1 Introduction
Let
$\mathcal {P}$
describe a property of a degree d algebraic hypersurface in
$\mathbb {P}^n$
. In algebraic geometry and adjacent fields, we are often interested in measuring the likelihood of the property
$\mathcal {P}$
for a ‘randomly chosen’ hypersurface. When working over an infinite field, we can use Zariski dense open sets to show that property
$\mathcal {P}$
holds generically. However, the situation over finite fields is more subtle since open sets in the relevant parameter space may not have any
$\mathbb {F}_q$
-points (despite being Zariski dense over
$\overline {\mathbb {F}_q}$
).
There are alternative methods to quantify how widespread property
$\mathcal {P}$
is for hypersurfaces over finite fields. One method is to count the proportion of degree d hypersurfaces over
$\mathbb {F}_q$
satisfying
$\mathcal {P}$
and consider the asymptotic proportion (either as
$d\to \infty $
or
$q\to \infty $
). As another natural metric, we can ask for the maximum size of a linear family
$\mathcal {P}$
can carry. More precisely, we can pose the following question for each finite field
$\mathbb {F}_q$
and positive integers d and n.
Question 1.1. What is the maximum value of
$t\in \mathbb {N}$
for which there exist
$\{F_i=0\}$
for
$i=0, 1,\ldots , t$
such that
$X_{[a_0:\cdots : a_t]} = \{a_0 F_0 + \cdots + a_t F_t = 0\}$
satisfies the property
$\mathcal {P}$
for all choices
$[a_0:a_1:\cdots :a_t]\in \mathbb {P}^{t}(\mathbb {F}_q)$
?
The question can be rephrased in the language of linear systems: what is the largest (projective) dimension of a linear system
$\mathcal {L}\cong \mathbb {P}^t$
of degree d hypersurfaces in
$\mathbb {P}^n$
such that each
$\mathbb {F}_q$
-member of
$\mathcal {L}$
satisfies
$\mathcal {P}$
? An answer to Question 1.1 measures the extent to which the property
$\mathcal {P}$
propagates linearly in the parameter space of all degree d hypersurfaces in
$\mathbb {P}^n$
. Larger values of t indicate higher levels of prevalence for
$\mathcal {P}$
.
Question 1.1 has been addressed for several specific choices of
$\mathcal {P}$
. For instance,
$\mathcal {P}$
may correspond to the property of being smooth [Reference Asgarli, Ghioca and Reichstein1], irreducible over
$\mathbb {F}_q$
[Reference Asgarli, Ghioca and Reichstein2], reducible over
$\mathbb {F}_q$
[Reference Asgarli, Ghioca and Reichstein2] or nonblocking with respect to
$\mathbb {F}_q$
-lines [Reference Asgarli, Ghioca and Yip3]. To illustrate some of these results, we specialise to the setting of cubic plane curves, that is,
$d = 3$
and
$n = 2$
. Below are two concrete examples from the recent literature where Question 1.1 has a known answer.
Theorem 1.2 [Reference Asgarli, Ghioca and Reichstein1].
Let
$\mathbb {F}_q$
be a finite field with characteristic
$p\neq 3$
. Then, there is a two-dimensional linear system
$\mathcal {L}_{\text {smo}}=\langle F_0, F_1, F_2\rangle \cong \mathbb {P}^2$
of cubic plane curves such that every
$\mathbb {F}_q$
-member of
$\mathcal {L}_{\text {smo}}$
is smooth. Moreover, no such three-dimensional system exists.
Theorem 1.3 [Reference Asgarli, Ghioca and Reichstein2].
Let
$\mathbb {F}_q$
be a finite field. Then, there is a three-dimensional linear system
$\mathcal {L}_{\text {irr}}=\langle F_0, F_1, F_2, F_3\rangle \cong \mathbb {P}^3$
of cubic plane curves such that every
$\mathbb {F}_q$
-member of
$\mathcal {L}_{\text {irr}}$
is irreducible over
$\mathbb {F}_q$
. Moreover, no such four-dimensional system exists.
