Pseudoregulus Type in Projective Geometry

• Pseudoregulus type is a classical family of maximum scattered Fq-linear sets defined on PG(1,q^t) and extended to higher dimensions, characterized by models like Frobenius graphs.
• The concept is rigorously analyzed via projection, field reduction, and ring geometry, ensuring detailed descriptions of equivalence, rigidity, and transversal structures.
• Its applications span translation planes, semifields, hyperovals, and MRD codes, establishing it as a benchmark for newer scattered linear set constructions.

Searching arXiv for recent and foundational papers on pseudoregulus type linear sets, scattered polynomials, and related translation planes. Pseudoregulus type denotes the classical family of maximum scattered $\mathbb F_q$-linear sets that first appears on the projective line $\mathrm{PG}(1,q^t)$ and extends to higher-dimensional spaces $\mathrm{PG}(2n-1,q^t)$. In the line case, it is represented by Frobenius-graph models such as $$\{(\lambda,\lambda^q)_{q^t}:\lambda\in \mathbb F_{q^t}^\ast\}$$; in the higher-dimensional case, it is characterized by a large family of pairwise disjoint weight-$t$ lines together with exactly two transversal $(n-1)$-spaces. Across the literature, pseudoregulus type is the prototype against which later families of scattered linear sets, scattered polynomials, translation planes, and rank-metric constructions are measured (Lunardon et al., 2012, Casarino et al., 2022).

1. Classical definition and standard models

On $\mathrm{PG}(1,q^t)$, a linear set of pseudoregulus type is, up to projective equivalence, the standard scattered set

$$L=\{(\lambda,\lambda^q)_{q^t}:\lambda\in\mathbb F_{q^t}^\ast\}.$$

Equivalent formulations replace $x^q$ by $x^{q^\nu}$ with $\mathrm{PG}(1,q^t)$0, or by the monomial

$\mathrm{PG}(1,q^t)$1

with $\mathrm{PG}(1,q^t)$2 chosen so that

$\mathrm{PG}(1,q^t)$3

The associated linear set is then

$\mathrm{PG}(1,q^t)$4

and after a projective change of coordinates all such sets are projectively equivalent (Casarino et al., 2022, Csajbók et al., 2015).

In this line setting, pseudoregulus type is a maximum scattered $\mathrm{PG}(1,q^t)$5-linear set of rank $\mathrm{PG}(1,q^t)$6. The standard model has exactly two transversal points, namely $\mathrm{PG}(1,q^t)$7 and $\mathrm{PG}(1,q^t)$8 (Csajbók et al., 2015). For $\mathrm{PG}(1,q^t)$9, the prototype is

$\mathrm{PG}(2n-1,q^t)$0

and the corresponding linear set $\mathrm{PG}(2n-1,q^t)$1 is explicitly identified as pseudoregulus type (Csajbók et al., 2017).

In higher dimension, the notion is defined on $\mathrm{PG}(2n-1,q^t)$2. A scattered $\mathrm{PG}(2n-1,q^t)$3-linear set $\mathrm{PG}(2n-1,q^t)$4 of rank $\mathrm{PG}(2n-1,q^t)$5 is of pseudoregulus type if there exist

$\mathrm{PG}(2n-1,q^t)$6

pairwise disjoint lines $\mathrm{PG}(2n-1,q^t)$7 with weight $\mathrm{PG}(2n-1,q^t)$8, and exactly two $\mathrm{PG}(2n-1,q^t)$9-dimensional subspaces $$\{(\lambda,\lambda^q)_{q^t}:\lambda\in \mathbb F_{q^t}^\ast\}$$0, disjoint from $$\{(\lambda,\lambda^q)_{q^t}:\lambda\in \mathbb F_{q^t}^\ast\}$$1, such that each $$\{(\lambda,\lambda^q)_{q^t}:\lambda\in \mathbb F_{q^t}^\ast\}$$2 meets both $$\{(\lambda,\lambda^q)_{q^t}:\lambda\in \mathbb F_{q^t}^\ast\}$$3 and $$\{(\lambda,\lambda^q)_{q^t}:\lambda\in \mathbb F_{q^t}^\ast\}$$4. The family $$\{(\lambda,\lambda^q)_{q^t}:\lambda\in \mathbb F_{q^t}^\ast\}$$5 is the $$\{(\lambda,\lambda^q)_{q^t}:\lambda\in \mathbb F_{q^t}^\ast\}$$6-pseudoregulus, and $$\{(\lambda,\lambda^q)_{q^t}:\lambda\in \mathbb F_{q^t}^\ast\}$$7 are the transversal spaces (Lunardon et al., 2012).

