma$ is spanned by points defined as in $(a)$, $(b)$
and $(c)$, it turns out to have equations

$\Gamma:\begin{cases}x_{00}=x_{10}=\cdots=x_{r-2,0}=0\\\ \sum_{i\in
I_{0}}a_{0i}x_{0,is}+\sum_{i\in I_{1}}a_{1i}x_{1,is}+\cdots+\sum_{i\in
I_{r-2}}a_{r-2,i}x_{r-2,is}=0\end{cases}$

with some $a_{ji}\in\mathbb{F}^{*}_{q^{n}}$, $j=0,\ldots,r-2$ and $i\in
I_{j}$. Consequently, $L\cong L_{F}$, where
$F(\textbf{x})=\sum_{j=0}^{r-2}\sum_{i\in I_{j}}a_{ji}x^{q^{si}}$. ∎

##  3 Maximum scattered linear sets of $\mathrm{PG}(1,q^{n})$

In this section, we will focus on maximum scattered linear sets of the
projective line $\mathrm{PG}(1,q^{n})$. These are covered as a special case in
the results of the previous section providing $r=2$. As expressed in Section
1, our aim is to characterize the families of these linear sets which were
introduced from 2018 onwards (see [11, 12, 23, 25, 28, 35]) by means of
certain properties of their projection vertices.

For this purpose, we will start by recalling some characterisation results
obtained in this direction, regarding the two classical and up to date most
investigated examples: pseudoregulus and Lunardon-Polverino type scattered
linear sets of the line.

Let $\ell$ be an integer such that $\ell\mid n$. In the following, we will
indicate by

$\mathrm{N}_{q^{n}/q^{\ell}}(x)=x^{\frac{q^{n}-1}{q^{\ell}-1}}\quad\textnormal{and}\quad\mathrm{Tr}_{q^{n}/q^{\ell}}(x)=\sum_{i=0}^{n/\ell-1}x^{q^{i\ell}},$

the norm and trace of $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q^{\ell}}$ of an
element $x\in\mathbb{F}_{q^{n}}$, respectively.

As noted in [14], two $\mathbb{F}_{q}$-subspaces of $\mathbb{F}_{q^{n}}^{2}$
defining the same scattered linear set in $\mathrm{PG}(1,q^{n})$ are not
necessarily on the same orbit under the action of $\Gamma\mathrm{L}(2,q^{n})$.
This leads in [10] to the introduction of the notion of
$\mathcal{Z}(\Gamma\mathrm{L})$-class and $\Gamma\mathrm{L}$-class of a linear
set of rank $n$ in $\mathrm{PG}(1,q^{n})$. Precisely,

######  Definition 3.1.

[10, Definition 2.4] Let $L_{U}$ be an $\mathbb{F}_{q}$-linear set of
$\mathrm{PG}(1,q^{n})$ of rank $n$ with maximum field of linearity
$\mathbb{F}_{q}$, i.e., $L_{U}$ is not an $\mathbb{F}_{q^{u}}$-linear set for
any $u>1$. Then $L_{U}$ has $\mathcal{Z}(\Gamma\mathrm{L})$-class $z$, if $z$
is the largest integer such that there exist $\mathbb{F}_{q}$-subspaces
$U_{1},U_{2},\ldots,U_{z}$ of $\mathbb{F}_{q^{n}}^{2}$ with

  1. 1.

$L_{U_{i}}=L_{U}$ for $i\in\\{1,2,\ldots,z\\}$, and

  2. 2.

there is no $\lambda\in\mathbb{F}_{q^{n}}^{*}$ such that $U_{i}=\lambda U_{j}$
for each $i\neq j$ and $i,j\in\\{1,2,...,z\\}$.

Similarly, the notion of $\Gamma\mathrm{L}$-class of an
$\mathbb{F}_{q}$-linear set was defined.

######  Definition 3.2.

[10, Definition 2.5]. Let $L_{U}$ be an $\mathbb{F}_{q}$-linear set of
$\mathrm{PG}(1,q^{n})$ of rank $n$ with maximum field of linearity
$\mathbb{F}_{q}$. Then $L_{U}$ is of $\Gamma\mathrm{L}$-class $c$, if $c$ is
the largest integer such that there exist $\mathbb{F}_{q}$-subspaces
$U_{1},U_{2},\ldots,U_{c}$ of $\mathbb{F}_{q^{n}}^{2}$ with

  1. 1.

$L_{U_{i}}=L_{U}$ for $i\in\\{1,2,\ldots,c\\}$, and

  2. 2.

there is no $\kappa\in\Gamma\mathrm{L}(2,q^{n})$ such that
$U_{i}=U_{j}^{\kappa}$ for each $i\neq j$, $i,j\in\\{1,2,\ldots,c\\}$.

In the following, we will denote the $\mathcal{Z}(\Gamma\mathrm{L})$-class and
the $\Gamma\mathrm{L}$-class of a linear set $L=L_{U}$ by
$c_{\mathcal{Z}(\Gamma\mathrm{L})}(L)$ and $c_{\Gamma\mathrm{L}}(L)$,
respectively. These integers are invariant under the action of the group
$\mathrm{P}\Gamma\mathrm{L}(2,q^{n})$ on $L$, see [10, Proposition 2.6], and
it is clear that $c_{\Gamma\mathrm{L}}(L)\leq
c_{\mathcal{Z}(\Gamma\mathrm{L})}(L)$. The linear set $L$ is called $simple$
if its $\Gamma\mathrm{L}$-class is one. Also, in [37], the
$\Gamma\mathrm{L}$-class of a linear set was rephrased via a group action
property on its projection vertices, see [10, Section 5.2], [14, Theorem 6 and
7] and [37, Theorem 3.4]. More precisely, the following was proved

######  Theorem 3.3.

Let $\Sigma$ be a canonical subgeometry and $\Lambda$ be a line of
$\Sigma^{*}=\mathrm{PG}(n-1,q^{n})$. The $\Gamma\mathrm{L}$-class of a linear
set $L\subset\Lambda$ is the number of orbits, under the action of
$\mathrm{Aut}(\Sigma)$ of $(n-3)$-subspaces $\Gamma$ of $\Sigma^{*}$ disjoint
from $\Sigma$ and from the line $\Lambda$ such that
$\mathrm{p}_{\Gamma,\Lambda}(\Sigma)$ is equivalent to $L$.

Let $L_{U}$ be a linear set of $\mathrm{PG}(1,q^{n})$ and let $\tau$ be a
polarity of the projective line. Then, it is always possible to construct
another linear set, which is called the dual linear set of $L_{U}$ with
respect to the polarity $\tau$, see [30]. In particular, if
$\beta:\mathbb{F}_{q^{n}}^{2}\times\mathbb{F}_{q^{n}}^{2}\longrightarrow\mathbb{F}_{q^{n}}$
is a non-degenerate reflexive sesquilinear form on $\mathbb{F}_{q^{n}}^{2}$
determining a polarity $\tau$, then the map
$\beta^{\prime}=\mathrm{Tr}_{q^{n}/q}\circ\beta$ is a non-degenerate reflexive
sesquilinear form on $\mathbb{F}_{q^{n}}^{2}$, when this is regarded as a
$2n$-dimensional vector space over $\mathbb{F}_{q}$.

Let $\perp_{\beta}$ and $\perp_{\beta^{\prime}}$ be the orthogonal complement
maps defined by $\beta$ and $\beta^{\prime}$ on the lattices of the
$\mathbb{F}_{q^{n}}$-subspaces and $\mathbb{F}_{q}$-subspaces of
$\mathbb{F}_{q^{n}}^{2}$, respectively. The dual linear set $L_{U}^{\tau}$ of
$L_{U}$ with respect to the polarity $\tau$ is the $\mathbb{F}_{q}$-linear set
of rank $n$ of $\mathrm{PG}(1,q^{n})$ defined by the orthogonal complement
$U^{\perp_{\beta^{\prime}}}$ and, up to projective equivalence, such a linear
set does not depend on $\tau$, see [30, Proposition 2.5].

As a direct consequence of [29, Theorem 1], we may state the following result:

######  Theorem 3.4.

Let $L_{f}$ be a maximum scattered linear set of $\mathrm{PG}(1,q^{n})$. If
$W$ is an $n$-dimensional $\mathbb{F}_{q}$-subspace of
$\mathbb{F}_{q^{n}}^{2}$ such that $L_{W}=L_{f}$ with
$f(x)=\sum_{i=1}^{n-1}a_{i}x^{q^{i}}$, then one of the following possibilities
may occur:

  1. 1.

$L_{f}$ is equivalent to a linear set of pseudoregulus type (see [6]), i.e.,

$L^{1,n}_{s}=\\{\langle(x,x^{q^{s}})\rangle_{\mathbb{F}_{q^{n}}}\,\colon\,x\in\mathbb{F}^{*}_{q^{n}}\\};$
(6)

  2. 2.

$W=\lambda U_{f}$ for some $\lambda\in\mathbb{F}^{*}_{q^{n}}$;

  3. 3.

$W=\lambda U_{\hat{f}}$ for some $\lambda\in\mathbb{F}_{q^{n}}^{*}$, where

$\hat{f}(x)=\sum_{i=1}^{n-1}a_{n-i}^{q^{i}}x^{q^{i}}.$

In particular, if $L_{f}$ is of pseudoregulus type
$c_{\Gamma\mathrm{L}}(L_{f})=\phi(n)/2$, where $\phi$ is the totient function,
otherwise $c_{\Gamma\mathrm{L}}(L_{f})\leq 2$.

In [21], Lunardon and Polverino presented a class of linear sets which was
generalised later by Sheekey in [32]. They are known in the literature as
linear sets of LP-type and an element of this family is
$\mathrm{P}\Gamma\mathrm{L}$-equivalent to

$L^{2,n}_{s,\eta}=\\{\langle(x,\eta
x^{q^{s}}+x^{q^{n-s}})\rangle_{\mathbb{F}_{q^{n}}}\,\colon\,x\in\mathbb{F}^{*}_{q^{n}}\\},$
(7)

where $\mathrm{N}_{{q^{n}}/q}(\eta)\notin\\{0,1\\}$, $\gcd(r,n)=1$ for any
$n\geq 3$ and $q>2$.

In the following, we will state two theorems containing characterization
results for the linear sets of pseudoregulus and LP-type by means of some
geometric properties of their vertices.

######  Theorem 3.5.

[15, Theorem 2.3] Let $\Sigma$ be a canonical subgeometry of
$\mathrm{PG}(n-1,q^{n})$, $q>2$, $n\geq 3$. Assume that $\Gamma$ and $\Lambda$
are $(n-3)$-subspace and a line of $\mathrm{PG}(n-1,q^{n})$, respectively,
such that $\Sigma\cap\Gamma=\Lambda\cap\Gamma=\emptyset$. Then the following
assertions are equivalent:

  1. 1.

The set $\mathrm{p}_{\Gamma,\Lambda}(\Sigma)$ is a scattered
$\mathbb{F}_{q}$-linear set of pseudoregulus type;

  2. 2.

there exists a generator $\sigma$ of the subgroup of
$\mathrm{P}\Gamma\mathrm{L}(n,q^{n})$ fixing $\Sigma$ pointwise such that
$\dim(\Gamma\cap\Gamma^{\sigma})=n-4$; furthermore $\Gamma$ is not contained
in the span of any hyperplane of $\Sigma$;

  3. 3.

There exists a point $P$ and a generator $\sigma$ of the subgroup of
$\mathrm{P}\Gamma\mathrm{L}(n,q^{n})$ fixing $\Sigma$ pointwise, such that
$\langle P,P^{\sigma},\ldots,P^{\sigma^{n-1}}\rangle=\mathrm{PG}(n-1,q^{n})$,
and

$\Gamma=\langle P,P^{\sigma},\ldots,P^{\sigma^{n-3}}\rangle.$

In the same spirit, later in [37], the authors characterized linear sets of
LP-type. In order to do this, they introduced the notion of intersection
number of a subspace with respect to a collineation fixing pointwise a
canonical subgeometry. More precisely, let $\Gamma$ be a non-empty
$k$-dimensional subspace of $\mathrm{PG}(n-1,q^{n})$, the intersection number
of $\Gamma$ with respect to $\sigma$, denoted by
$\mathrm{intn}_{\sigma}(\Gamma)$, is defined as the least positive integer
$\gamma$ satisfying

$\dim(\Gamma\cap\Gamma^{\sigma}\cap\ldots\cap\Gamma^{\sigma^{\gamma}})>k-2\gamma.$

It is easy to see that $\mathrm{intn}_{\sigma}(\Gamma)$ is invariant under the
action of $\mathrm{Aut}(\Sigma)$. By Theorem 3.5, a maximum scattered linear
set is of pseudoregulus type if and only if $\mathrm{intn}_{\sigma}(\Gamma)=1$
for some $\sigma$ fixing $\Sigma$ pointwise. Also in [37] the following result
was proven.

######  Theorem 3.6.

[37, Theorem 3.2] Let $\Sigma$ be a canonical subgeometry of
$\mathrm{PG}(n-1,q^{n})$, $q>2$ and $n\geq 4$. Let $L$ be a scattered linear
set in a line $\Lambda$ of $\mathrm{PG}(n-1,q^{n})$. Then $L$ is a linear set
of LP-type if and only if

  1. 1.

there exists an $(n-3)$-subspace $\Gamma$ of $\mathrm{PG}(n-1,q^{n})$ such
that $\Gamma\cap\Sigma=\Gamma\cap\Lambda=\emptyset$ and
$L=\mathrm{p}_{\Gamma,\Lambda}(\Sigma)$;

  2. 2.

there exists a generator $\sigma$ of the subgroup of
$\mathrm{P}\Gamma\mathrm{L}(n,q^{n})$ fixing $\Sigma$ pointwise such that
$\mathrm{intn}_{\sigma}(\Gamma)=2$;

  3. 3.

there exists a unique point $P\in\mathrm{PG}(n-1,q^{n})$ and some point $Q$
such that

$\Gamma=\langle P,P^{\sigma},\dots,P^{\sigma^{n-4}},Q\rangle;$

  4. 4.

the line $\langle P^{\sigma^{n-1}},P^{\sigma^{n-3}}\rangle$ meets $\Gamma$.

Moreover, as a consequence of Theorem 3.4 we get a slight strengthening of
[37, Theorem 3.5], indeed removing the assumptions $n\leq 6$ and $n=8$.
Precisely, we may state the following result.

######  Theorem 3.7.

Let $L$ be a maximum scattered linear set in a line $\Lambda$ of
$\mathrm{PG}(n-1,q^{n})$ for any $n\geq 4$ and $q>2$. Then $L$ is a linear set
of LP-type if and only if for each $(n-3)$-subspace $\Gamma$ of
$\mathrm{PG}(n-1,q^{n})$ such that $L=\mathrm{p}_{\Gamma,\Lambda}(\Sigma)$,
the following holds:

  * $i)$

there exists a generator $\sigma$ of the subgroup of
$\mathrm{P}\Gamma\mathrm{L}(n,q^{n})$ fixing $\Sigma$ pointwise such that
$\mathrm{intn}_{\sigma}(\Gamma)=2$;

  * $(ii)$

if $P$ is the unique point of $\mathrm{PG}(n-1,q^{n})$ such that

$\Gamma=\langle P,P^{\sigma},\ldots,P^{\sigma^{n-4}},Q\rangle$

then the line $\langle P^{\sigma^{n-1}},P^{\sigma^{n-3}}\rangle$ meets
$\Gamma$.