Let us compare the two properties in Theorems 1.2 and 1.3. The smoothness condition refers to geometric smoothness, while irreducibility over
$\mathbb {F}_q$
does not necessarily imply geometric irreducibility (that is, irreducibility over
$\overline {\mathbb {F}_q}$
). This distinction naturally raises the goal of establishing a version of Theorem 1.3, where the conclusion is strengthened from irreducibility over
$\mathbb {F}_q$
to irreducibility over
$\overline {\mathbb {F}_q}$
. The purpose of the present paper is to address this objective.
To summarise, we address Question 1.1 when
$\mathcal {P}$
stands for ‘is geometrically irreducible’ (that is, irreducible over
$\overline {\mathbb {F}_q}$
) for cubic plane curves:
$d=3$
and
$n=2$
. In this special case, every linear system
$\mathcal {L}$
of (projective) dimension
$4$
has an
$\mathbb {F}_q$
-member that is a reducible plane cubic over
$\mathbb {F}_q$
by Theorem 1.3 (see [Reference Asgarli, Ghioca and Reichstein2, Theorem 1.3(d)] for further details). Hence, the answer to Question 1.1 in this setting is at most
$3$
. We predict that the answer is exactly
$3$
.
Conjecture 1.4. There exists a linear system
$\mathcal {L}=\langle F_0, F_1, F_2, F_3\rangle \cong \mathbb {P}^3$
of cubic plane curves where each
$\mathbb {F}_q$
-member of
$\mathcal {L}$
is geometrically irreducible.
As mentioned earlier, Conjecture 1.4 strengthens Theorem 1.3. As partial progress, we establish the following theorem.
Theorem 1.5. There exists a linear system
$\mathcal {L}=\langle F_0, F_1, F_2, F_3\rangle $
of cubic plane curves where each
$\mathbb {F}_q$
-member of
$\mathcal {L}$
is irreducible over
$\mathbb {F}_q$
and there is at most one geometrically reducible
$\mathbb {F}_q$
-member of
$\mathcal {L}$
.
To further motivate the problem and explain its arithmetic origin, we explain why the analogue of Conjecture 1.4 fails when the base field
$\mathbb {F}_q$
is replaced by an algebraically closed field
$\mathbb {K}$
. Since a cubic form in three variables is described by 10 coefficients, the parameter space of cubic plane curves is
$\mathbb {P}^9(\mathbb {K})$
. Let
${\mathcal {L} = \langle F_0, F_1, F_2, F_3 \rangle }$
be a linear system where each
$F_i \in \mathbb {K}[x,y,z]$
is an homogeneous polynomial of degree 3.
The set of reducible cubic curves forms a seven-dimensional subvariety Y of
$\mathbb {P}^9(\mathbb {K})$
. This can be seen as follows: any reducible cubic polynomial can be factored as
$$ \begin{align} (a_0 x + a_1 y + a_2 z)(b_0 x^2 + b_1 y^2 + b_2 z^2 + b_3 xy + b_4 yz + b_5 zx). \end{align} $$
The variety Y is the image of the natural map
$\mathbb {P}^2 \times \mathbb {P}^5 \to \mathbb {P}^9$
induced by the multiplication from (1.1). Since
$\dim (Y) + \dim (\mathcal {L}) = 7 + 3 = 10$
, the intersection
$C : = Y \cap \mathcal {L}$
has dimension at least 1 in
$\mathbb {P}^9$
. Because
$\mathbb {K}$
is algebraically closed, we have
$C(\mathbb {K}) \neq \emptyset $
, meaning that
$\mathcal {L}$
has at least one reducible
$\mathbb {K}$
-member. In particular, Conjecture 1.4 does not hold over
$\mathbb {K}$
.