2. Geometric characterizations via projection, field reduction, and ring geometry

A central characterization realizes pseudoregulus type as a projection of a canonical subgeometry. Let $$\{(\lambda,\lambda^q)_{q^t}:\lambda\in \mathbb F_{q^t}^\ast\}$$8 be a canonical subgeometry of $$\{(\lambda,\lambda^q)_{q^t}:\lambda\in \mathbb F_{q^t}^\ast\}$$9, let $t$0 be a $t$1-subspace disjoint from $t$2, and let $t$3 be a line disjoint from $t$4. Then

$t$5

is a scattered linear set of pseudoregulus type precisely when there exists a generator $t$6 of the subgroup of $t$7 fixing $t$8 pointwise such that

$t$9

with $(n-1)$0 not contained in the span of any hyperplane of $(n-1)$1. Equivalently, there exist an imaginary point $(n-1)$2 and such a generator $(n-1)$3 for which

$(n-1)$4

and the orbit $(n-1)$5 spans the ambient space (Csajbók et al., 2015). In the line case $(n-1)$6, the same vertex criterion is commonly written as

$(n-1)$7

or

$(n-1)$8

for a generator $(n-1)$9 of the pointwise stabilizer of the canonical subgeometry (Grimaldi et al., 2024).

Field reduction gives a second description. For a pseudoregulus-type linear set in $\mathrm{PG}(1,q^t)$0, the field-reduced image lies on the hypersurface

$\mathrm{PG}(1,q^t)$1

where $\mathrm{PG}(1,q^t)$2. This hypersurface is partitioned by $\mathrm{PG}(1,q^t)$3-spaces

$\mathrm{PG}(1,q^t)$4

and the field-reduction image of the pseudoregulus-type linear set is one such $\mathrm{PG}(1,q^t)$5. The same objects also appear as exterior splashes of canonical subgeometries on exterior lines (Csajbók et al., 2015).

A third characterization uses the projective line over the endomorphism ring $\mathrm{PG}(1,q^t)$6. If a scattered linear set $\mathrm{PG}(1,q^t)$7 corresponds to a point $\mathrm{PG}(1,q^t)$8, then $\mathrm{PG}(1,q^t)$9 is of pseudoregulus type if and only if there exists a projectivity $$L=\{(\lambda,\lambda^q)_{q^t}:\lambda\in\mathbb F_{q^t}^\ast\}.$$0 of $$L=\{(\lambda,\lambda^q)_{q^t}:\lambda\in\mathbb F_{q^t}^\ast\}.$$1 such that

$$L=\{(\lambda,\lambda^q)_{q^t}:\lambda\in\mathbb F_{q^t}^\ast\}.$$2

Equivalently, pseudoregulus type is exactly the case in which the two rulings arising from $$L=\{(\lambda,\lambda^q)_{q^t}:\lambda\in\mathbb F_{q^t}^\ast\}.$$3 are projectively interchangeable in the ring-geometric model (Havlicek et al., 2016).

3. Equivalence, orbit structure, and rigidity

On $$L=\{(\lambda,\lambda^q)_{q^t}:\lambda\in\mathbb F_{q^t}^\ast\}.$$4, all linear sets of pseudoregulus type are $$L=\{(\lambda,\lambda^q)_{q^t}:\lambda\in\mathbb F_{q^t}^\ast\}.$$5-equivalent, so the family is projectively uniform (Csajbók et al., 2015, Csajbók et al., 2015). This projective uniformity does not collapse the finer semilinear orbit structure. For a pseudoregulus-type maximum scattered linear set $$L=\{(\lambda,\lambda^q)_{q^t}:\lambda\in\mathbb F_{q^t}^\ast\}.$$6, the $$L=\{(\lambda,\lambda^q)_{q^t}:\lambda\in\mathbb F_{q^t}^\ast\}.$$7-class is

$$L=\{(\lambda,\lambda^q)_{q^t}:\lambda\in\mathbb F_{q^t}^\ast\}.$$8

where $$L=\{(\lambda,\lambda^q)_{q^t}:\lambda\in\mathbb F_{q^t}^\ast\}.$$9 is Euler’s totient function (Casarino et al., 2022). In the higher-dimensional family of pseudoregulus type in $x^q$0, projective equivalence is controlled by the companion automorphism of the defining semilinear map, up to inversion, and there are

$x^q$1

projective orbits for $x^q$2, $x^q$3 (Lunardon et al., 2012).