In the last decade, many new constructions of maximum scattered linear sets of
the projective line were exhibited. In the following, in the light of what was
done for the examples belonging to classes (6) and (7), we will provide
characterization results similar to Theorems 3.5 and 3.6, for all the other
constructions mentioned above.

Toward this aim we first premise a complete list of examples that have
appeared from 2018 onward.

When $n\in\\{6,8\\}$, in [12], the following family of $\mathbb{F}_{q}$-linear
sets was presented

$L^{3,n}_{\ell,\eta}=\\{\langle(x,\eta
x^{q^{\ell}}+x^{q^{\ell+n/2}})\rangle_{\mathbb{F}_{q^{n}}}\,\colon\,x\in\mathbb{F}^{*}_{q^{n}}\\},$
(8)

where $\gcd(\ell,n/2)=1$ and $\eta\in\mathbb{F}_{q^{n}}$ such that
$\mathrm{N}_{q^{n}/q^{n/2}}(\eta)\not\in\\{0,1\\}$. Moreover in [12], it was
shown that, for some choices of $q$ and $\eta$, $L_{\ell,\eta}^{3,n}$ turns
out to be scattered, see [12, Theorem 7.1 and Theorem 7.2]. Later, necessary
and sufficient conditions for these examples in order to be scattered, were
also provided. Precisely,

  * -

if $n=6$ and $q>2$, $L_{2,\eta}^{3,n}$ is scattered if and only if the
equation

$Y^{2}-(\mathrm{Tr}_{q^{3}/q}(\gamma)-1)Y+\mathrm{N}_{q^{3}/q}(\gamma)=0,$

where $\gamma=-\frac{\eta^{q^{3}+1}}{1-\eta^{q^{3}+1}}$, admits two solutions
in $\mathbb{F}_{q},$ see [2] and [31, Theorem 7.3],

  * -

if $n=8$ and $q\leq 11$ or $q\geq 1039891$ odd, $L_{\ell,\eta}^{3,n}$ is
scattered if and only $\mathrm{N}_{q^{8}/q^{4}}(\eta)=-1$, see [36].

In 2018, another example was found in the case when $n=6$ and $q$ is odd,
consisting of linear sets with the following shape:

${}L^{4,6}_{\delta}=\\{\langle(x,x^{q}+x^{q^{3}}+\delta
x^{q^{5}})\rangle_{\mathbb{F}_{q^{6}}}\,\colon\,x\in\mathbb{F}^{*}_{q^{6}}\\},$
(9)

where $\delta\in\mathbb{F}_{q^{6}}^{*}$ such that $\delta^{2}+\delta=1;$ see
[11] and [27]. Very recently, it has been proven that for a different set of
conditions on $\delta$, $L_{\delta}^{4,6}$ is scattered also for $q$ even and
large enough, see [4].
In the series of articles [5, 17, 23, 25, 28, 35, 37], various authors studied
the class of linear sets defined as

$L_{m,h,s}^{5,t}=\\{\langle(x,\psi_{t,m,h,s}(x))\rangle_{\mathbb{F}_{q^{2t}}}:x\in\mathbb{F}_{q^{2t}}^{*}\\}$
(10)

where

$\psi_{t,m,h,s}(x)=x^{q^{s(t-1)}}+h^{1-q^{s(2t-1)}}x^{q^{s(2t-1)}}+m\left(x^{q^{s}}-h^{1-q^{s(t+1)}}x^{q^{s(t+1)}}\right)$

for $q$ odd, $t\geq 3$, $\gcd(s,2t)=1$ and $m,h\in\mathbb{F}_{q^{2t}}^{*}$
with $\mathrm{N}_{q^{2t}/q^{t}}(h)=-1$.

The linear set $L_{m,h,s}^{5,t}$ turns out to be scattered under one of the
following hypotheses:

  1. $(i)$

$m=1$ and $h\in\mathbb{F}_{q^{t}}^{*}$, see [5, 25, 28, 37].

  2. $(ii)$

$m=1$ and $h\in\mathbb{F}_{q^{2t}}\setminus\mathbb{F}_{q^{t}}$, see [5, 17,
23, 28].

  3. $(iii)$

$m$ belongs to $\mathbb{F}_{q^{t}}$ and it is neither a $(q-1)$-th power nor a
$(q+1)$-th power of an element belonging to $\ker\mathrm{Tr}_{q^{2t}/q^{t}}$
and $h=1$, see [35].

Note that if $h\in\mathbb{F}_{q^{t}}^{*}$, $t\geq 3$ odd and
$\mathrm{N}_{q^{2t}/q^{t}}(h)=-1$ implies that $q\equiv 1\pmod{4}$.

Moreover, in [5], the authors showed that $L^{5,3}_{1,h,1}$ does not lie in
Classes (6),(7),(8) and (9), except when $q$ is a power of $5$, and
$h\in\mathbb{F}_{q}^{*}$. In this case, the linear set is equivalent to
$L^{4,6}_{\delta}$ for some $\delta\in\mathbb{F}_{q^{6}}^{*}$, see [5,
Corollary 3.11]. In [17], it was proven that $L^{5,4}_{1,h,s}$ is not
$\mathrm{P}\Gamma\mathrm{L}$-equivalent to any linear sets of
$\mathrm{PG}(1,q^{8})$ belonging to classes in (6),(7),(8), (9), see [17,
Theorem 4.4].

As we have seen before, in [15, 37], linear sets of pseudoregulus and LP-type
were characterized exploiting the intersection number of their vertex of
projection. This approach does not work for scattered linear sets belonging to
the other classes described above. Indeed in the following we show that there
exist projection vertices of inequivalent linear sets having the same
intersection number. To this aim, let us consider

$\Sigma=\\{\langle(x,x^{q},\dots,x^{q^{5}})\rangle_{\mathbb{F}_{q^{6}}}:x\in\mathbb{F}_{q^{6}}^{*}\\},$

the canonical subgeometry of $\mathrm{PG}(5,q^{6})$ fixed by the collineation

$\sigma:\langle(x_{0},x_{1},\dots,x_{5})\rangle_{\mathbb{F}_{q^{6}}}\in\mathrm{PG}(5,q^{6})\longrightarrow\langle(x^{q}_{5},x^{q}_{0},x^{q}_{1},\dots,x^{q}_{4})\rangle_{\mathbb{F}_{q^{6}}}\in\mathrm{PG}(5,q^{6}).$

Let $L^{3,6}_{1,\eta}$ be defined as in (8). It is easy to see that this
example is equivalent to a linear set obtained as the projection of $\Sigma$
from the solid $\Gamma_{\eta}$ to the line $\Lambda_{\eta}$ having the
following equations:

$\displaystyle\Gamma_{\eta}:\begin{cases}x_{0}=0\\\ x_{1}+\eta
x_{4}=0\end{cases}$ $\displaystyle\Lambda_{\eta}:x_{1}=x_{2}=x_{3}=x_{5}=0.$
(11)

We have $\Gamma_{\eta}\cap\Sigma=\emptyset=\Gamma_{\eta}\cap\Lambda_{\eta}$.
Moreover, $\dim(\Gamma_{\eta}\cap\Gamma_{\eta}^{\sigma})=1$ and
$\dim(\Gamma_{\eta}\cap\Gamma_{\eta}^{\sigma}\cap\Gamma_{\eta}^{\sigma^{2}})=-1$,
hence $\mathrm{intn}_{\sigma}(\Gamma_{\eta})=3$. Similarly, the linear set
$L^{5,3}_{1,h,1}$ is equivalent to the projection of $\Sigma$ from the vertex
$\Gamma_{h}$ on the line $\Lambda_{h}$ with equations

$\displaystyle\Gamma_{h}:\begin{cases}x_{0}=0\\\
h^{q-1}x_{1}-h^{q^{2}-1}x_{2}+x_{4}+x_{5}=0\end{cases}$
$\displaystyle\Lambda_{h}:x_{1}=x_{2}=x_{3}=x_{4}=0$ (12)

and as shown in [5, Section 3.2], $\mathrm{intn}_{\sigma}(\Gamma_{h})=3$ as
well.

Since, Theorem 2.1 will be crucial in the subsequent part, we will state it
when $r=2$, i.e., for linear sets of the projective line
$\mathrm{PG}(1,q^{n})$.

######  Proposition 3.8.

Let $L_{f}$ be a linear set of rank $n$ of the projective line
$\mathrm{PG}(1,q^{n})$ as defined in (2) with
$f(x)=\sum_{i=1}^{n-1}a_{i}x^{q^{si}}$, $m=\deg_{q^{s}}f(x)$ and
$I=\mathrm{supp}_{q^{s}}f(x)$. Then, there exist a canonical subgeometry
$\Sigma$ of $\Sigma^{*}=\mathrm{PG}(n-1,q^{n})$ and an imaginary point $P$ of
$\Sigma^{*}$ (wrt $\Sigma$) such that $L_{f}$ is equivalent to
$\mathrm{p}_{\Gamma,\Lambda}(\Sigma)$ with

$\Gamma_{f}=\langle P^{\sigma^{j}},Q_{i}\colon j\not\in I,j\neq
0\,\textnormal{and}\,i\in I\setminus\\{m\\}\rangle,$ (13)

where $Q_{i}\in\langle P^{\sigma^{i}},P^{\sigma^{m}}\rangle$, $i\in
I\setminus\\{m\\}$ and $\Lambda=\langle P,P^{\sigma^{m}}\rangle$.

We observe that in light of Theorems 3.3, 3.4 and Proposition 3.8, the result
above can be further specialised for maximum scattered linear sets not of
pseudoregulus type as follows.

######  Theorem 3.9.

Let $L_{f}$ be a maximum scattered linear set of the projective line
$\mathrm{PG}(1,q^{n})$ not of pseudoregulus type with
$f(x)=\sum_{i=1}^{n-1}a_{i}x^{q^{si}}$, $m=\deg_{q^{s}}f(x)$ and
$I=\mathrm{supp}_{q^{s}}f(x)$. Then, there exist a canonical subgeometry
$\Sigma$ of $\Sigma^{*}=\mathrm{PG}(n-1,q^{n})$ and an imaginary point $P$ of
$\Sigma^{*}$ (wrt $\Sigma)$ such that

  1. 1.

$L_{f}$ is equivalent to $\mathrm{p}_{\Gamma_{f},\Lambda_{f}}(\Sigma)$ with

$\Gamma_{f}=\langle P^{\sigma^{j}},Q_{i}\colon j\not\in I,j\neq
0\,\textnormal{and}\,i\in I\setminus\\{m\\}\rangle\,\,\textnormal{ and
}\Lambda_{f}=\langle P,P^{\sigma^{m}}\rangle$ (14)

where $Q_{i}\in\langle P^{\sigma^{i}},P^{\sigma^{m}}\rangle$ and $i\in
I\setminus\\{m\\}$,

  2. 2.

$L_{f}$ is equivalent to
$\mathrm{p}_{\Gamma_{\hat{f}},\Lambda_{\hat{f}}}(\Sigma)$ with

$\Gamma_{\hat{f}}=\langle P^{\sigma^{j}},Q_{i}\colon j\not\in\hat{I},j\neq
0\,\textnormal{and}\,i\in\hat{I}\setminus\\{\hat{m}\\}\rangle\,\,\textnormal{
and }\Lambda_{\hat{f}}=\langle P,P^{\sigma^{\hat{m}}}\rangle$ (15)

where $Q_{i}\in\langle P^{\sigma^{i}},P^{\sigma^{\hat{m}}}\rangle$,
$i\in\hat{I}\setminus\\{\hat{m}\\}$,
$\hat{I}=\mathrm{supp}_{q^{s}}\hat{f}(x)$, and
$\hat{m}=\deg_{q^{s}}\hat{f}(x)$.

In particular, if $c_{\Gamma\mathrm{L}}(L_{f})=2$ and
$\Lambda_{f}=\Lambda_{\hat{f}}$ then $\Gamma_{f}$ and $\Gamma_{\hat{f}}$ do
not belong to the same orbit under the action of $\mathrm{Aut}(\Sigma)$,
otherwise they do.

######  Remark 3.10.

We stress here the fact that in previous statment, the vertex
$\Gamma_{\hat{f}}$ can be spanned by the same $P^{\sigma^{j}}$’s and the
points $T_{i}\in\Gamma_{\hat{f}}\cap\langle
P^{\sigma^{i}},P^{\sigma^{\tilde{m}}}\rangle,$ where
$i\in\hat{I}\setminus\\{\tilde{m}\\}$ and $\tilde{m}=\min\hat{I}$, as well.

Putting together Theorem 3.9 above with [14, Theorem 2] and [19, Theorem 3],
we derive the following result.

######  Theorem 3.11.

Let $\Sigma\cong\mathrm{PG}(n-1,q)$ be a canonical subgeometry of
$\Sigma^{*}=\mathrm{PG}(n-1,q^{n})$. Let $L$ be a linear set contained in a
line $\Lambda$ of $\Sigma^{*}$ such that
$L=\mathrm{p}_{\Gamma,\Lambda}(\Sigma)$ where $\Gamma$ is an $(n-3)$-subspace
of $\Sigma^{*}$ such that $\Gamma\cap\Sigma=\emptyset=\Gamma\cap\Lambda$. Let
us consider $L_{f}$ be a maximum scattered linear set of
$\mathrm{PG}(1,q^{n})$ not of pseudoregulus type and $\Gamma_{f}$,
$\Gamma_{\hat{f}}$, $\Lambda_{f}$ and $\Lambda_{\hat{f}}$ as defined in (14)
and $\eqref{gammahatf}$. Then, if $L$ is simple or
$\Lambda_{f}=\Lambda_{\hat{f}}$, the following statements are equivalent

  * $(i)$

$L_{f}\cong L$;

  * $(ii)$

there exists a collineation $\beta$ of $\Sigma^{*}$ such that
$\mathbf{\Sigma}^{\beta}=\Sigma$ (cf. (3)) and either
$\Gamma=\Gamma^{\beta}_{f}$ or $\Gamma=\Gamma^{\beta}_{\hat{f}}$.