For example, if we take
$\mathbb {K} = \overline {\mathbb {F}_q}$
, then any three-dimensional linear system
${\mathcal {L} = \langle F_0, F_1, F_2, F_3 \rangle \cong \mathbb {P}^3}$
contains a reducible
$\overline {\mathbb {F}_q}$
-member. The subtlety of Conjecture 1.4 lies in the fact that, when
$\mathcal {L}$
is defined over
$\mathbb {F}_q$
, the variety C, which is generically a curve, may lack
$\mathbb {F}_q$
-points. In fact, Conjecture 1.4 is equivalent to the following statement: there exists an
$\mathbb {F}_q$
-linear subspace
$\mathcal {L} \cong \mathbb {P}^3$
in the parameter space
$\mathbb {P}^9$
such that
$Y \cap \mathcal {L}$
has no
$\mathbb {F}_q$
-points, where Y is the locus of reducible cubics. Viewed through this lens, the difficulty of Conjecture 1.4 is tied to finding a specific ‘pointless’ curve inside a large-dimensional projective space. Furthermore, the above analysis also shows that the proportion of geometrically irreducible cubic plane curves defined over
$\operatorname {\mathbb {F}}_q$
tends to
$1$
as
$q\to \infty $
(since the number of geometrically reducible plane cubics defined over
$\operatorname {\mathbb {F}}_q$
is
$O(q^7)$
, while we have
$O(q^9)$
plane cubics defined over
$\operatorname {\mathbb {F}}_q$
).
While we focus on the case of cubic plane curves in the present paper, the same question applies to hypersurfaces of degree d in
$\mathbb {P}^n$
for any d and n.
Problem 1.6. Determine the maximum (projective) dimension of a linear system
$\mathcal {L}$
of degree d hypersurfaces in
$\mathbb {P}^n$
such that each
$\mathbb {F}_q$
-member is geometrically irreducible.
By [Reference Asgarli, Ghioca and Reichstein2, Theorem 1.3(d)], every linear system of dimension
$\binom {n+d-1}{n-1}$
has an
$\mathbb {F}_q$
-member that is reducible over
$\mathbb {F}_q$
. Hence, the answer to Problem 1.6 is at most
${\binom {n+d-1}{n-1}-1}$
. However, by [Reference Asgarli, Ghioca and Reichstein1, Theorem 2], when
$p=\operatorname {char}(\mathbb {F}_q) \nmid \gcd (d, n+1)$
, there exists an n-dimensional linear system whose
$\mathbb {F}_q$
-members are all smooth, and hence geometrically irreducible. Hence, the answer to Problem 1.6 is at least n. We expect the true answer to Problem 1.6 to be closer to the upper bound
$\binom {n+d-1}{n-1}-1$
. After all, we expect most geometrically reducible hypersurfaces defined over
$\mathbb {F}_q$
to be reducible over
$\mathbb {F}_q$
.
We also have a related open problem with the condition ‘geometrically irreducible’ relaxed to ‘not containing a linear component over
$\overline {\mathbb {F}_q}$
’.
Problem 1.7. Determine the maximum (projective) dimension of a linear system
$\mathcal {L}$
of degree d hypersurfaces in
$\mathbb {P}^n$
such that each
$\mathbb {F}_q$
-member has no linear factor.
By definition, the answer to Problem 1.6 is less than or equal to the answer to Problem 1.7. It is reasonable to expect that the two answers agree, at least for all sufficiently large q (as a function of n and d). The heuristic is that most reducible hypersurfaces (over
$\overline {\mathbb {F}_q})$
have a linear factor. Note that Conjecture 1.4 concerns the case
$n=2$
and
$d=3$
for which Problems 1.6 and 1.7 coincide.