The family is rigid, but not in the strongest possible sense. For $x^q$4 or $x^q$5, a pseudoregulus-type linear set on $x^q$6 can be obtained as the projection of two different canonical subgeometries $x^q$7 from the same center $x^q$8 to the same axis, while no ambient collineation $x^q$9 satisfies

$x^{q^\nu}$0

Accordingly, the existence of an ambient collineation between projecting configurations is not a necessary condition for equivalence of the projected linear sets in general; the exact criterion is the paper’s condition (A) (Csajbók et al., 2015).

This combination of projective uniqueness and semilinear multiplicity is one of the distinctive features of pseudoregulus type. It explains why the family is simultaneously the most symmetric scattered family and a nontrivial source of inequivalent defining subspaces.

4. Position in the classification of maximum scattered linear sets

In the classification of maximum scattered linear sets on projective lines, pseudoregulus type is the baseline family. For $x^{q^\nu}$1, the complete classification states that the only maximum scattered $x^{q^\nu}$2-linear sets are those of pseudoregulus type and those of Lunardon–Polverino type. Every maximum scattered set is projectively equivalent to

$x^{q^\nu}$3

with $x^{q^\nu}$4; the case $x^{q^\nu}$5 is exactly pseudoregulus type (Csajbók et al., 2017).

For $x^{q^\nu}$6, the projection model becomes the organizing principle. Every maximum scattered linear set is the projection of a canonical $x^{q^\nu}$7-subgeometry $x^{q^\nu}$8 from a plane $x^{q^\nu}$9. If $\mathrm{PG}(1,q^t)$00 generates the cyclic group fixing $\mathrm{PG}(1,q^t)$01 pointwise and

$\mathrm{PG}(1,q^t)$02

then the abstract states that if $\mathrm{PG}(1,q^t)$03 and $\mathrm{PG}(1,q^t)$04 are not both points, the projected linear set is of pseudoregulus type. In the equivalent vertex language, pseudoregulus type is exactly the case

$\mathrm{PG}(1,q^t)$05

Once pseudoregulus type is excluded, the remaining cases split into LP type and a narrow residual case with $\mathrm{PG}(1,q^t)$06; exhaustive computation shows that for $\mathrm{PG}(1,q^t)$07 no new maximum scattered linear set exists (Lia et al., 31 Jul 2025).

At the level of scattered polynomials, pseudoregulus type is the monomial family

$\mathrm{PG}(1,q^t)$08

Recent work still treats this as one of the three known families that exist for infinitely many values of $\mathrm{PG}(1,q^t)$09 and $\mathrm{PG}(1,q^t)$10, alongside Lunardon–Polverino-type binomials and a family of quadrinomials (Giannoni et al., 14 Jan 2026). The same comparison appears in the study of new scattered quadrinomials, where pseudoregulus type is listed as family (i) among the previously known families for $\mathrm{PG}(1,q^t)$11 (Smaldore et al., 2024).

5. Translation planes, semifields, hyperovals, and MRD codes

Pseudoregulus type entered finite geometry in a decisive way through the construction of translation planes. Lunardon and Polverino started from a pseudoregulus-type scattered linear set in $\mathrm{PG}(1,q^t)$12 and performed a hyper-regulus replacement in the Desarguesian spread

$\mathrm{PG}(1,q^t)$13

This yields an André translation plane. In the later generalization to arbitrary scattered linearized polynomials, the paper explicitly notes that for $\mathrm{PG}(1,q^t)$14, the plane $\mathrm{PG}(1,q^t)$15 arising from the polynomial $\mathrm{PG}(1,q^t)$16 is exactly the André $\mathrm{PG}(1,q^t)$17-plane from the original construction, up to projective equivalence (Casarino et al., 2022). In the stabilizer-based analysis of scattered polynomials, pseudoregulus type is precisely the monomial case

$\mathrm{PG}(1,q^t)$18

with

$\mathrm{PG}(1,q^t)$19

and if $\mathrm{PG}(1,q^t)$20 is of pseudoregulus type then the associated translation plane $\mathrm{PG}(1,q^t)$21 is an André plane (Longobardi et al., 2022).