###### Proof.

$(i)\Rightarrow(ii)$ Let us suppose that $L_{f}\cong L$. If the
$\Gamma\mathrm{L}$-class of $L_{f}$, and hence of $L$, is one then the result
follows from [14, Theorem 6]. Suppose that
$c_{\Gamma\mathrm{L}}(L_{f})=c_{\Gamma\mathrm{L}}(L)=2$ and
$\Lambda_{f}=\Lambda_{\hat{f}}$. Let us put $\Gamma_{1}=\Gamma_{f}$,
$\Gamma_{2}=\Gamma_{\hat{f}}$ and
$\boldsymbol{\Lambda}=\Lambda_{f}=\Lambda_{\hat{f}}$ and consider
$L_{1}=\mathrm{p}_{\Gamma_{1},\boldsymbol{\Lambda}}(\boldsymbol{\Sigma})$ and
$L_{2}=\mathrm{p}_{\Gamma_{2},\boldsymbol{\Lambda}}(\boldsymbol{\Sigma})$. By
Theorem 3.9, there exists
$\alpha_{i}:\boldsymbol{\Lambda}\longrightarrow\Lambda$ such that
$L_{i}^{\alpha_{i}}=L$, $i=1,2$. Then $\alpha_{i}$ can be extended to an
element, which we indicate with the same symbol $\alpha_{i}$, belonging to
$\mathrm{P}\Gamma\mathrm{L}(n,q^{n})$ such that
$\Gamma_{i}^{\alpha_{i}}=\Gamma$, $i=1,2$. Thus,
$L=L_{i}^{\alpha_{i}}=\mathrm{p}_{\Gamma,{\Lambda}}(\boldsymbol{\Sigma}^{\alpha_{i}})$,
$i=1,2$.
In order to get $(ii)$, it is enough to prove the existence of either a map
$\phi\in\mathrm{P}\Gamma\mathrm{L}(n,q^{n})$ such that $\Gamma^{\phi}=\Gamma$,
$\boldsymbol{\Sigma}^{\alpha_{1}\phi}=\Sigma$ or of a map
$\psi\in\mathrm{P}\Gamma\mathrm{L}(n,q^{n})$ such that $\Gamma^{\psi}=\Gamma$,
$\boldsymbol{\Sigma}^{\alpha_{2}\psi}=\Sigma$. Indeed, if such a $\phi$ or
$\psi$ did exist, then we would get the result by putting
$\beta=\alpha_{1}\phi$ or $\beta=\alpha_{2}\psi$, respectively. In other
words, it is enough to prove the assertion when $\boldsymbol{\Lambda}=\Lambda$
and either $\Gamma_{1}=\Gamma$ and $L_{1}=L$ or $\Gamma_{2}=\Gamma$ and
$L_{2}=L$. Let $\boldsymbol{\Sigma}=L_{\boldsymbol{V}}$ with

$\boldsymbol{V}=\\{(x,x^{q},\ldots,x^{q^{n-1}})\colon
x\in\mathbb{F}_{q^{n}}\\}$

and $V$ be an $n$-dimensional $\mathbb{F}_{q}$-subspace of
$\mathbb{F}_{q^{n}}^{n}$ such that $\Sigma=L_{V}$. Since $\boldsymbol{\Sigma}$
and $\Sigma$ are canonical subgeometries, we have that
$\langle\boldsymbol{V}\rangle_{\mathbb{F}_{q^{n}}}=\langle
V\rangle_{\mathbb{F}_{q^{n}}}=\mathbb{F}_{q^{n}}^{n}$. Moreover, let
$\boldsymbol{V}=\langle\textbf{v}_{1},\textbf{v}_{2},\ldots,\textbf{v}_{n}\rangle_{\mathbb{F}_{q}}$
and $V=\langle v_{1},v_{2},\ldots,v_{n}\rangle_{\mathbb{F}_{q}}$. Also let
$\Gamma_{i}=\mathrm{PG}(Z_{i},q^{n})$,
$\boldsymbol{\Lambda}=\mathrm{PG}(R,q^{n})$.
Note that $\boldsymbol{W}_{i}=(\boldsymbol{V}+Z_{i})\cap R$ and $W=(V+Z)\cap
R$ are two $n$-dimensional $\mathbb{F}_{q}$-vector spaces in $R$. As either
$\Gamma_{1}=\Gamma$ and
$L_{1}=\mathrm{p}_{\Gamma_{1},\boldsymbol{\Lambda}}(\boldsymbol{\Sigma})=\mathrm{p}_{\Gamma,\boldsymbol{\Lambda}}(\Sigma)=L$
or $\Gamma_{2}=\Gamma$ and
$L_{2}=\mathrm{p}_{\Gamma_{2},\boldsymbol{\Lambda}}(\boldsymbol{\Sigma})=\mathrm{p}_{\Gamma,\boldsymbol{\Lambda}}(\Sigma)=L$,
by Theorem 3.4, it follows that there exists a non-singular
$\mathbb{F}_{q^{n}}$-semilinear map $\gamma:R\rightarrow R$ such that either
$\boldsymbol{W}^{\gamma}_{1}=W$ or
$\boldsymbol{W}_{2}^{\gamma}=(\boldsymbol{W}_{1}^{\perp_{\beta^{\prime}}})^{\gamma}=W$
where $\perp_{\beta^{\prime}}$ is the orthogonal complement map defined by a
sesquilinear form $\beta^{\prime}$ on the lattices of the
$\mathbb{F}_{q}$-subspaces of $R\cong\mathbb{F}_{q^{n}}^{2}$ seen as a
$2n$-dimensional space over $\mathbb{F}_{q}$. Moreover, $\boldsymbol{W}_{1}$
and $\boldsymbol{W}_{2}$ are not on the same orbit under the action of the
group of semilinear maps of $R$ in itself, otherwise $L_{f}$, and hence also
$L$, would not have $\Gamma\mathrm{L}$-class two.
Assume first that $\boldsymbol{W}_{1}^{\gamma}=W$ and let
$\Gamma\in\\{\Gamma_{1},\Gamma_{2}\\}$ and, hence, $Z\in\\{Z_{1},Z_{2}\\}$. We
show that the case $\boldsymbol{W}_{1}^{\gamma}=W$ and $\Gamma_{2}=\Gamma$
cannot occur. Indeed, if $\boldsymbol{W}_{1}^{\gamma}=W$ and $Z=Z_{2}$, then
the map $\gamma$ can be extended to non-singular semilinear map, saying
$\gamma$ again, from $\mathbb{F}^{n}_{q^{n}}$ in itself such that
$Z_{2}^{\gamma}=Z_{2}$. Then for each vector $v\in\mathbb{F}_{q^{n}}^{n}$ and
$\mu\in\mathbb{F}_{q^{n}}$, we have $(\mu
v)^{\gamma}=\mu^{p^{\ell}}v^{\gamma}$ for some integer $\ell$. Since $R+Z$ and
$\boldsymbol{W}_{2}+Z$ are direct sums, for any
$\textbf{v}\in\boldsymbol{V}\setminus\\{\textbf{0}\\}$ and $z\in Z$, the
intersection $(\langle\textbf{v}\rangle_{\mathbb{F}_{q}}+Z)\cap R$ is a one-
dimensional $\mathbb{F}_{q}$-subspace, say
$\langle\textbf{w}\rangle_{\mathbb{F}_{q}}$ with
$\textbf{w}\in\boldsymbol{W}_{2}$. This implies
$h\textbf{w}=\textbf{v}+z^{\prime}$ for some $z^{\prime}\in Z$ and
$h\in\mathbb{F}_{q}$ and hence
$\textbf{v}+z=h\textbf{w}+(z-z^{\prime})\in\boldsymbol{W}_{2}+Z$. Then,
$\boldsymbol{W}_{2}+Z=\boldsymbol{V}+Z$and hence
$\boldsymbol{W}_{1}\subseteq\boldsymbol{V}+Z_{2}=\boldsymbol{W}_{2}+Z_{2}$.
Since $\boldsymbol{W}_{1}\subseteq R$ we get that
$\boldsymbol{W}_{1}\subset(\boldsymbol{V}+Z_{2})\cap R=\boldsymbol{W}_{2}$ and
hence $\boldsymbol{W}_{1}=\boldsymbol{W}_{2}$, a contradiction.
Hence, let us suppose that $\boldsymbol{W}_{1}^{\gamma}=W$ and $Z=Z_{1}$.
Similar argument shows that $\boldsymbol{W}_{1}+Z=\boldsymbol{V}+Z$ and
$W+Z=V+Z$. Then

$V+Z=W+Z=\boldsymbol{W}^{\gamma}_{1}+Z^{\gamma}=(\boldsymbol{W}_{1}+Z)^{\gamma}=(\boldsymbol{V}+Z)^{\gamma}$
(16)

and hence for any $i=1,\ldots,n$,
$\textbf{v}^{\gamma}_{i}=v^{\prime}_{i}+z_{i}$ for some $v^{\prime}_{i}\in V$
and $z_{i}\in Z$. We will show that $v_{1}^{\prime},\ldots,v_{n}^{\prime}$ is
an $\mathbb{F}_{q}$ basis of $V$. Indeed, suppose that
$\sum_{i=1}^{n}\lambda_{i}v^{\prime}_{i}=0$ for
$\lambda_{i}\in\mathbb{F}_{q}$. Therefore,
$\sum_{i=1}^{n}\lambda_{i}\textbf{v}_{i}^{\gamma}=\sum_{i=0}^{n}\lambda_{i}z_{i}$.
As on the left-hand since
$\textbf{v}^{\gamma}_{1},\textbf{v}_{2}^{\gamma},\ldots,\textbf{v}_{n}^{\gamma}$
are $\mathbb{F}_{q}$ independent, it follows that either $\lambda_{i}=0$ or,
saying
$z=\sum_{i=1}^{n}\lambda_{i}\textbf{v}_{i}^{\gamma}=\sum_{i=0}^{n}\lambda_{i}z_{i}$,
then $z\in\boldsymbol{V}^{\gamma}\cap Z$. Hence in the latter case
$z^{\gamma^{-1}}\in\boldsymbol{V}\cap Z$, a contradiction. Since
$\Sigma=L_{V}$ is a subgeometry,
$v^{\prime}_{1},v^{\prime}_{2},\ldots,v^{\prime}_{n}$ are linearly independent
over $\mathbb{F}_{q^{n}}$. Let $\rho$ be the $\mathbb{F}_{q^{n}}$-semilinear
map of $\mathbb{F}_{q^{n}}^{n}$ such that
$\textbf{v}_{i}^{\rho}=v^{\prime}_{i}$ for each $i=1,2,\ldots,n$ and
$(\mu\textbf{v})^{\rho}=\mu^{p^{\ell}}\textbf{v}^{\rho}$ for any
$\mu\in\mathbb{F}_{q^{n}}$. If
$P=\langle\textbf{z}\rangle_{\mathbb{F}_{q^{n}}}\in\Gamma$, then we have
$\textbf{z}=\sum^{n}_{i=1}a_{i}\textbf{v}_{i}$ for some
$a_{i}\in\mathbb{F}_{q^{n}}$. Then

$\begin{split}\textbf{z}^{\rho}=\sum_{i=1}^{n}a_{i}^{p^{\ell}}v^{\prime}_{i}=\sum_{i=1}^{n}a_{i}^{p^{\ell}}(\textbf{v}_{i}^{\gamma}-z_{i})=\textbf{z}^{\gamma}-\sum_{i=1}^{n}a_{i}^{p^{\ell}}z_{i}\in
Z\end{split}$

and hence the collineation induced by $\rho$ maps $\Gamma$ in $\Gamma$ and
$\boldsymbol{\Sigma}$ into $\Sigma$. Finally, a similar argument applies to
the case $\boldsymbol{W}_{2}^{\gamma}=W$.
$(ii)\Rightarrow(i)$. It follows [14, Theorem 2] and [19, Theorem 3]. ∎

Let $\Sigma_{1}$ and $\Sigma_{2}$ be two canonical subgeometries of
$\Sigma^{*}$ and let us consider $\mathbb{G}_{1}=\langle\sigma_{1}\rangle$ and
$\mathbb{G}_{2}$ the subgroups of $\mathrm{P}\Gamma\mathrm{L}(n,q^{n})$ fixing
pointwise the canonical subgeometries $\Sigma_{1}$ and $\Sigma_{2}$ of
$\Sigma^{*}$, respectively. If $\beta$ is a collineation of $\Sigma^{*}$ such
that $\Sigma_{1}^{\beta}=\Sigma_{2}$, then
$\mathbb{G}_{2}=\mathbb{G}_{1}^{\beta}$, i.e. they are conjugate under
$\beta$. Indeed, for any point $P\in\Sigma_{2}$ we have

$P^{\beta^{-1}\sigma_{1}\beta}=((P^{\beta^{-1}})^{\sigma_{1}})^{\beta}=P^{\beta^{-1}\beta}=P$

Analogously, it is straightforward to see that
$\mathrm{Aut}(\Sigma_{2})=\mathrm{Aut}(\Sigma_{1})^{\beta}$. Hence, as
byproduct of Theorem 3.9 and Theorem 3.11, we can state the following.

######  Proposition 3.12.

Let $\Sigma\cong\mathrm{PG}(n-1,q)$ be a canonical subgeometry of
$\Sigma^{*}=\mathrm{PG}(n-1,q^{n})$. Let $L$ be a linear set contained in a
line $\Lambda$ of $\Sigma^{*}$ such that
$L=\mathrm{p}_{\Gamma,\Lambda}(\Sigma)$, where $\Gamma$ is an $(n-3)$-subspace
of $\Sigma^{*}$ such that $\Gamma\cap\Sigma=\emptyset=\Gamma\cap\Lambda$. If
$L$ is equivalent to a maximum scattered linear set of $\mathrm{PG}(1,q^{n})$
not of pseudoregulus type $L_{f}$, with $f(x)=\sum_{i\in I}a_{i}x^{q^{si}}$,
$m=\deg_{q^{s}}f(x)$ and $I=\mathrm{supp}_{q^{s}}f(x)$ such that either
$L_{f}$ is simple or $\Lambda_{f}=\Lambda_{\hat{f}}$ (cf. (14) and (15)), then
there exists a collineation $\sigma$ of $\Sigma^{*}$ fixing $\Sigma$ pointwise
and an imaginary point $P$ of $\Sigma^{*}$ (wrt $\Sigma)$ such that either

$\Gamma=\langle P^{\sigma^{j}},Q_{i}\colon j\not\in I,j\neq
0\,\textnormal{and}\,i\in I\setminus\\{m\\}\rangle,$ (17)

where $Q_{i}\in\langle P^{\sigma^{i}},P^{\sigma^{m}}\rangle$, $i\in
I\setminus\\{m\\}$, or

$\Gamma=\langle P^{\sigma^{j}},Q_{i}\colon j\not\in\hat{I},j\neq
0\,\textnormal{and}\,i\in\hat{I}\setminus\\{\hat{m}\\}\rangle,$ (18)

where $Q_{i}\in\langle P^{\sigma^{i}},P^{\sigma^{\hat{m}}}\rangle$,
$i\in\hat{I}\setminus\\{\hat{m}\\}$ and $\hat{I}=\mathrm{supp}\hat{f}(x)$,
$\hat{m}=\deg_{q^{s}}\hat{f}(x)$.