While these open problems are new, we note that the study of reducible members in a linear system of algebraic hypersurfaces has a rich literature. One case that has been investigated thoroughly is the number of reducible (or totally reducible) hypersurfaces in a pencil of hypersurfaces [Reference Lorenzini4, Reference Pereira and Yuzvinsky5, Reference Vistoli7]. The setting between the cited work and the present work differs in a few places. We only consider
$\mathbb {F}_q$
-members while the previous work is about controlling reducibility over
$\overline {\mathbb {F}_q}$
-members. However, we do not restrict our attention to pencils and allow large-dimensional linear systems.
We provide two proofs for Theorem 1.5. In Section 2, we provide a nonconstructive proof in the spirit of [Reference Asgarli, Ghioca and Reichstein2], while in Section 3, we provide an explicit construction of a three-dimensional linear system as desired for the conclusion of Theorem 1.5. We believe both proofs (which are quite different in their approach) could be useful for pursuing Conjecture 1.4. Appendix A provides numerical evidence (computed using SageMath) that supports Conjecture 1.4 for all
$q\leq 11$
.
2 A nonconstructive proof
In this section, we discuss the construction in our previous paper [Reference Asgarli, Ghioca and Reichstein2], joint with Reichstein, in the special case of plane cubic curves; this allows us to provide a first (nonconstructive) proof of our Theorem 1.5.
We begin by recalling the motivation behind the main result of [Reference Asgarli, Ghioca and Reichstein2] specialised to our context. Recall that the parameter space of plane conics corresponds to the projective space
$\mathbb {P}^5$
, as each conic can be represented by the equation
$$ \begin{align*} b_0 x^2 + b_1 y^2 + b_2 z^2 + b_3 xy + b_4 yz + b_5 zx= 0 \end{align*} $$
for some coefficients
$b_0, \ldots , b_5$
. Imposing the additional condition that the conic passes through a specific point
$Q_1 = [x_1 : y_1 : z_1]$
introduces one linear constraint on the coefficients
$b_0, \ldots , b_5$
. As a result, the space of conics passing through
$Q_1$
forms a projective subspace of dimension
$4$
, namely
$\mathbb {P}^4$
. Each time we require the conic to pass through a new point Q, we add another linear constraint on the coefficients. However, it is not guaranteed that these constraints will be linearly independent. We say that the points
$Q_1, \ldots , Q_s$
(with
$s<7$
) are in general position with respect to conics if the
$\mathbb {F}_q$
-vector space dimension of the space of conics passing through
$Q_1, \ldots , Q_s$
is exactly
$6 - s$
. Over the algebraic closure, the set of such s-tuples forms a Zariski-dense open subset. In particular, when
$s = 6$
, if the six points are chosen from
$\mathbb {P}^2(\overline {\mathbb {F}_q})$
in general position, then no conic passes through all of them. The main contribution of [Reference Asgarli, Ghioca and Reichstein2] is to show that points in general position can be modelled as Galois orbits in a suitable sense. In the specific case of conics, it is possible to construct a set of six points in general position as the Galois orbit of a single point of degree 6, that is, a point with coordinates in
$\mathbb {F}_{q^6}$
.
More precisely, [Reference Asgarli, Ghioca and Reichstein2, Theorem 1.1] asserts the existence of a point
$P\in \mathbb {P}^2(\mathbb {F}_{q^6})$
such that P is not contained in any degree
$2$
curve C over
$\mathbb {F}_q$
. Equivalently, no conic defined over
$\mathbb {F}_q$
contains the Galois orbit
$S=\{P, P^{\sigma }, \ldots , P^{\sigma ^{5}}\}$
. Here,
$P^{\sigma }$
denotes the image of the point P under the Frobenius map
$[x:y:z]\mapsto [x^q:y^q:z^q]$
. For simplicity, let us write
$P_i=P^{\sigma ^{i}}$
so that
$S=\{P_0, \ldots , P_5\}$
.