The family is also central in semifield geometry. In $\mathrm{PG}(1,q^t)$22, maximum scattered linear sets of pseudoregulus type are used to characterize the associated linear sets of Generalized Twisted Fields. In $\mathrm{PG}(1,q^t)$23, the corresponding pseudoregulus-type linear sets with transversal lines in the two reguli of the hyperbolic quadric $\mathrm{PG}(1,q^t)$24 characterize the Knuth semifields $\mathrm{PG}(1,q^t)$25 and $\mathrm{PG}(1,q^t)$26 (Lunardon et al., 2012).

Coding-theoretically, pseudoregulus type provides both reference examples and input data for new constructions. A new family of MRD codes is obtained by starting from maximum scattered linear sets of pseudoregulus type in $\mathrm{PG}(1,q^t)$27 and transferring the corresponding $\mathrm{PG}(1,q^t)$28-subspaces to $\mathrm{PG}(1,q^t)$29. For $\mathrm{PG}(1,q^t)$30 and suitable parameters, the resulting linear sets are maximum scattered and give new MRD codes with parameters $\mathrm{PG}(1,q^t)$31 for $\mathrm{PG}(1,q^t)$32 and $\mathrm{PG}(1,q^t)$33 for odd $\mathrm{PG}(1,q^t)$34 (Csajbók et al., 2017).

A further application occurs in the André/Bruck–Bose representation of translation hyperovals. The affine point sets of translation hyperovals in $\mathrm{PG}(1,q^t)$35 are precisely those whose direction sets are scattered $\mathrm{PG}(1,q^t)$36-linear sets of pseudoregulus type in $\mathrm{PG}(1,q^t)$37 (D'haeseleer et al., 2019).

6. Generalizations and adjacent constructions

The most direct generalization is $\mathrm{PG}(1,q^t)$38-pseudoregulus type. In $\mathrm{PG}(1,q^t)$39, an $\mathrm{PG}(1,q^t)$40-linear set of rank $\mathrm{PG}(1,q^t)$41 is of $\mathrm{PG}(1,q^t)$42-pseudoregulus type if there exist

$\mathrm{PG}(1,q^t)$43

pairwise disjoint $\mathrm{PG}(1,q^t)$44-subspaces of weight $\mathrm{PG}(1,q^t)$45, together with exactly $\mathrm{PG}(1,q^t)$46 transversal $\mathrm{PG}(1,q^t)$47-spaces satisfying the incidence conditions of Definition 3.1. These objects arise by projecting a canonical subgeometry from the span of all but $\mathrm{PG}(1,q^t)$48 director spaces of a Desarguesian spread. Among them, the maximum $\mathrm{PG}(1,q^t)$49-scattered examples are exactly those whose associated exponent set is a Moore exponent set (Napolitano et al., 2020).

Pseudoregulus type also reappears in the one-sided theory of partially scattered polynomials. If $\mathrm{PG}(1,q^t)$50 and

$\mathrm{PG}(1,q^t)$51

then $\mathrm{PG}(1,q^t)$52 is $\mathrm{PG}(1,q^t)$53-$\mathrm{PG}(1,q^t)$54-partially scattered if and only if the associated $\mathrm{PG}(1,q^t)$55-linear set $\mathrm{PG}(1,q^t)$56 is of pseudoregulus type in $\mathrm{PG}(1,q^t)$57. In the same family, weak equivalence classes are governed by the exponent $\mathrm{PG}(1,q^t)$58, and there are exactly $\mathrm{PG}(1,q^t)$59 weak equivalence classes (Bartoli et al., 2021).

Recent papers constructing new scattered quadrinomials distinguish their families from pseudoregulus type rather than extending it. One paper proves that $\mathrm{PG}(1,q^t)$60 is never $\mathrm{PG}(1,q^t)$61-equivalent to a monomial $\mathrm{PG}(1,q^t)$62 of pseudoregulus type (Smaldore et al., 2024). Another compares the new quadrinomials $\mathrm{PG}(1,q^t)$63 with the classical benchmark families and shows geometrically that pseudoregulus type corresponds to vertex intersection number $\mathrm{PG}(1,q^t)$64, LP type to $\mathrm{PG}(1,q^t)$65, while the quadrinomial family has vertex intersection number at least $\mathrm{PG}(1,q^t)$66 (Giannoni et al., 14 Jan 2026).

Taken together, these developments show that pseudoregulus type remains the reference geometry in the subject: it is the uniquely rigid monomial family on $\mathrm{PG}(1,q^t)$67, the first case detected in vertex-based classification, the model for one-sided partial scatteredness, and the comparison orbit from which newer scattered families are separated.