###### Proof.

By Theorem 3.11, there exists a collineation $\beta$ of $\Sigma^{*}$ such that
$\mathbf{\Sigma}^{\beta}=\Sigma$ and either $\Gamma=\Gamma^{\beta}_{f}$ or
$\Gamma=\Gamma^{\beta}_{\hat{f}}$ where $\Gamma_{f}$ and $\Gamma_{\hat{f}}$
are defined as in (14) or (15). If $\Gamma=\Gamma_{f}^{\beta}$, the result
follows putting $\sigma=\beta^{-1}\boldsymbol{\sigma}^{s}\beta$ (cf. (2)),
$P=\boldsymbol{P}^{\beta}$ and $Q_{i}=\boldsymbol{Q}^{\beta}_{i}$ where
$\boldsymbol{P}$ and $\boldsymbol{Q}_{i}$ are the points as in (14) of the
statement of Theorem 3.9, and noting that
$P^{\sigma^{j}}=\boldsymbol{P}^{\beta\sigma^{j}}=\boldsymbol{P}^{\boldsymbol{\sigma}^{sj}\beta}$,
$j=1,\ldots,n-1$. If otherwise $\Gamma=\Gamma_{\hat{f}}^{\beta}$ the result
instead follows by putting
$\sigma=\beta^{-1}\boldsymbol{\sigma}^{s(n-1)}\beta$,
$P={\boldsymbol{P}}^{\beta}$, ${\boldsymbol{T}}^{\beta}_{i}=Q_{i}$ and taking
into account Remark 3.10. ∎

In particular, if $L\subset\Lambda$ is simple, then
$\mathrm{P}\Gamma\mathrm{L}(n,q^{n})$ acts transitively on the vertices of any
linear set equivalent to $L$, while the group $\mathrm{Aut}(\Sigma)$ acts
transitively on the subspaces $\Gamma$ such that
$L=\mathrm{p}_{\Gamma,\Lambda}(\Sigma)$.

Finally, we will state Theorem 2.4 in the case of a linear set contained in a
projective line of $\mathrm{PG}(n-1,q^{n})$.

######  Theorem 3.13.

Let $I\subseteq\\{1,\dots,n-1\\}$ and $m$ denotes the maximum integer in $I$.
Let $\Sigma$ be a canonical subgeometry of $\Sigma^{*}=\mathrm{PG}(n-1,q^{n})$
and consider an $(n-3)$-dimensional subspace $\Gamma$ and a line $\Lambda$ of
$\Sigma^{*}$ such that $\Gamma\cap\Sigma=\emptyset=\Lambda\cap\Gamma$.
Assume $L=\mathrm{p}_{\Gamma,\Lambda}(\Sigma)$ is a linear set
$(1,n-2)$-evasive of rank $n$. If there exists a point $P$ with the property
that

$\Gamma=\langle P^{\sigma^{j}},Q_{i}\leavevmode\nobreak\ |\leavevmode\nobreak\
j\notin I,\,j\neq 0\,\text{ and }i\in I\setminus\\{m\\}\rangle$

where $Q_{i}\in\langle P^{\sigma^{m}},P^{\sigma^{i}}\rangle$ and $\sigma$ is a
generator of the subgroup of $\mathrm{P}\Gamma\mathrm{L}(n,q^{n})$ fixing
$\Sigma$ pointwise, then $P$ is an imaginary point (wrt $\Sigma)$ and $L\cong
L_{f}$ where $f(x)=\sum_{i\in I}a_{i}x^{q^{si}}$ and $\gcd(s,n)=1$ for some
integer $1\leq s\leq n-1$.

######  Remark 3.14.

  1.   2. $(i)$

Note that if $L_{f}$ is equivalent to $L=\mathrm{p}_{\Gamma,\Lambda}(\Sigma)$
with $\Gamma$ and $\Lambda$ as Proposition 3.8, $f$ is a permutation
polynomial if and only if $\Sigma\cap\langle\Gamma,P\rangle=\emptyset$.

  3. $(ii)$

The imaginary point $P$ (wrt $\Sigma$) in Theorem 3.13 is not unique. For
instance, let $L^{4,6}_{\delta}$ be as in (9), $\delta^{2}+\delta=1$, then
$\delta\in\mathbb{F}_{q^{2}}$, and let us consider the point $A$ with
homogeneous coordinates $(0,-1,0,-1,0,1+\delta)$ in the frame
$(P,P^{\sigma},\ldots,P^{\sigma^{5}},U)$ with $P=\langle
v\rangle_{\mathbb{F}_{q^{6}}}$ and $U=\langle
v+v^{\sigma}+\ldots+v^{\sigma^{n-1}}\rangle_{\mathbb{F}_{q^{6}}}$. Then,

$\begin{vmatrix}0&-1&0&-1&0&1+\delta\\\ 1+\delta^{q}&0&-1&0&-1&0\\\
0&1+\delta&0&-1&0&-1\\\ -1&0&1+\delta^{q}&0&-1&0\\\ 0&-1&0&1+\delta&0&-1\\\
-1&0&-1&0&1+\delta^{q}&0\end{vmatrix}=(1-\delta)(2+\delta)^{2}(\delta^{q}-1)(2+\delta^{q})^{2}\neq
0.$

and hence $\dim\mathcal{L}_{A}=5$ and the solid $\Gamma$ as in (13) is spanned
by the points $A^{\sigma^{2}},A^{\sigma^{4}}$,
$Q^{\prime}_{1}=\langle(0,0,-2-\delta^{q},0,\delta^{q}+1,0)\rangle_{\mathbb{F}_{q^{6}}}$
and
$Q^{\prime}_{2}=\langle(0,0,2+\delta^{q},0,-2-\delta^{q},0)\rangle_{\mathbb{F}_{q^{6}}}$,
where $Q_{1}^{\prime}\in\langle A^{\sigma},A^{\sigma^{5}}\rangle$ and
$Q_{2}^{\prime}\in\langle A^{\sigma^{3}},A^{\sigma^{5}}\rangle$.

##  4 Maximum scattered linear sets which are neither of pseudoregulus nor of
LP type

As we have seen in the previous section if
$L=\mathrm{p}_{\Gamma,\Lambda}(\Sigma)$ for a given subgeometry $\Sigma$ and
suitable subspaces $\Gamma$ and $\Lambda$ of $\mathrm{PG}(n-1,q^{n})$, then,
in general, the intersection number of $\Gamma$ does not characterise $L$ by
its own.

In this section, we will deepen the study of projection vertices of linear
sets of $PG(1,q^{n})$ known to date, which are neither of pseudoregulus nor of
LP type, in order to provide characterization results for all of them in term
of certain properties of relevant vertices.

In this regard, we will use a notion from classical projective geometry: the
cross-ratio of four points of the projective line.
Let $A,B,C,D$ be four points of $\mathrm{PG}(1,q^{n})$ such that $A,B,C$ are
distinct; the cross-ratio $(A;B;C;D)$ of these points is defined as

$(A;B;C;D)=\frac{\begin{vmatrix}a_{0}&c_{0}\\\
a_{1}&c_{1}\par\end{vmatrix}\begin{vmatrix}b_{0}&d_{0}\\\
b_{1}&d_{1}\end{vmatrix}}{\begin{vmatrix}a_{0}&d_{0}\\\
a_{1}&d_{1}\end{vmatrix}\begin{vmatrix}b_{0}&c_{0}\\\
b_{1}&c_{1}\end{vmatrix}},$

where
$A=\langle(a_{0},a_{1})\rangle_{\mathbb{F}_{q^{n}}},B=\langle(b_{0},b_{1})\rangle_{\mathbb{F}_{q^{n}}},C=\langle(c_{0},c_{1})\rangle_{\mathbb{F}_{q^{n}}},D=\langle(d_{0},d_{1})\rangle_{\mathbb{F}_{q^{n}}}$
, respectively. Setting $\infty:=\frac{1}{0}$, we have

$(A;B;C;A)=\infty,\,(A;B;C;B)=0,\,(A;B;C;C)=1$

Moreover, by the definition we have

$(A;B;C;D)=(B;A;D;C)=(C;D;A;B)=(D;C;B;A);$

and if $(A;B;C;D)=\kappa$, all the possible values of the cross-ratio are

$\begin{matrix}\noindent(A;B;D;C)=1/\kappa,&(A;C;B;D)=1-\kappa,&(A;D;B;C)=(\kappa-1)/\kappa,\vspace{0.2cm}\\\
(A;D;C;B)=\kappa/(\kappa-1),&(A;C;D;B)=1/(1-\kappa).\end{matrix}$

We will say that the points pair $\\{A,B\\}$ harmonically separates the points
pair $\\{C,D\\}$ if the cross-ratio $(A;B;C;D)=-1$, see [8, Chapter 4] and
[18]. Finally, it is straightforward to see that if
$\phi\in\mathrm{P}\Gamma\mathrm{L}(2,q^{n})$ with companion automorphism
$\tau$, then

$(A^{\phi};B^{\phi};C^{\phi};D^{\phi})=(A;B;C;D)^{\tau}.$

###  4.1 A characterization of $L_{\ell,\eta}^{3,n}$, $n\in\\{6,8\\}$

Let $L_{\ell,\eta}^{3,n}$ be the linear set defined as in (8) and denote by
$U_{\ell,\eta}^{3,n}\subseteq\mathbb{F}_{q^{n}}\times\mathbb{F}_{q^{n}}$ its
underlying vector space. According to [12], $U^{3,n}_{\ell,\eta}$ is
$\mathrm{GL}(2,q^{n})$-equivalent to $U^{3,n}_{n-\ell,\eta^{q^{n-s}}}$ and to
$U^{3,n}_{\ell+n/2,\eta^{-1}}$ as well. Thus, as indicated in [11, Section 4],
we may assume $\ell<n/4$, and $\gcd(\ell,n/2)=1$ and hence reduce the study to
linear sets $L^{3,n}_{\ell,\eta}$ with $\ell=1$. Moreover, in [11, Proposition
4.1 and 4.2], it is proven that in both cases, the relevant linear set is
simple.

######  Theorem 4.1.

Let $L$ be a $(1,4)_{q}$-evasive linear set of rank $6$ contained in a line
$\Lambda$ of $\mathrm{PG}(5,q^{6})$ and let $\Sigma$ be a canonical
subgeometry of $\mathrm{PG}(5,q^{6})$. Then $L$ is equivalent to a maximum
scattered linear set of type $L_{1,\eta}^{3,6}$, for some
$\eta\in\mathbb{F}_{q^{6}}$, if and only if for each solid $\Gamma$ of
$\mathrm{PG}(5,q^{6})$ such that $L=\mathrm{p}_{\Gamma,\Lambda}(\Sigma)$, the
following holds:

  1. 1.

there exist a generator $\sigma$ of the subgroup of
$\mathrm{P}\Gamma\mathrm{L}(6,q^{6})$ fixing $\Sigma$ pointwise and a point
$P\in\mathrm{PG}(5,q^{6})$ such that

$\Gamma=\langle P^{\sigma^{i}},Q:i\in\\{2,3,5\\}\rangle$

with $Q\in\langle P^{\sigma},P^{\sigma^{4}}\rangle$,

  2. 2.

the points $P^{\sigma},P^{\sigma^{4}},Q,Q^{\sigma^{3}}$ are collinear, and the
equation

$Y^{2}-(\mathrm{Tr}_{q^{3}/q}(\gamma)-1)Y+\mathrm{N}_{q^{3}/q}(\gamma)=0,$
(19)

where $\gamma=(P^{\sigma};Q;Q^{\sigma^{3}};P^{\sigma^{4}})^{\tau}$ for some
$\tau\in\mathrm{Aut}(\mathbb{F}_{q^{6}})$, admits two solutions in
$\mathbb{F}_{q}$.

###### Proof.

Let us suppose that $L$ is equivalent to a maximum scattered linear set of
type $L_{1,\eta}^{3,6}$. By Proposition 3.12 and since $L$ is simple, Point 1
holds true. By Theorem 3.9, let
$\Gamma_{f}=\langle\boldsymbol{P}^{\boldsymbol{\sigma}^{i}},\boldsymbol{Q}:i\in\\{2,3,5\\}\rangle$,
where $f(x)=\eta x^{q}+x^{q^{4}}$, $\boldsymbol{P}$ an imaginary point of
$\mathrm{PG}(5,q^{6})$ (wrt $\boldsymbol{\Sigma}$),
$\boldsymbol{Q}\in\langle\boldsymbol{P}^{\boldsymbol{\sigma}},\boldsymbol{P}^{\boldsymbol{\sigma}^{4}}\rangle$,
$P=\boldsymbol{P}^{{\beta}}$ and $Q=\boldsymbol{Q}^{\beta}$ where $\beta$ is
the collineation of $\mathrm{PG}(5,q^{6})$ such that
$\Gamma_{f}^{\beta}=\Gamma$ and $\boldsymbol{\Sigma}^{\beta}=\Sigma$.
Clearly since $L_{1,\eta}^{3,6}$ is simple, all possible projection vertices
for such a linear set are equivalent to $\Gamma_{f}$ under the action of
$\mathrm{P}\Gamma\mathrm{L}(5,q^{6})$. Since $\beta$ preserves the
collinearity of points, in order to show Point 2, it is enough to prove that
$\boldsymbol{P}^{\boldsymbol{\sigma}},\boldsymbol{P}^{\boldsymbol{\sigma}^{4}},\boldsymbol{Q},\boldsymbol{Q}^{\boldsymbol{\sigma}^{3}}$
are collinear. Indeed, note that $\boldsymbol{\sigma}^{3}$ fixes the line
$\langle\boldsymbol{P}^{\boldsymbol{\sigma}},\boldsymbol{P}^{\boldsymbol{\sigma}^{4}}\rangle$,
therefore
$\boldsymbol{Q}^{\boldsymbol{\sigma}^{3}}\in\langle\boldsymbol{P}^{\boldsymbol{\sigma}},\boldsymbol{P}^{\boldsymbol{\sigma}^{4}}\rangle$.
Let us suppose that
$\boldsymbol{P}=\langle\textbf{v}\rangle_{\mathbb{F}_{q^{6}}}$ with v a non-
zero vector belonging to $\mathbb{F}_{q^{6}}^{6}$. If
$\boldsymbol{Q}=\langle\textbf{v}^{\boldsymbol{\sigma}}-\eta\textbf{v}^{\boldsymbol{\sigma}^{4}}\rangle_{\mathbb{F}_{q^{6}}}$,
then
$\boldsymbol{Q}^{\boldsymbol{\sigma}^{3}}=\langle\textbf{v}^{\boldsymbol{\sigma}^{4}}-\eta^{q^{3}}\textbf{v}^{\boldsymbol{\sigma}}\rangle_{\mathbb{F}_{q^{6}}}$.
Hence,

$(\boldsymbol{P}^{\boldsymbol{\sigma}};\boldsymbol{P}^{\boldsymbol{\sigma}^{4}};\boldsymbol{Q};\boldsymbol{Q}^{\boldsymbol{\sigma}^{3}})=\frac{\begin{vmatrix}1&1\\\
0&-\eta\end{vmatrix}\begin{vmatrix}0&-\eta^{q^{3}}\\\
1&1\end{vmatrix}}{\begin{vmatrix}1&-\eta^{q^{3}}\\\
0&1\end{vmatrix}\begin{vmatrix}0&1\\\ 1&-\eta\end{vmatrix}}=\eta^{q^{3}+1}.$

