Pseudoregulus-Type Monomials in Finite Geometry

• Pseudoregulus-type monomials are maximum scattered linear sets in PG(2n-1, q^t) characterized by a distinct configuration of lines (the pseudoregulus) and two unique transversal (n-1)-spaces.
• Their construction uses an explicit monomial form via semilinear maps, linking the geometry of Segre varieties with classical semifield structures.
• The classification proves that every such linear set is projectively equivalent to a monomial model L₍p, s₎, providing a unifying framework for scattered linear sets in finite geometry.

A pseudoregulus-type monomial is a particular form of maximum scattered $\mathbb{F}_q$-linear set in the projective space $PG(2n-1,q^t)$, $t \geq 3$, characterized by the existence of a distinguished configuration of lines (the pseudoregulus) and two transversal $(n-1)$-spaces. Every such linear set can be written, up to projective equivalence, in an explicit monomial form corresponding to a specific choice of semilinear map and parameter, subsuming all known pseudoregulus-type examples, and providing a unified connection to the geometry of the Segre variety ${\cal S}_{n,n}$ as well as classical semifield structures (Lunardon et al., 2012).

1. Foundations: Scattered Linear Sets and Pseudoregulus Structure

Let $V$ be a $2n$-dimensional vector space over $\mathbb{F}_{q^t}$ and $A = PG(V,\mathbb{F}_{q^t}) = PG(2n-1,q^t)$. An $\mathbb{F}_q$-linear set of $A$ of rank $k$ is given as

$$L_U = \{ \langle u \rangle_{\mathbb{F}_{q^t}} : u \in U \setminus \{0\} \},$$

where $U \leq V$ is an $\mathbb{F}_q$-subspace of dimension $k$. If $|L_U| = q^{k-1} + \cdots + q + 1$, all points have weight $1$ and the linear set is called scattered. A theorem of Blokhuis–Lavrauw bounds scattered ranks in $PG(r-1, q^t)$ by $k \leq rt/2$, with the upper bound yielding maximum scattered sets. For $PG(2n-1, q^t)$, this maximum is $nt$.

A maximum scattered $\mathbb{F}_q$-linear set $L \subseteq PG(2n-1, q^t)$, $t \geq 3$, is of pseudoregulus type if there exists a family $\mathcal{P}_L$ of $(q^t-1)/(q-1)$ pairwise disjoint lines of $A$, each meeting $L$ in $q^{t-1}+\cdots+q+1$ points, and exactly two $(n-1)$-spaces $T_1, T_2$ disjoint from $L$ but meeting every line of $\mathcal{P}_L$.

For $t \geq 3$, the lines of weight $t$ in $L$ always constitute such a pseudoregulus, and the transversals $T_1, T_2$ are unique [(Lunardon et al., 2012), Proposition 3.2, Corollary 3.3].

2. Monomial Construction via Semilinear Maps

Given a decomposition $V = U_1 \oplus U_2$ with each $U_i$ of dimension $n$ over $\mathbb{F}_{q^t}$, define $T_1 = PG(U_1)$ and $T_2 = PG(U_2)$. Let $\varphi: U_1 \to U_2$ be an invertible semilinear map whose companion automorphism $\sigma \in \mathrm{Aut}(\mathbb{F}_{q^t})$ satisfies $\mathrm{Fix}(\sigma) = \mathbb{F}_q$ and $\gcd(\operatorname{ord} \sigma, t)=1$. For any $p \in \mathbb{F}_{q^t}^*$,

$$W_{p,\varphi} := \{ u + p \cdot \varphi(u) : u \in U_1 \} \subset V$$

gives the $\mathbb{F}_q$-subspace leading to the linear set $L_{p,\varphi}$.

The properties (Theorem 3.5) are:

• $L_{p,\varphi}$ is maximum scattered (rank $nt$), with all weight-1 points.
• The lines $$s_u = \langle \langle u \rangle, \langle \varphi(u) \rangle \rangle$$ ($u \in U_1 \setminus \{0\}$) each meet $L_{p,\varphi}$ in $q^{t-1}+\cdots+1$ points and together form the pseudoregulus $\mathcal{P}_L$.
• The only two $(n-1)$-spaces disjoint from $L_{p,\varphi}$ and meeting each $s_u$ are $T_1$ and $T_2$.

Monomial Specialization

Identifying $U_1 \cong U_2 \cong \mathbb{F}_{q^{nt}}$ as $\mathbb{F}_{q^t}$-vector spaces, the semilinear map becomes $\varphi(x) = x^{q^s}$, $0 < s < t$, $\gcd(s,t)=1$. The explicit form is

$$L_{p,s} = \{ \langle (u, p u^{q^s}) \rangle_{\mathbb{F}_{q^t}} : u \in \mathbb{F}_{q^{nt}}^* \} \subset PG(2n-1, q^t).$$

The subspaces $T_1 = PG(\{(u,0)\})$, $T_2 = PG(\{(0,v)\})$ are the unique transversals. Choices of $p$ yield disjoint but projectively equivalent families [(Lunardon et al., 2012), Remark 3.6].