Next, we follow [Reference Asgarli, Ghioca and Reichstein2, Theorem 1.3(c)] to construct a linear system of cubics
$\mathcal {L}\cong \mathbb {P}^3$
, where each
$\mathbb {F}_q$
-member of
$\mathcal {L}$
is irreducible over
$\mathbb {F}_q$
. We will show that the same linear system
$\mathcal {L}$
has at most one geometrically reducible
$\mathbb {F}_q$
-member, proving Theorem 1.5. To construct
$\mathcal {L}$
, recall that the dimension of the
$\mathbb {F}_q$
-vector space of cubic forms in three variables is
$10$
. Imposing the condition that a cubic passes through a specific point imposes at most one linear condition on the coefficients. Since
$S=\{P_0, \ldots , P_5\}$
has six points and S is defined over
$\mathbb {F}_q$
(despite the fact that each
$P_i$
is not individually defined over
$\mathbb {F}_q$
), the
$\mathbb {F}_q$
-vector subspace of all cubics passing through S has dimension at least
$10-6=4$
. Let
$F_0, F_1, F_2, F_3$
denote four linearly independent cubic forms in
$\mathbb {F}_q[x,y,z]$
each passing through all points of S. Let
$\mathcal {L}=\langle F_0, F_1, F_2, F_3\rangle \cong \mathbb {P}^3$
denote the three-dimensional linear system of cubic curves passing through S.
Let C be a reducible cubic curve (over
$\overline {\mathbb {F}_q}$
) which is an
$\mathbb {F}_q$
-member of
$\mathcal {L}$
. There are two ways in which a reducible cubic
$C=L\cup Q$
can pass through the set S.
-
(a) Let
$L_{ij}$
be the line joining
$P_i$
and
$P_{j}$
. Then Q can vary in
$\mathbb {P}^1$
-worth of conics passing through the remaining four points. -
(b) Let
$Q_{i}$
be the conic passing through five points in the set
$S\setminus \{P_i\}$
. Then, L can vary in
$\mathbb {P}^1$
-worth of lines passing through the remaining point
$P_{i}$
.
However, if C is defined over
$\mathbb {F}_q$
, it must be the case that C is a union of three
$\mathbb {F}_{q^3}$
-lines, Galois conjugated by
$\operatorname {Gal}(\mathbb {F}_{q^3}/\mathbb {F}_q)$
; note that C is assumed to be geometrically reducible, while, by [Reference Asgarli, Ghioca and Reichstein2, Theorem 1.3(c)], C is irreducible over
$\operatorname {\mathbb {F}}_q$
since the entire
$\operatorname {\mathbb {F}}_q$
-linear space
$\mathcal {L}$
consists of
$\operatorname {\mathbb {F}}_q$
-irreducible plane cubics. It is straightforward to see that exactly one of these curves, namely
$\overline {P_0 P_3}\cup \overline {P_1 P_4} \cup \overline {P_2 P_5}$
, is defined over
$\mathbb {F}_q$
. Hence, all
$\mathbb {F}_q$
-members of
$\mathcal {L}$
are irreducible over
$\mathbb {F}_q$
and exactly one
$\mathbb {F}_q$
-member of
$\mathcal {L}$
fails to be geometrically irreducible. This completes the proof of Theorem 1.5.
3 An explicit construction
The first proof of Theorem 1.5 relies on the existence of a point
$P\in \mathbb {P}^2(\mathbb {F}_{q^6})$
which does not lie on any conic defined over
$\mathbb {F}_q$
. The proof of this assertion in [Reference Asgarli, Ghioca and Reichstein2, Theorem 1.1] was obtained by an intricate counting argument and hence is nonconstructive by its nature. In this section, we offer an alternative proof of Theorem 1.5 which has the advantage of providing an explicit construction. This latter method offers a new perspective on the problem and suggests future avenues for addressing higher-degree systems (as in Problems 1.6 and 1.7) through explicit methods.
We start with a lemma on reducible cubic curves containing only the monomials
$x^2y, y^2z, z^2x, xyz$
.