Now, the linear set $L\cong L_{1,\eta}^{3,6}$ is equivalent to
$L^{3,6}_{2,\eta^{-q^{5}}}$ which is scattered as well. Then by [31, Theorem
7.3], the equation

$Y^{2}-(\mathrm{Tr}_{q^{3}/q}(\alpha)-1)Y+\mathrm{N}_{q^{3}/q}(\alpha)=0$

admits two solutions in $\mathbb{F}_{q}$, where
$\alpha=-\frac{(\eta^{-q^{5}})^{q^{3}+1}}{1-(\eta^{-q^{5}})^{q^{3}+1}}=\frac{1}{1-\eta^{q^{2}+q^{5}}}$.
Moreover, we have
$\alpha^{q}=(\boldsymbol{P}^{\boldsymbol{\sigma}};\boldsymbol{Q};\boldsymbol{Q}^{\boldsymbol{\sigma}^{3}};\boldsymbol{P}^{\boldsymbol{\sigma}^{4}})$.
Also, if $\rho$ is the companion automorphism of $\beta$, then

$(P^{\sigma};Q;Q^{\sigma^{3}};P^{\sigma^{4}})=(\boldsymbol{P}^{\beta\sigma};\boldsymbol{Q}^{\beta};\boldsymbol{Q}^{\beta{\sigma}^{3}};\boldsymbol{P}^{\beta\sigma^{4}})=(\boldsymbol{P}^{\boldsymbol{\sigma}\beta};\boldsymbol{Q}^{\beta};\boldsymbol{Q}^{\boldsymbol{\sigma}^{3}\beta};\boldsymbol{P}^{\boldsymbol{\sigma}^{4}\beta})=(\boldsymbol{P}^{\boldsymbol{\sigma}};\boldsymbol{Q};\boldsymbol{Q}^{\boldsymbol{\sigma}^{3}};\boldsymbol{P}^{\boldsymbol{\sigma}^{4}})^{\rho}.$

Finally, since
$\mathrm{Tr}_{q^{3}/q}(\alpha)=\mathrm{Tr}_{q^{3}/q}(\alpha^{q})$ and
$\mathrm{N}_{q^{3}/q}(\alpha)=\mathrm{N}_{q^{3}/q}(\alpha^{q}),$ Point 2
follows with $\gamma=\alpha^{q}$ and $\tau=\rho^{-1}$.

Suppose now that Points 1 and 2 hold true. Theorem 3.13 implies that
$P=\langle v\rangle_{\mathbb{F}_{q^{6}}}$ is an imaginary point (wrt
$\Sigma$), $Q=\langle v^{\sigma}-\eta
v^{\sigma^{4}}\rangle_{\mathbb{F}_{q^{6}}}$ and $L\cong L_{f}$, $f(x)=\eta
x^{q^{s}}+x^{q^{4s}}$ for some $\eta\in\mathbb{F}_{q^{6}}^{*}$ and
$s\in\\{1,5\\}$. Since $P,P^{\sigma},Q,Q^{\sigma^{3}}$ are distinct,
$\mathrm{N}_{q^{6}/q^{3}}(\eta)\neq 1$. If $s=1$, since $L_{f}=L_{\hat{f}}$ is
equivalent to $L_{g_{1}}$ with $g_{1}(x)=\eta^{-q^{5}}x^{q^{2}}+x^{q^{5}}$.
Point 2 implies that the equation

$Y^{2}-(\mathrm{Tr}_{q^{3}/q}(\alpha_{1})-1)Y+\mathrm{N}_{q^{3}/q}(\alpha_{1})=0$

admits two solutions in $\mathbb{F}_{q}$, where
$\alpha_{1}=-\frac{(\eta^{-q^{5}})^{q^{3}+1}}{1-(\eta^{-q^{5}})^{q^{3}+1}}$.
Hence, by [31, Theorem 7.3], $L_{g_{1}}$ is scattered and $L$ is equivalent to
a scattered linear set of type $L_{1,\eta}^{3,6}$. If $s=5$ and hence
$f(x)=\eta x^{q^{5}}+x^{q^{2}}$, $L_{f}\cong L_{g_{2}}$ with
$g_{2}(x)=x^{q^{5}}+\eta^{-1}x^{q^{2}}$. Point 2 implies that the equation

$Y^{2}-(\mathrm{Tr}_{q^{3}/q}(\alpha_{2})-1)Y+\mathrm{N}_{q^{3}/q}(\alpha_{2})=0$

admits two roots in $\mathbb{F}_{q}$ with
$\alpha_{2}=-\frac{\eta^{-(q^{3}+1)}}{1-\eta^{-(q^{3}+1)}}$. Then, by [31,
Theorem 7.3], $g_{2}(x)$ is scattered and hence $L$ is scattered as well, and
it is of the type $L^{3,6}_{2,\eta^{-1}}\cong L_{1,\eta^{q}}^{3,6}$. ∎

In the same way we get

######  Theorem 4.2.

Let $L$ be an $(1,6)_{q}$-evasive linear set of rank $8$ contained in a line
$\Lambda$ of $\mathrm{PG}(7,q^{8})$, with $q\leq 11$ or $q\geq 1039891$ odd.
Then, $L$ is equivalent to a maximum scattered linear set of type
$L_{r,\eta}^{3,8}$, for some $\eta\in\mathbb{F}_{q^{8}}$ and $\gcd(s,8)=1$, if
and only if for each $5$-dimensional subspace $\Gamma$ of
$\mathrm{PG}(7,q^{8})$ such that $L=\mathrm{p}_{\Gamma,\Lambda}(\Sigma)$, the
following hold:

  1. 1.

there exists a generator $\sigma$ of the subgroup of
$\mathrm{P}\Gamma\mathrm{L}(8,q^{8})$ fixing $\Sigma$ pointwise and a point
$P\in\mathrm{PG}(7,q^{8})$ such that

$\Gamma=\langle P^{\sigma^{i}},Q:i\in\\{2,3,4,6,7\\}\rangle$

with $Q\in\langle P^{\sigma},P^{\sigma^{5}}\rangle$,

  2. 2.

the points $P^{\sigma},P^{\sigma^{5}},Q,Q^{\sigma^{4}}$ are collinear and pair
$\\{Q,Q^{\sigma^{4}}\\}$ harmonically separates the pair
$\\{P^{\sigma},P^{\sigma^{5}}\\}$ .

###### Proof.

Let $L$ be equivalent to a maximum scattered linear set of type
$L_{1,\eta}^{3,8}$. Since $L$ is simple, by Proposition 3.12, Point 1 holds
true. By Theorem 3.9, let
$\Gamma_{f}=\langle\boldsymbol{P}^{\boldsymbol{\sigma}^{i}},\boldsymbol{Q}:i\in\\{2,3,4,6,7\\}\rangle$,
$f(x)=\eta x^{q}+x^{q^{5}}$, with $\boldsymbol{P}$ an imaginary point of
$\mathrm{PG}(7,q^{8})$ (wrt $\boldsymbol{\Sigma}$),
$\boldsymbol{Q}\in\langle\boldsymbol{P}^{\boldsymbol{\sigma}},\boldsymbol{P}^{\boldsymbol{\sigma}^{5}}\rangle$,
$P=\boldsymbol{P}^{{\beta}}$ and $Q=\boldsymbol{Q}^{\beta}$ where $\beta$ is
the collineation of $\mathrm{PG}(7,q^{8})$ such that
$\boldsymbol{\Sigma}^{\beta}=\Sigma$ and $\Gamma_{f}^{\beta}=\Gamma$.
As before, all possible projection vertices for such a linear set are
equivalent to $\Gamma_{f}$ under the action of
$\mathrm{P}\Gamma\mathrm{L}(7,q^{8})$. Since $\beta$ preserves collinearity of
points and the property for a pair of points to harmonically separate another
pair, in order to show Point 2, it is enough to prove that
$\boldsymbol{P}^{\boldsymbol{\sigma}},\boldsymbol{P}^{\boldsymbol{\sigma}^{5}},\boldsymbol{Q},\boldsymbol{Q}^{\boldsymbol{\sigma}^{4}}$
are collinear and
$(\boldsymbol{P}^{\boldsymbol{\sigma}};\boldsymbol{P}^{\boldsymbol{\sigma}^{5}};\boldsymbol{Q};\boldsymbol{Q}^{\boldsymbol{\sigma}^{4}})=-1$.
Clearly, $\boldsymbol{\sigma}^{4}$ fixes the line
$\langle\boldsymbol{P}^{\boldsymbol{\sigma}},\boldsymbol{P}^{\boldsymbol{\sigma}^{5}}\rangle$
and
$\boldsymbol{Q}^{\boldsymbol{\sigma}^{4}}\in\langle\boldsymbol{P}^{\boldsymbol{\sigma}},\boldsymbol{P}^{\boldsymbol{\sigma}^{5}}\rangle$.
Now, let us suppose that
$\boldsymbol{P}=\langle\textbf{v}\rangle_{\mathbb{F}_{q^{6}}}$ with v a non-
zero vector belonging to $\mathbb{F}_{q^{8}}^{8}$. If
$\boldsymbol{Q}=\langle\textbf{v}^{\boldsymbol{\sigma}}-\eta\textbf{v}^{\boldsymbol{\sigma}^{5}}\rangle_{\mathbb{F}_{q^{8}}}$,
then
$\boldsymbol{Q}^{\boldsymbol{\sigma}^{4}}=\langle\textbf{v}^{\boldsymbol{\sigma}^{5}}-\eta^{q^{4}}\textbf{v}^{\boldsymbol{\sigma}}\rangle_{\mathbb{F}_{q^{8}}}$
with $\eta\in\mathbb{F}_{q^{8}}^{*}$. Since we have assumed that either $q\leq
11$ or $q\geq 1039891$, by [36, Theorem 1.1] we have
$\mathrm{N}_{q^{8}/q^{4}}(\eta)=-1$. Then

$(\boldsymbol{P}^{\boldsymbol{\sigma}};\boldsymbol{P}^{\boldsymbol{\sigma}^{5}};\boldsymbol{Q};\boldsymbol{Q}^{\boldsymbol{\sigma}^{4}})=\frac{\begin{vmatrix}1&1\\\
0&-\eta\end{vmatrix}\begin{vmatrix}0&-\eta^{q^{4}}\\\
1&1\end{vmatrix}}{\begin{vmatrix}1&-\eta^{q^{4}}\\\
0&1\end{vmatrix}\begin{vmatrix}0&1\\\
1&-\eta\end{vmatrix}}=\eta^{q^{4}+1}=-1.$

and hence Point 2.
Now suppose Point 1 and 2 hold true. By Theorem 3.13, since $L$ is
$(1,6)_{q}$-evasive, $L\cong L_{f}$ with $f(x)=\eta x^{q^{s}}+x^{q^{5s}}$,
$s\in\\{1,3,5,7\\}$ and $\eta\in\mathbb{F}_{q^{8}}^{*}$. Moreover, the point
$P=\langle v\rangle_{\mathbb{F}_{q^{8}}}$ is an imaginary point (wrt
$\Sigma$). Since $Q=\langle v^{\sigma}-\eta
v^{\sigma^{5}}\rangle_{\mathbb{F}_{q^{8}}}$ and
$-1=(P^{\sigma};P^{\sigma^{5}};Q;Q^{\sigma^{4}})=\mathrm{N}_{q^{8}/q^{4}}(\eta)$,
again by [36, Theorem 1.1], $L_{f}$, and hence $L$, is equivalent to a
scattered linear set of type $L_{r,\eta}^{3,8}$. ∎

###  4.2 A characterization of $L_{\delta}^{4,6}$

Let $L_{\delta}^{4,6}$ be the linear set defined as in (9) for $q$ odd and
$\delta^{2}+\delta=1$. In [11], it is showed that for $q\equiv 0,\pm
1\pmod{5}$, this is scattered. Remaining cases for $q$ odd were treated later
in [27].

In the following, exploiting same arguments used by Csajbók, Marino and Zullo
in [11, Thereom 5.7]), we firstly determine conditions under which the maximum
scattered linear set $L_{\delta}^{4,6}$ is simple.

######  Proposition 4.3.

The linear set $L_{\delta}^{4,6}$, with $q=p^{e}$ odd and
$\delta^{2}+\delta=1$, is simple if and only if either $q\equiv 0\pmod{5}$ or
$p\equiv\pm 2\pmod{5}$ for some prime $p$.

###### Proof.

To this aim it is enough to observe that the missing case in [11, Thereom
5.7]) is when $q=p^{2h+1}$ and $p\equiv\pm 2\pmod{5}$. In this case,
$\delta\in\mathbb{F}_{p^{2}}\setminus\mathbb{F}_{p}$. So, $\delta$ and
$\delta^{p}$ are two distinct roots of $X^{2}+X-1=0$ and $X^{q}+X+1=0$ and we
get that $U_{f}$ and $U_{\hat{f}}$ with $f(x)=x^{q}+x^{q^{3}}+\delta
x^{q^{5}}$ are $\Gamma\mathrm{L}(2,q^{6})$-equivalent. ∎

Hence, we can prove the following results.

######  Theorem 4.4.

Let $L$ be a $(1,4)_{q}$-evasive linear set of rank $6$ contained in a line
$\Lambda$ of $\mathrm{PG}(5,q^{6})$ and let $\Sigma$ be a canonical
subgeometry of $\mathrm{PG}(5,q^{6})$, $q$ odd. Then $L$ is equivalent to a
maximum scattered linear set of type $L_{\delta}^{4,6}$ if and only if for
each solid $\Gamma$ of $\mathrm{PG}(5,q^{6})$ with
$\Gamma\cap\Sigma=\Gamma\cap\Lambda=\emptyset$ such that
$L=\mathrm{p}_{\Gamma,\Lambda}(\Sigma)$, the following hold:

  1. 1.

there exist a generator $\sigma$ of the subgroup of
$\mathrm{P}\Gamma\mathrm{L}(6,q^{6})$ fixing $\Sigma$ pointwise and a point
$P=\langle v\rangle_{\mathbb{F}_{q^{6}}}$, $v$ a non-zero vector of
$\mathbb{F}_{q^{6}}^{6}$, such that

$\Gamma=\langle P^{\sigma^{2}},P^{\sigma^{4}},Q,R\rangle$ (20)

with $Q\in\langle P^{\sigma},P^{\sigma^{5}}\rangle$ and $R\in\langle
P^{\sigma^{3}},P^{\sigma^{5}}\rangle$;

  2. 2.

the point $C=\langle v^{\sigma}-v^{\sigma^{3}}\rangle_{\mathbb{F}_{q^{6}}}$
belongs to $\Gamma$;

  3. 3.

the points $P^{\sigma},P^{\sigma^{5}},C^{\sigma^{4}}$ and $Q$ are collinear
and the cross-ratio $(P^{\sigma};P^{\sigma^{5}};C^{\sigma^{4}};Q)$ is in
$\mathbb{F}_{q^{2}}$;

  4. 4.

the points $C,P^{\sigma^{5}}$ and $\langle Q,Q^{\sigma^{2}}\rangle\cap\langle
R,R^{\sigma^{2}}\rangle$ are collinear.