3. Classification: Uniqueness of the Monomial Model

Theorem 3.13 establishes that, for $n \geq 2$, $t \geq 3$, every maximum scattered $\mathbb{F}_q$-linear set of pseudoregulus type in $PG(2n-1, q^t)$ is projectively equivalent to some $L_{p,\varphi}$. The equivalence classes are parametrized by $\varphi(t)/2$, where $\varphi$ denotes Euler’s totient function: these correspond to field automorphisms of $\mathbb{F}_{q^t}$ fixing $\mathbb{F}_q$, modulo inversion.

Proof outline:

• Any pseudoregulus-type set arises as a projection of $PG(n-1,q) \subset PG(nt-1, q^t)$ from a suitable center to a $(2n-1)$-space.
• The center selects two director $(n-1)$-spaces of a Desarguesian $(t-1)$-spread; the projection is precisely $W_{p,\varphi}$ with the unique semilinear collineation and companion automorphism as above.
• Counting orbits under $\mathrm{P}\Gamma\mathrm{L}(2n, q^t)$ reveals that $L_{p,\varphi}$ and $L_{p,\varphi'}$ are equivalent iff their companion field automorphisms are inverses.

The monomial form $L_{p,s}$ thus comprehensively classifies all pseudoregulus-type examples.

4. Relation to the Segre Variety $\mathcal{S}_{n,n}(q)$

$PG(2n-1, q^t)$ (denoted $A$) naturally embeds as a $D$-subspace in $PG(n^2-1,q)$, the ambient space of the Segre variety $\mathcal{S}_{n,n}(q)$. Here, $PG(n^2-1,q)$ can be represented as $PG(\mathrm{End}(\mathbb{F}_{q^n}, \mathbb{F}_q))$, whose Desarguesian spread decomposes into two disjoint director spaces $I, I'$. The union of lines connecting corresponding points of $I$ and $I'$ yields a copy of $PG(2n-1, q^n)$, within which $A = PG(2n-1, q^t)$ (with $t|n$) embeds as a secant $D$-subspace.

Within this geometric framework, the pseudoregulus lines $s_u$ correspond to the intersections of $A$ with the maximal $(n-1)$-spaces from one ruling of the Segre variety, i.e., the subvariety of rank-1 endomorphisms in $PG(n^2-1,q)$. The transversals $T_1$ and $T_2$ arise as intersections with these director subspaces. Sections 5.1–5.4 show that various semifield-spread-sets (linear sets disjoint from the determinantal hypersurface $\det = 0$) naturally arise as monomial pseudoreguli within this context (Lunardon et al., 2012).

5. Connections to Classical Semifields

Generalized Twisted Fields

Albert’s Generalized Twisted Fields (GTF) are semifields on $F_{q^{nt}}$ with multiplication

$x \star y = yx - c\, y^{q^m} x^{q^\ell}$

where $\gcd(m, t) = \gcd(\ell, n) = 1$ and $c \in F_{q^{nt}}$. The left-multiplication maps $y \mapsto x \star y$ comprise the subspace

$$W = \{ (u, -c u^{q^m}) : u \in F_{q^{nt}} \} \subset PG(2n-1, q^t),$$

which is of the form $L_{p,s}$ for suitable $p, s$. Theorem 5.6 asserts that any pseudoregulus-type linear set in $PG(2n-1, q^t)$ and disjoint from the determinantal hypersurface arises from a GTF.

2-dimensional Knuth Semifields

Within $PG(3, q^t)$, the Knuth semifield families $K_{17}, K_{19}$ produce linear sets of the form

$$\begin{aligned} L(K_{17}) &= \{ \langle (x, y, f y^{q^\sigma}, x^{q^\sigma} + g y^{q^\sigma}) \rangle : (x, y) \neq (0,0) \}, \ L(K_{19}) &= \{ \langle (x, y, f y^{q^\sigma}, x^{q^\sigma} + g y^{q^\sigma}) \rangle : (x, y) \neq (0,0) \}, \end{aligned}$$

with coordinate differences distinguishing the two families. Proposition 5.9 and Theorem 5.10 confirm that such sets are precisely the pseudoregulus-type examples in $PG(3, q^t)$ with both transversals lying in the same generator family of the hyperbolic quadric $Q^+(3, q^t)$.

Thus, the monomial description $L_{p,s}$ unifies all known large scattered pseudoregulus-type linear sets, encapsulating their $\mathrm{P}\Gamma\mathrm{L}$-classification, geometric significance within $\mathcal{S}_{n,n}$, and ties to classical semifields (Lunardon et al., 2012).

6. Summary Table: Key Structural Elements

Object	Definition/Role	Reference in (Lunardon et al., 2012)
$L_{p,s}$	Monomial maximum scattered linear set	Thm 3.5, Sect. 3
$\mathcal{P}_L$	Family of $(q^t-1)/(q-1)$ disjoint lines of weight $t$	Prop. 3.2, Cor. 3.3
$T_1, T_2$	Unique transversal $(n-1)$-spaces	Def. pseudoregulus type
GTF/KTF correspondences	Specific semifields realized as $L_{p,s}$	Thm 5.6, Thm 5.10
Segre variety $\mathcal{S}_{n,n}$	Geometric embedding highlighting pseudoregulus structure	Sect. 5.1–5.4

The existence and monomial parametrization of pseudoregulus-type linear sets constitute a comprehensive classification and bridge between finite geometry, semifield theory, and algebraic combinatorics, as established in (Lunardon et al., 2012).