Moreover, the vertices $\Gamma$ such that
$L=\mathrm{p}_{\Gamma,\Lambda}(\Sigma)$ belong to the same orbit under the
action of $\mathrm{Aut}(\Sigma)$ if and only if $q\equiv 0\pmod{5}$ or
$p\equiv\pm 2\pmod{5}$ for some prime $p$.

###### Proof.

Let us suppose that $L$ is equivalent to a maximum scattered linear set of
type $L_{\delta}^{4,6}$, $\delta^{2}+\delta=1$. Since
$\Lambda_{f}=\Lambda_{\hat{f}}$ where $f(x)=x^{q}+x^{q^{3}}+\delta x^{q^{5}}$,
by Proposition 3.12, there exists a collineation $\beta$ such that
$\boldsymbol{\Sigma}^{\beta}=\Sigma$ and either $\Gamma_{f}^{\beta}=\Gamma$ or
$\Gamma_{\hat{f}}^{\beta}=\Gamma$, an imaginary point
$\boldsymbol{P}=\langle\textbf{v}\rangle_{\mathbb{F}_{q}^{6}}$ such that

$\Gamma_{f}=\langle\boldsymbol{P}^{\boldsymbol{\sigma^{2}}},\boldsymbol{P}^{\boldsymbol{\sigma}^{4}},\boldsymbol{Q}_{1},\boldsymbol{R}_{1}\rangle$

where
$\boldsymbol{Q}_{1}=\langle\textbf{v}^{\boldsymbol{\sigma}^{5}}-\delta\textbf{v}^{\boldsymbol{\sigma}}\rangle_{\mathbb{F}_{q^{6}}}$
and
$\boldsymbol{R}_{1}=\langle\textbf{v}^{\boldsymbol{\sigma}^{5}}-\delta\textbf{v}^{\boldsymbol{\sigma}^{3}}\rangle_{\mathbb{F}_{q^{6}}}$
and

$\Gamma_{\hat{f}}=\langle\boldsymbol{P}^{\boldsymbol{\sigma^{2}}},\boldsymbol{P}^{\boldsymbol{\sigma}^{4}},\boldsymbol{Q}_{2},\boldsymbol{R}_{2}\rangle$

where
$\boldsymbol{Q}_{2}=\langle\delta^{q}\textbf{v}^{\boldsymbol{\sigma}^{5}}-\textbf{v}^{\boldsymbol{\sigma}}\rangle_{\mathbb{F}_{q^{6}}}$
and
$\boldsymbol{R}_{2}=\langle\textbf{v}^{\boldsymbol{\sigma}^{5}}-\textbf{v}^{\boldsymbol{\sigma}^{3}}\rangle_{\mathbb{F}_{q^{6}}}$.
Let us consider the point
$\boldsymbol{C}_{1}=\langle\textbf{v}^{\boldsymbol{\sigma}}-\textbf{v}^{\boldsymbol{\sigma}^{3}}\rangle_{\mathbb{F}_{q^{6}}}$.
Clearly $\boldsymbol{C}_{1}\in\Gamma_{f}$ and it is easy to see that the point
$\boldsymbol{P}^{\boldsymbol{\sigma}},\boldsymbol{P}^{\boldsymbol{\sigma}^{5}},\boldsymbol{C}_{1}^{\boldsymbol{\sigma}^{4}}$
and $\boldsymbol{Q}_{1}$ are collinear. The cross-ratio
$(\boldsymbol{P}^{\boldsymbol{\sigma}};\boldsymbol{P}^{\boldsymbol{\boldsymbol{\sigma}}^{5}};\boldsymbol{C}_{1}^{\boldsymbol{\boldsymbol{\sigma}}^{4}};\boldsymbol{Q}_{1})=\delta\in\mathbb{F}_{q^{2}}$
and since $\delta^{2}+\delta=1$, elementary argument also shows that point 4
holds true for the points
$\boldsymbol{C}_{1},\boldsymbol{P}^{\boldsymbol{\sigma}^{5}}$ and
$\langle\boldsymbol{Q}_{1},\boldsymbol{Q}^{\boldsymbol{\sigma}^{2}}_{1}\rangle\cap\langle\boldsymbol{R}_{1},\boldsymbol{R}_{1}^{\boldsymbol{\sigma}^{2}}\rangle$
are collinear. Then, if $\Gamma_{f}^{\beta}=\Gamma$, putting
$\sigma=\boldsymbol{\sigma}^{\beta}$, $\boldsymbol{P}^{\beta}=P$,
$\boldsymbol{Q}_{1}^{\beta}=Q,\boldsymbol{R}_{1}^{\beta}=R$ and
$\boldsymbol{C}_{1}^{\beta}=C$, Point (1-4) hold true for $\Gamma$ as well.
If $\Gamma^{\beta}_{\hat{f}}=\Gamma$, let us consider the point
$\boldsymbol{S}=\langle\delta^{q}\textbf{v}^{\boldsymbol{\sigma}^{3}}-\textbf{v}^{\boldsymbol{\sigma}}\rangle_{\mathbb{F}_{q^{6}}}\in\Gamma_{\hat{f}}\cap\langle\boldsymbol{P}^{\boldsymbol{\sigma}},\boldsymbol{P}^{\boldsymbol{\sigma}^{3}}\rangle_{\mathbb{F}_{q^{6}}}$.
Clearly, the solid
$\Gamma_{\hat{f}}=\langle\boldsymbol{P}^{\boldsymbol{\sigma^{2}}},\boldsymbol{P}^{\boldsymbol{\sigma}^{4}},\boldsymbol{Q}_{2},\boldsymbol{S}\rangle$.
Then, it is easy to see that the point
$\boldsymbol{P}^{\boldsymbol{\sigma}},\boldsymbol{P}^{\boldsymbol{\sigma}^{5}},\boldsymbol{R}_{2}^{\boldsymbol{\sigma}^{2}}$
and $\boldsymbol{Q}_{2}$ are collinear and their cross-ratio is equal to
$\delta^{-q}$ which belongs to $\mathbb{F}_{q^{2}}$. Since
$\delta^{2}+\delta=1$, elementary argument shows that the points
$\boldsymbol{R}_{2},\boldsymbol{P}^{\boldsymbol{\sigma}}$ and
$\langle\boldsymbol{Q}_{2},\boldsymbol{Q}_{2}^{\boldsymbol{\sigma}^{4}}\rangle\cap\langle\boldsymbol{S},\boldsymbol{S}^{\boldsymbol{\sigma}^{4}}\rangle$
are collinear.

Then, if $\Gamma_{\hat{f}}^{\beta}=\Gamma$, putting
$\sigma=\boldsymbol{\sigma}^{5\beta}$, $\boldsymbol{P}^{\beta}=P$,
$\boldsymbol{Q}^{\beta}_{2}=Q,\boldsymbol{S}^{\beta}=R$ and
$\boldsymbol{R}_{2}^{\beta}=C$, Point (1-4) hold for $\Gamma$.
Now we suppose that Point $(\textit{1-4})$ hold true then by Theorem 3.13,
$L\cong L_{f}$ with $f(x)=x^{q^{s}}+\gamma x^{q^{3s}}+\delta
x^{q^{5s}}\in\mathbb{F}_{q^{6}}[x]$ with $s\in\\{1,5\\}$ and $P=\langle
v\rangle_{\mathbb{F}_{q^{6}}}$ is an imaginary point. Then $Q=\langle
v^{\sigma^{5}}-\delta v^{\sigma}\rangle_{\mathbb{F}_{q^{6}}}$ and
$R=\langle\gamma v^{\sigma^{5}}-\delta
v^{\sigma^{3}}\rangle_{\mathbb{F}_{q^{6}}}$. Since $C=\langle
v^{\sigma}-v^{\sigma^{3}}\rangle_{\mathbb{F}_{q^{6}}}\in\Gamma$, then
$\gamma=1$. By Point 3, it is easy to see that the cross-ratio
$(P^{\sigma};P^{\sigma^{5}};C^{\sigma^{4}};Q)=\delta$, and hence
$\delta\in\mathbb{F}_{q^{2}}$. Moreover, by Point 4, the lines $\langle
Q,Q^{\sigma^{2}}\rangle$ and $\langle R,R^{\sigma^{2}}\rangle$ meet at the
point $\langle
v^{\sigma}-(\delta^{2}+\delta)v^{\sigma^{3}}+v^{\sigma^{5}}\rangle_{\mathbb{F}_{q^{6}}}$
and the collinearity of this point, $C$ and $P^{\sigma^{5}}$ implies

$0=\begin{vmatrix}0&0&1\\\ 1&-1&0\\\
1&-(\delta^{2}+\delta)&1\end{vmatrix}=-\delta^{2}-\delta+1,$ (21)

obtaining $L\cong L_{\delta}^{4,6}$. ∎

The family of linear sets of type $L^{4,6}_{\delta}$ was investigated for also
for $q$ even in [4]. The conditions assuring the property of being scattered
in this case are rather demanding, and it seems quite difficult to translate
them in geometric terms, see [4, Theorem 2.9]. However, by using arguments
similar to those used in the theorem above, we can state the following result.

######  Theorem 4.5.

Let $L$ be a $(1,4)_{q}$-evasive linear set of rank $6$ contained in a line
$\Lambda$ of $\mathrm{PG}(5,q^{6})$, $q$ even and let $\Sigma$ be a canonical
subgeometry of $\mathrm{PG}(5,q^{6})$, $q$ even. If $L$ is equivalent to a
maximum scattered linear set of type $L_{\delta}^{4,6}$, then for each solid
$\Gamma$ of $\mathrm{PG}(5,q^{6})$ with
$\Gamma\cap\Sigma=\Gamma\cap\Lambda=\emptyset$ such that
$L=\mathrm{p}_{\Gamma,\Lambda}(\Sigma)$, the following holds:

  1. 1.

there exist a generator $\sigma$ of the subgroup of
$\mathrm{P}\Gamma\mathrm{L}(6,q^{6})$ fixing $\Sigma$ pointwise and a point
$P=\langle v\rangle_{\mathbb{F}_{q^{6}}}$, $v$ a nonzero vector of
$\mathbb{F}_{q^{6}}^{6}$, such that the solid

$\Gamma=\langle P^{\sigma^{2}},P^{\sigma^{4}},Q,R\rangle$ (22)

with $Q\in\langle P^{\sigma},P^{\sigma^{5}}\rangle$ and $R\in\langle
P^{\sigma^{3}},P^{\sigma^{5}}\rangle$;

  2. 2.

the point $C=\langle v^{\sigma}-v^{\sigma^{3}}\rangle_{\mathbb{F}_{q^{6}}}$
belongs to $\Gamma$;

  3. 3.

the point $P^{\sigma},P^{\sigma^{5}},C^{\sigma^{4}}$ and $Q$ are collinear and
the cross-ratio $(P^{\sigma};P^{\sigma^{5}};C^{\sigma^{4}};Q)$ does not belong
to any subfield of $\mathbb{F}_{q{{}^{6}}}$.

###  4.3 A characterization of $L_{s,m,h}^{5,t}$

This section is devoted to characterize via geometrical properties of their
projection vertex the scattered linear set of type $L_{s,m,h}^{5,t}$ as in
(10) with

$\psi_{t,s,m,h}(x)=x^{q^{s(t-1)}}+h^{1-q^{s(2t-1)}}x^{q^{s(2t-1)}}+m\left(x^{q^{s}}-h^{1-q^{s(t+1)}}x^{q^{s(t+1)}}\right),$

where $t\geq 3$ and
$(m,h)\in\mathbb{F}_{q^{t}}^{*}\times\mathbb{F}_{q^{2t}}^{*}$. We will proceed
by splitting our discussion in two cases according to suitable conditions for
the parameters $m$ and $h$ are satisfied, precisely:

  1. $(i)$

$m=1$ and $h\in\mathbb{F}_{q^{2t}}$ with $\mathrm{N}_{q^{2t}/q^{t}}(h)=-1$

  2. $(ii)$

$m$ neither a $(q-1)$-th power nor a $(q+1)$-th power of an element belonging
to $\ker\mathrm{Tr}_{q^{2t}/q^{t}}$ and $q\equiv 1\pmod{4}$ if $t$ is odd.

Actually, for $q\leq 5$ odd and $t\in\\{3,4,5,6\\}$, Magma computation
suggests a wider choice of parameters for which the linear set
$L_{s,m,h}^{5,t}$ is scattered (see [16]).

#### The linear sets of the type $L_{s,m,h}^{5,t}$, $m=1$

In order to investigate the geometrical properties of the projection vertex of
a linear set of type $L_{s,h}^{5,t}:=L_{s,1,h}^{5,t}$ in
$\mathrm{PG}(1,q^{2t})$, we first observe that, by Theorem 3.9, [25,
Proposition 3.1] and [28, Proposition 4.17], for any $t\geq 5$
$\psi_{t,h,s}(x):=\psi_{t,s,1,h}(x)$ is
$\Gamma\mathrm{L}(2t,q^{2t})$-equivalent to its adjoint. For $t=3$, using the
same argument as done in [23, Theorem 4.7], the simplicity of $L_{s,h}^{5,t}$
follows. Therefore, we get the following.

######  Theorem 4.6.

The linear set $L_{s,h}^{5,t}$ of $\mathrm{PG}(1,q^{2t})$ is simple, for $t=3$
or $t\geq 5$.

Now, as we did for the previous examples, we prove the following result.

######  Theorem 4.7.

Let $L$ be an $(1,2(t-1))$-evasive linear set of rank $2t$ in a line $\Lambda$
of $\mathrm{PG}(2t-1,q^{2t})$. Let $\Sigma$ be a canonical subgeometry of
$\mathrm{PG}(2t-1,q^{2t})$, $q$ odd and $t\geq 3$. Then $L$ is equivalent to a
scattered linear set of the type $L_{s,h}^{5,t}$ if and only if for each
$(2t-3)$ dimensional subspace $\Gamma$ of $\mathrm{PG}(2t-1,q^{2t})$ such that
$L=\mathrm{p}_{\Gamma,\Lambda}(\Sigma)$, the following holds:

  1. 1.

there exist a generator $\sigma$ of the subgroup of
$\mathrm{P}\Gamma\mathrm{L}(2t,q^{2t})$ fixing $\Sigma$ pointwise and a point
$P=\langle v\rangle_{\mathbb{F}_{q^{2t}}}\in\mathrm{PG}(2t-1,q^{2t})$ such
that

$\Gamma=\langle P^{\sigma^{i}},Q,R,S:i\notin\\{0,1,t-1,t+1,2t-1\\}\rangle$

with $Q\in\langle P^{\sigma},P^{\sigma^{2t-1}}\rangle$, $R\in\langle
P^{\sigma^{t-1}},P^{\sigma^{2t-1}}\rangle$ and $S\in\langle
P^{\sigma^{t+1}},P^{\sigma^{2t-1}}\rangle$,

  2. 2.

the point $X=\langle v^{\sigma}-v^{\sigma^{t-1}}\rangle_{\mathbb{F}_{q^{2t}}}$
belongs to $\Gamma$,

  3. 3.

there exists
$h\in\mathcal{H}=\\{z\in\mathbb{F}_{q^{2t}}\,\colon\,\mathrm{N}_{q^{2t}/q^{t}}(z)=-1\\}$
such that

    1. (a)

the points $P^{\sigma},P^{\sigma^{2t-1}},D,Q$ where $D=\langle
v^{\sigma}+v^{\sigma^{2t-1}}\rangle_{\mathbb{F}_{q^{2t}}}$ are collinear and
the cross-ratio $(P^{\sigma};P^{\sigma^{2t-1}};D;Q)=-h^{1-q^{s(2t-1)}}$,

    2. (b)

the points $P^{\sigma},P^{\sigma^{t+1}},R^{\sigma^{2}},Y$, where $Y=\langle
Q,S\rangle\cap\langle P^{\sigma},P^{\sigma^{t+1}}\rangle$, are collinear and
the cross-ratio $(P^{\sigma};P^{\sigma^{t+1}};R^{\sigma^{2}};Y)=h^{1+q^{2s}}$.

###### Proof.

Let us suppose that $L$ is equivalent to a maximum scattered linear set of
type $L^{5,t}_{s,h}$. By Theorem 3.11, there exists $\beta$ such that
$\boldsymbol{\Sigma}^{\beta}=\Sigma$ and either
$\Gamma_{\psi_{t,h,s}}^{\beta}=\Gamma$ or
$\Gamma_{\hat{\psi}_{t,h,s}}^{\beta}=\Gamma$. If $t=3$ or $t\geq 5$, since
$L^{5,t}_{s,h}$ is simple, it is enough to prove the statement supposing that
$\Gamma_{\psi_{t,h,s}}^{\beta}=\Gamma$.
Let $\boldsymbol{\rho}=\boldsymbol{\sigma}^{s}$ and
$\boldsymbol{P}=\langle\textbf{v}\rangle_{\mathbb{F}_{q^{2t}}}$ be an
imaginary point of $\mathrm{PG}(2t-1,q^{2t})$ (wrt $\boldsymbol{\Sigma}$),
$\boldsymbol{Q}\in\langle\boldsymbol{P}^{\boldsymbol{\rho}},\boldsymbol{P}^{\boldsymbol{\rho}^{2t-1}}\rangle$,
$\boldsymbol{R}\in\langle\boldsymbol{P}^{\boldsymbol{\rho}^{t-1}},\boldsymbol{P}^{\boldsymbol{\rho}^{2t-1}}\rangle$
and
$\boldsymbol{S}\in\langle\boldsymbol{P}^{\boldsymbol{\rho}^{t+1}},\boldsymbol{P}^{\boldsymbol{\rho}^{2t-1}}\rangle$
and

$\Gamma_{\psi_{t,h,s}}=\langle\boldsymbol{P}^{\boldsymbol{\rho}^{i}},\boldsymbol{Q},\boldsymbol{R},\boldsymbol{S}:i\not\in\\{1,t-1,t+1,2t-1\\}\rangle$

where

$\boldsymbol{Q}=\langle\textbf{v}^{\boldsymbol{\rho}^{2t-1}}-h^{1-q^{s(2t-1)}}\textbf{v}^{\boldsymbol{\rho}}\rangle_{\mathbb{F}_{q^{2t}}},\boldsymbol{R}=\langle\textbf{v}^{\boldsymbol{\rho}^{2t-1}}-h^{1-q^{s(2t-1)}}\textbf{v}^{\boldsymbol{\rho}^{t-1}}\rangle_{\mathbb{F}_{q^{2t}}},\textnormal{
and }$ $\boldsymbol{S}=\langle
h^{1-q^{s(t+1)}}\textbf{v}^{\boldsymbol{\rho}^{2t-1}}+h^{1-q^{s(2t-1)}}\textbf{v}^{\boldsymbol{\rho}^{t+1}}\rangle_{\mathbb{F}_{q^{2t}}}.$

It is straightforward to check that the lines
$\langle\boldsymbol{Q},\boldsymbol{R}\rangle$ and
$\langle\boldsymbol{P}^{\boldsymbol{\rho}},\boldsymbol{P}^{\boldsymbol{\rho}^{t-1}}\rangle$
meet at the point
$\boldsymbol{X}=\langle\textbf{v}^{\boldsymbol{\rho}^{t-1}}-\textbf{v}^{\boldsymbol{\rho}}\rangle_{\mathbb{F}_{q^{2t}}}$
and this belongs to $\Gamma_{\psi_{t,h,s}}$. Also, the lines
$\langle\boldsymbol{Q},\boldsymbol{S}\rangle$ and
$\langle\boldsymbol{P}^{\boldsymbol{\rho}},\boldsymbol{P}^{\boldsymbol{\rho}^{t+1}}\rangle$
meet at the point $\boldsymbol{Y}=\langle
h^{1-q^{s(t+1)}}\textbf{v}^{\boldsymbol{\rho}}+\textbf{v}^{\boldsymbol{\rho}^{t+1}}\rangle_{\mathbb{F}_{q^{2t}}}$.
Let
$\boldsymbol{D}=\langle\textbf{v}^{\boldsymbol{\rho}}+\textbf{v}^{\boldsymbol{\rho}^{2t-1}}\rangle_{\mathbb{F}_{q^{2t}}}$,
then the points
$\boldsymbol{P}^{\boldsymbol{\rho}},\boldsymbol{P}^{\boldsymbol{\rho}^{2t-1}},\boldsymbol{D},\boldsymbol{Q}$
and
$\boldsymbol{P}^{\boldsymbol{\rho}},\boldsymbol{P}^{\boldsymbol{\rho}^{t+1}},\boldsymbol{R}^{\boldsymbol{\rho}^{2}},\boldsymbol{Y}$
are collinear. Moreover,

$(\boldsymbol{P}^{\boldsymbol{\rho}};\boldsymbol{P}^{\boldsymbol{\rho}^{2t-1}};\boldsymbol{D};\boldsymbol{Q})=\frac{\begin{vmatrix}1&1\\\
0&1\end{vmatrix}\begin{vmatrix}0&1\\\
1&-h^{1-q^{s(2t-1)}}\end{vmatrix}}{\begin{vmatrix}1&1\\\
0&-h^{1-q^{s(2t-1)}}\end{vmatrix}\begin{vmatrix}0&1\\\
1&1\end{vmatrix}}=-h^{1-q^{s(2t-1)}}$

and

$(\boldsymbol{P}^{\boldsymbol{\rho}};\boldsymbol{P}^{\boldsymbol{\rho}^{t+1}};\boldsymbol{R}^{\boldsymbol{\rho}^{2}};\boldsymbol{Y})=\frac{\begin{vmatrix}1&1\\\
0&-h^{q^{2s}-q^{s}}\\\ \end{vmatrix}\begin{vmatrix}0&h^{1-q^{s(t+1)}}\\\
1&1\end{vmatrix}}{\begin{vmatrix}1&h^{1-q^{s(t+1)}}\\\
0&1\end{vmatrix}\begin{vmatrix}0&1\\\ 1&-h^{q^{2s}-q^{s}}\\\
\end{vmatrix}}=h^{q^{2s}+1}.$

Therefore, since $\beta$ preserve the collinearity and if
$(A_{1};A_{2};A_{3};A_{4})\in\mathcal{H}$ then
$(A^{\beta}_{1};A^{\beta}_{2};A^{\beta}_{3};A^{\beta}_{4})$ belongs to
$\mathcal{H}$, the Points 1-3 hold true.
For $t=4$, we have

$\Gamma_{\hat{\psi}_{t,h,s}}=\langle\boldsymbol{P}^{\boldsymbol{\rho}^{i}},\boldsymbol{Q},\boldsymbol{R},\boldsymbol{S}:i\not\in\\{1,3,5,7\\}\rangle$

where

$\boldsymbol{Q}=\langle\textbf{v}^{\boldsymbol{\rho}}-h^{q^{s}-1}\textbf{v}^{\boldsymbol{\rho}^{7}}\rangle_{\mathbb{F}_{q^{8}}},\quad\boldsymbol{R}=\langle\textbf{v}^{\boldsymbol{\rho}^{3}}+h^{q^{3s}-1}\textbf{v}^{\boldsymbol{\rho}^{7}}\rangle_{\mathbb{F}_{q^{8}}},\textnormal{
and }$
$\boldsymbol{S}=\langle\textbf{v}^{\boldsymbol{\rho}^{7}}-\textbf{v}^{\boldsymbol{\rho}^{5}}\rangle_{\mathbb{F}_{q^{8}}}.$

If $\boldsymbol{\Sigma}^{\beta}=\Sigma$ and
$\Gamma_{\hat{\psi}_{t,h,s}}^{\beta}=\Gamma$, let
$\boldsymbol{D}=\langle\textbf{v}^{\rho}+\textbf{v}^{\boldsymbol{\rho}^{7}}\rangle_{\mathbb{F}_{q^{8}}}$,
$\boldsymbol{T}=\langle
h^{q^{3s}-q^{s}}\textbf{v}^{\boldsymbol{\rho}}-\textbf{v}^{\boldsymbol{\rho}^{3}}\rangle_{\mathbb{F}_{q^{8}}}$
and
$\boldsymbol{Z}=\langle\textbf{v}^{\boldsymbol{\rho}}-h^{1-q^{s}}\textbf{v}^{\boldsymbol{\rho}^{5}}\rangle_{\mathbb{F}_{q^{8}}}$.
The result follows noting that

$\Gamma_{\hat{\psi}_{t,h,s}}=\langle\boldsymbol{P}^{\boldsymbol{\rho}^{i}},\boldsymbol{Q},\boldsymbol{Z},\boldsymbol{T}:i\not\in\\{1,3,5,7\\}\rangle$

and putting $\sigma=\beta^{-1}\boldsymbol{\rho}^{7}\beta$,
$P=\boldsymbol{P}^{\beta}$, $Q=\boldsymbol{Q}^{\beta}$,
$R=\boldsymbol{Z}^{\beta}$, $S=\boldsymbol{T}^{\beta}$,
$X=\boldsymbol{S}^{\beta}$ and $D=\boldsymbol{D}^{\beta}$.
Now, let suppose that Points $\textit{1}-\textit{3}$ hold true. By Theorem
3.13, there exist a subgeometry $\Sigma$, a collineation $\sigma$ fixing
$\Sigma$ pointwise and an imaginary point (wrt $\Sigma$) $P=\langle
v\rangle_{\mathbb{F}_{q^{2t}}}$, such that $L\cong L_{f}$ with
$f(x)=x^{q^{s}}+\alpha x^{q^{s(t-1)}}+\beta x^{q^{s(t+1)}}+\gamma
x^{q^{s(2t-1)}}\in\mathbb{F}_{q^{n}}[x]$, where $Q=\langle
v^{\sigma^{2t-1}}-\gamma v^{\sigma}\rangle_{\mathbb{F}_{q^{2t}}}$,
$R=\langle\alpha v^{\sigma^{2t-1}}-\gamma
v^{\sigma^{t-1}}\rangle_{\mathbb{F}_{q^{2t}}}$ and $S=\langle\beta
v^{\sigma^{2t-1}}-\gamma v^{\sigma^{t+1}}\rangle_{\mathbb{F}_{q^{2t}}}$.
Again, it is straightforward to see that $X=\langle v^{\sigma^{t-1}}-\alpha
v^{\sigma}\rangle_{\mathbb{F}_{q^{2t}}}$ and, hence by 2, $\alpha=1$. By Point
3(b), saying $D=\langle
v^{\sigma}+v^{\sigma^{2t-1}}\rangle_{\mathbb{F}_{q^{{\color[rgb]{0,0,1}2}t}}}$,
we get $\gamma=h^{1-q^{s(2t-1)}}$ for some $h\in\mathbb{F}_{q^{2t}}$ with
$\mathrm{N}_{q^{2t}/q^{t}}(h)=-1$. Finally, since $Y=\langle\beta
v^{\sigma}-v^{\sigma^{t+1}}\rangle_{\mathbb{F}_{q^{2t}}}$, 3(c) implies that
$\beta\gamma^{q^{2s}}=h^{1+q^{2s}}$, so
$\beta=\frac{h^{1+q^{2s}}}{h^{q^{2s}-q^{s}}}=h^{1+q^{s}}=-h^{1-q^{s(t+1)}}$.
Therefore, $L\cong L_{s,h}^{5,t}$. ∎

By Point $3(a-b)$ of Theorem above, if $h\in\mathbb{F}_{q^{t}}$, then the
cross-ratio $(P^{\sigma},P^{\sigma^{2t-1}},D,Q)$ and
$(P^{\sigma},P^{\sigma^{t+1}},R^{\sigma^{2}},Y)$ are equal to $-1$. Thus,
setting $L^{5,t}_{s}:=L^{5,t}_{h,s}$ with $h\in\mathbb{F}_{q^{t}}$, we can
state the following.

######  Corollary 4.8.

Let $L$ be an $(1,2(t-1))$-evasive linear set of rank $2t$ in a line $\Lambda$
of $\mathrm{PG}(2t-1,q^{2t})$. Let $\Sigma$ be a canonical subgeometry of
$\mathrm{PG}(2t-1,q^{2t})$, $q$ odd. Then $L$ is equivalent to a scattered
linear set of the type $L_{s}^{5,t}$ if and only if for each $(2t-3)$
dimensional subspace $\Gamma$ of $\mathrm{PG}(2t-1,q^{2t})$ such that
$L=\mathrm{p}_{\Gamma,\Lambda}(\Sigma)$, the following holds:

  1. 1.

there exists a generator $\sigma$ of the subgroup of
$\mathrm{P}\Gamma\mathrm{L}(2t,q^{2t})$ fixing $\Sigma$ pointwise and a point
$P=\langle v\rangle_{\mathbb{F}_{q^{2t}}}\in\mathrm{PG}(2t-1,q^{2t})$ such
that

$\Gamma=\langle P^{\sigma^{i}},Q,R,S:i\notin\\{0,1,t-1,t+1,2t-1\\}\rangle$

with $Q\in\langle P^{\sigma},P^{\sigma^{2t-1}}\rangle$, $R\in\langle
P^{\sigma^{t-1}},P^{\sigma^{2t-1}}\rangle$ and $S\in\langle
P^{\sigma^{t+1}},P^{\sigma^{2t-1}}\rangle$,

  2. 2.

the point $X=\langle v^{\sigma}-v^{\sigma^{t-1}}\rangle_{\mathbb{F}_{q^{2t}}}$
belongs to $\Gamma$.

  3. 3.

    1. (a)

the pair $\\{P^{\sigma},P^{\sigma^{2t-1}}\\}$ separates $\\{D,Q\\}$
harmonically, where $D=\langle
v^{\sigma}+v^{\sigma^{2t-1}}\rangle_{\mathbb{F}_{q^{2t}}}$.

    2. (b)

the pair $\\{P^{\sigma},P^{\sigma^{t+1}}\\}$ separates
$\\{R^{\sigma^{2}},Y\\}$ harmonically, where $Y=\langle Q,S\rangle\cap\langle
P^{\sigma},P^{\sigma^{t+1}}\rangle$.

  4. 4.

If $t$ is odd, then $q\equiv 1\pmod{4}$.

#### The linear sets of the type $L_{m,s}^{5,t}$, for
$m\in\mathbb{F}_{q^{t}}\setminus\\{0,1\\}$

In this subsection we will deal with the family of scattered linear sets of
$\mathrm{PG}(1,q^{2t})$, $t\geq 3$ and $q$ odd, recently studied in [35] (cf.
(10)). The elements in this class have the following fashion:

$L_{m,s}^{5,t}=\\{\langle(x,m(x^{q^{s}}-x^{q^{s(t+1)}})+x^{q^{s(t-1)}}+x^{q^{s(2t-1)}}\rangle_{\mathbb{F}_{q^{2t}}}:x\in\mathbb{F}_{q^{2t}}^{*}\\},$

where $\gcd(s,2t)=1$. Here, $m\in\mathbb{F}_{q^{t}}^{*}$ is neither a
$(q-1)$-th power nor a $(q+1)$-th power of any element of the vector space
$\ker\mathrm{Tr}_{q^{2t}/q^{t}}$ and if $t$ is odd, $q\equiv 1\pmod{4}$, see
[35, Theorem 2.3]. Firstly, by [35, Theorem 4.5] and Theorem 3.9, for this
class of scattered linear sets with parameter $m\in\mathbb{F}_{q^{t}}^{*}$ as
above, we can state the following.

######  Theorem 4.9.

For $t>4$, the $\Gamma\mathrm{L}$-class of the scattered linear set
$L_{m,s}^{5,t}$ is two.

Similarly to what did before, we are now ready to characterise the projection
vertex of a linear set equivalent an element of the type $L_{m,s}^{5,t}$.

######  Theorem 4.10.

Let $L$ be an $(1,2(t-1))$-evasive linear set of rank $2t$ in a line $\Lambda$
of $\mathrm{PG}(2t-1,q^{2t})$. Let $\Sigma$ be a canonical subgeometry of
$\mathrm{PG}(2t-1,q^{2t})$. Then $L$ is equivalent to a maximum scattered
linear set of the type $L_{m,s}^{5,t}$ if and only if for each $(2t-3)$
dimensional subspace $\Gamma$ of $\mathrm{PG}(2t-1,q^{2t})$ such that
$L=\mathrm{p}_{\Gamma,\Lambda}(\Sigma)$, the following holds:

  1. 1.

there exist a generator $\sigma$ of the subgroup of
$\mathrm{P}\Gamma\mathrm{L}(2t,q^{2t})$ fixing $\Sigma$ pointwise and a point
$P=\langle v\rangle_{\mathbb{F}_{q^{2t}}}\in\mathrm{PG}(2t-1,q^{2t})$ such
that

$\Gamma=\langle P^{\sigma^{i}},Q,R,S:i\notin\\{0,1,t-1,t+1,2t-1\\}\rangle$

with $Q\in\langle P^{\sigma},P^{\sigma^{2t-1}}\rangle$, $R\in\langle
P^{\sigma^{t-1}},P^{\sigma^{2t-1}}\rangle$ and $S\in\langle
P^{\sigma^{t+1}},P^{\sigma^{2t-1}}\rangle$,

  2. 2.

the point $C=\langle
v^{\sigma}+v^{\sigma^{t+1}}\rangle_{\mathbb{F}_{q^{2t}}}\in\Gamma$,

  3. 3.

there exists
$\mu\in\mathcal{W}=\\{z\in\mathbb{F}_{q^{t}}\colon\nexists\,w\in\ker\mathrm{Tr}_{q^{2t}/q^{t}}\textnormal{
s.t. }z=w^{q-1}\textnormal{ or }z=w^{q+1}\\}$ and

    1. (a)

the points $P^{\sigma},P^{\sigma^{2t-1}}$, $X$, $Q$, where $X=\langle
v^{\sigma}-v^{\sigma^{2t-1}}\rangle_{\mathbb{F}_{q^{2t}}}$, are collinear and
the cross ratio $(P^{\sigma};P^{\sigma^{2t-1}};X;Q)=\mu$,

    2. (b)

the points $P^{\sigma^{t-1}},P^{\sigma^{2t-1}}$, $Y$, $R$, where $Y=\langle
v^{\sigma^{t-1}}+v^{\sigma^{2t-1}}\rangle_{\mathbb{F}_{q^{2t}}}$, are
collinear and the pair $\\{P^{\sigma^{t-1}},P^{\sigma^{2t-1}}\\}$ separates
the pair $\\{Y,R\\}$ harmonically.

###### Proof.

Let us suppose that $L$ is equivalent to $L_{m,s}^{5,t}$. By Theorem 3.11,
there exists $\beta$ such that $\boldsymbol{\Sigma}^{\beta}=\Sigma$, saying
$\psi_{t,m,s}:=\psi_{t,m,h,s}$ with $h\in\mathbb{F}_{q^{t}}$, and either
$\Gamma_{\psi_{t,m,s}}^{\beta}=\Gamma$ or
$\Gamma_{\hat{\psi}_{t,m,s}}^{\beta}=\Gamma$.
If $\beta\in\mathrm{P}\Gamma\mathrm{L}(2t,q^{2t})$ such that
$\Gamma_{\psi_{t,m,s}}^{\beta}=\Gamma$, then let
$\boldsymbol{\rho}=\boldsymbol{\sigma}^{s}$ with $\gcd(s,2t)=1$,
$\boldsymbol{P}=\langle\textbf{v}\rangle_{\mathbb{F}_{q^{2t}}}$ be an
imaginary point of $\mathrm{PG}(2t-1,q^{2t})$ (wrt $\boldsymbol{\Sigma}$) and

$\Gamma_{\psi_{t,m,s}}=\langle\boldsymbol{P}^{\boldsymbol{\rho}^{i}},\boldsymbol{Q},\boldsymbol{R},\boldsymbol{S}:i\not\in\\{1,t-1,t+1,2t-1\\}\rangle$

where

$\boldsymbol{Q}=\langle
m\textbf{v}^{\boldsymbol{\rho}^{2t-1}}-\textbf{v}^{\boldsymbol{\rho}}\rangle_{\mathbb{F}_{q^{2t}}},\quad\boldsymbol{R}=\langle\textbf{v}^{\boldsymbol{\rho}^{2t-1}}-\textbf{v}^{\boldsymbol{\rho}^{t-1}}\rangle_{\mathbb{F}_{q^{2t}}},\textnormal{
and }$ $\boldsymbol{S}=\langle
m\textbf{v}^{\boldsymbol{\rho}^{2t-1}}+\textbf{v}^{\boldsymbol{\rho}^{t+1}}\rangle_{\mathbb{F}_{q^{2t}}}.$

Let us consider the point
$\boldsymbol{C}=\langle\textbf{v}^{\boldsymbol{\rho}}+\textbf{v}^{\boldsymbol{\rho}^{t+1}}\rangle_{\mathbb{F}_{q^{2t}}}$;
it belongs to $\Gamma_{\psi_{t,m,s}}$. Moreover, let
$\boldsymbol{X}=\langle\textbf{v}^{\boldsymbol{\rho}}-\textbf{v}^{\boldsymbol{\rho}^{2t-1}}\rangle_{\mathbb{F}_{q^{2t}}}$.
Clearly,
$\boldsymbol{X},\boldsymbol{Q}\in\langle\boldsymbol{P}^{\boldsymbol{\rho}};\boldsymbol{P}^{\boldsymbol{\rho}^{2t-1}}\rangle$
and
$(\boldsymbol{P}^{\boldsymbol{\rho}};\boldsymbol{P}^{\boldsymbol{\rho}^{2t-1}};\boldsymbol{X};\boldsymbol{Q})=1/m$.
Finally,
$(\boldsymbol{P}^{\boldsymbol{\rho}^{t-1}};\boldsymbol{P}^{\boldsymbol{\rho}^{2t-1}};\boldsymbol{Y};\boldsymbol{R})=-1$
where
$\boldsymbol{Y}=\langle\textbf{v}^{\boldsymbol{\rho}^{t-1}}+\textbf{v}^{\boldsymbol{\rho}^{2t-1}}\rangle_{\mathbb{F}_{q^{2t}}}$.
By setting $\sigma=\beta^{-1}\boldsymbol{\rho}\beta$,
$P=\boldsymbol{P}^{\beta}$, $Q=\boldsymbol{Q}^{\beta}$,
$R=\boldsymbol{R}^{\beta}$, $S=\boldsymbol{S}^{\beta}$,
$C=\boldsymbol{C}^{\beta}$ $X=\boldsymbol{X}^{\beta}$ and
$Y=\boldsymbol{Y}^{\beta}$ and noting that $\beta$ preserves the collinearity,
being harmonically separated and if $(A_{1};A_{2};A_{3};A_{4})\in\mathcal{W}$
then
$(A^{\beta}_{1};A^{\beta}_{2};A^{\beta}_{3};A^{\beta}_{4})\in\mathcal{W}$, the
Point(1–3) follow.
If $\Gamma_{\hat{\psi}_{t,m,s}}^{\beta}=\Gamma$, by Remark 3.10,

$\Gamma_{\hat{\psi}_{t,m,s}}=\langle\boldsymbol{P}^{\boldsymbol{\rho}^{i}},\boldsymbol{Q},\boldsymbol{R},\boldsymbol{S}:i\notin\\{0,1,t-1,t+1,2t-1\\}\rangle$

where

$\boldsymbol{Q}=\langle
m^{\boldsymbol{\rho}^{t-1}}\textbf{v}^{\boldsymbol{\rho}}+\textbf{v}^{\boldsymbol{\rho}^{t-1}}\rangle_{\mathbb{F}_{q^{2t}}},\quad\boldsymbol{R}=\langle\textbf{v}^{\boldsymbol{\rho}}-\textbf{v}^{\boldsymbol{\rho}^{t+1}}\rangle_{\mathbb{F}_{q^{2t}}},\textnormal{
and }$ $\boldsymbol{S}=\langle
m^{\boldsymbol{\rho}^{2t-1}}\textbf{v}^{\boldsymbol{\rho}}-\textbf{v}^{\boldsymbol{\rho}^{2t-1}}\rangle_{\mathbb{F}_{q^{2t}}}$

Let us consider the point
$\boldsymbol{C}=\langle\textbf{v}^{\boldsymbol{\rho}^{2t-1}}+\textbf{v}^{\boldsymbol{\rho}^{t-1}}\rangle_{\mathbb{F}_{q^{2t}}}$.
Since $m\in\mathbb{F}_{q^{t}}$,
$\boldsymbol{C}\in\Gamma_{\hat{\psi}_{t,m,s}}$. Moreover, let consider the
point
$\boldsymbol{X}=\langle\textbf{v}^{\boldsymbol{\rho}}-\textbf{v}^{\boldsymbol{\rho}^{2t-1}}\rangle_{\mathbb{F}_{q^{2t}}}$.
Similarly to the previous case,
$\boldsymbol{X},\boldsymbol{S}\in\langle\boldsymbol{P}^{\boldsymbol{\rho}};\boldsymbol{P}^{\boldsymbol{\rho}^{2t-1}}\rangle$
and
$(\boldsymbol{P}^{\boldsymbol{\rho}};\boldsymbol{P}^{\boldsymbol{\rho}^{2t-1}};\boldsymbol{X};\boldsymbol{S})=m^{\boldsymbol{\rho}^{2t-1}}$.
If
$\boldsymbol{Y}=\langle\textbf{v}^{\boldsymbol{\rho}^{t+1}}+\textbf{v}^{\boldsymbol{\rho}}\rangle_{\mathbb{F}_{q^{2t}}}$,
the cross-ratio
$(\boldsymbol{P}^{\boldsymbol{\rho}^{t-1}};\boldsymbol{P}^{\boldsymbol{\rho}^{2t-1}};\boldsymbol{Y};\boldsymbol{R})=-1$.
As we have done before, the result follows by setting
$\sigma=\beta^{-1}\boldsymbol{\rho}^{2t-1}\beta$, $P=\boldsymbol{P}^{\beta}$,
$Q=\boldsymbol{S}^{\beta}$, $R=\boldsymbol{R}^{\beta}$,
$S=\boldsymbol{Q}^{\beta}$, $C=\boldsymbol{C}^{\beta}$,
$X=\boldsymbol{X}^{\beta}$ and $Y=\boldsymbol{Y}^{\beta}$.
Now, let us suppose that Point (1–3) hold true. Then, by Theorem 3.13, $L\cong
L_{f}$ with $f(x)=\alpha x^{q^{s}}+\beta x^{q^{s(t-1)}}+\gamma
x^{q^{s(t+1)}}+x^{q^{s(2t-1)}}\in\mathbb{F}_{q^{2t}}[x]$ for some
$\alpha,\beta,\gamma\in\mathbb{F}_{q^{2t}}$ and some integer $1\leq s\leq
2t-1$ such that $\gcd(s,2t)=1$. Furthermore there exist $P=\langle
v\rangle_{\mathbb{F}_{q^{2t}}}$, an imaginary point with respect to the
subgeometry $\Sigma$ fixed pointwise by a collineation
$\sigma\in\mathrm{P}\Gamma\mathrm{L}(2t,q^{2t})$. Thus, let consider the
points $Q=\langle\alpha
v^{\sigma^{2t-1}}-v^{\sigma}\rangle_{\mathbb{F}_{q^{2t}}}$, $R=\langle\beta
v^{\sigma^{2t-1}}-v^{\sigma^{t-1}}\rangle_{\mathbb{F}_{q^{2t}}}$ and
$S=\langle\gamma
v^{\sigma^{2t-1}}-v^{\sigma^{t+1}}\rangle_{\mathbb{F}_{q^{2t}}}$. Since
$C=\langle v^{\sigma}+v^{\sigma^{t+1}}\rangle_{\mathbb{F}_{q^{2t}}}\in\Gamma$,
then $\alpha=-\gamma$. By Point 3(a), the cross-ratio
$(P^{\sigma};P^{\sigma^{2t-1}};X;Q)=1/{\alpha}\in\mathcal{W}$. By Point 3(b),
we get $\beta=1$. Since, if $1/\alpha\in\mathcal{W}$, then
$\alpha\in\mathcal{W}$, we get $f(x)=\psi_{t,\alpha,s}(x)$ and hence $L\cong
L_{\alpha,s}^{5,t}.$ ∎

## Acknowledgments

The authors express their gratitude to Prof. Giuseppe Marino for helpful
discussions and technical advices. Moreover, they wish to thank the Italian
National Group for Algebraic and Geometric Structures and their Applications
(GNSAGA– INdAM) for supporting this research.

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Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”
Università degli Studi di Napoli Federico II,
Via Cintia, Monte S. Angelo I-80126 Napoli, Italy.
email addresses:
{giovannigiuseppe.grimaldi, somi.gupta,
giovanni.longobardi<EMAIL_ADDRESS>