Lemma 3.1. Suppose
$ax^2y + by^2z + cz^2x + dxyz=0$
is a geometrically reducible cubic curve. Then,
$abc=0$
.
Proof. The reducible cubic has a linear factor L. Without loss of generality,
$L=x+\beta y + \gamma z$
for some scalars
$\beta , \gamma $
. If
$a=0$
, then we are done. Hence, we may assume
$a=1$
after scaling. We have
$$ \begin{align} x^2 y + b y^2 z + c z^2 x + d xyz = L Q \end{align} $$
for some quadratic factor Q. We match the coefficients on both sides of (3.1) to prove that
$b=0$
. We proceed in five steps.
-
(1) The cubic has no
$x^3$
term, so Q has no
$x^2$
term. The term
$x^2 y$
can only be constructed from multiplying x from L with a term in
$xy$
from Q; thus, the coefficient of
$xy$
in Q must be
$1$
. -
(2) If
$\beta \ne 0$
, then Q has no
$y^2$
term; in that case,
$LQ$
has the term
$(\beta y)\cdot xy$
which leads to the term
$xy^2$
in the cubic that cannot be cancelled, which is a contradiction. Therefore,
$\beta =0$
. -
(3) The cubic has no
$xy^2$
term and
$\beta = 0$
, so Q has no
$y^2$
term. The cubic has no
$x^2z$
term and Q has no
$x^2$
term, so Q has no
$xz$
term. -
(4) So,
$Q=xy + \delta _1 yz + \delta _2 z^2$
and
$L=x+\gamma z$
. From (3.1), we see
$\gamma \delta _1 =0$
. If
$\gamma = 0$
, then x divides the cubic, implying that
$b=0$
, as desired. -
(5) If
$\gamma \neq 0$
, then we have
$\delta _1=0$
. In this case, (3.1) reads We obtain
$$ \begin{align*} (x + \gamma z)(xy + \delta_2 z^2) = x^2y + by^2z + cz^2x + dxyz. \end{align*} $$
$b=0$
, as desired.
Thus, any geometrically reducible cubic of the form
$ax^2y + by^2z + cz^2x + dxyz=0$
satisfies
$abc=0$
.
We will now present the second proof of our main theorem.
Proof of Theorem 1.5.
Consider the linear system
$\mathcal {L}_1 = \langle x^2 y, y^2 z, z^2 x, xyz\rangle $
. By the normal basis theorem, there exists an element
$\alpha \in \mathbb {F}_{q^3}$
such that
$\alpha , \alpha ^q, \alpha ^{q^2}$
form a basis of
$\mathbb {F}_{q^3}$
as an
$\mathbb {F}_q$
-vector space. We construct a new linear system from
$\mathcal {L}_1$
where x, y and z are replaced by appropriate linear forms. Let
$$ \begin{align*} F &= (\alpha x + \alpha^q y + \alpha^{q^2} z)^2 (\alpha^q x + \alpha^{q^2} y + \alpha z), \\[-1pt] G &= (\alpha^q x + \alpha^{q^2} y + \alpha z)^2 (\alpha^{q^2} x + \alpha y + \alpha^q z), \\[-1pt] H &= (\alpha^{q^2} x + \alpha y + \alpha^q z)^2(\alpha x + \alpha^q y + \alpha^{q^2} z), \\[-1pt] T &= (\alpha x + \alpha^q y + \alpha^{q^2} z)(\alpha^q x + \alpha^{q^2} y + \alpha z)(\alpha^{q^2} x + \alpha y + \alpha^q z). \end{align*} $$
Consider the linear system
$\mathcal {L}_2=\langle F, G, H, T\rangle $
. The Frobenius map
$t\mapsto t^{q}$
sends
${F\mapsto G\mapsto H\mapsto F}$
and fixes T. Thus, the linear system
$\mathcal {L}_2$
is defined over
$\mathbb {F}_q$
, meaning that we can find new generators
$R_0, R_1, R_2, R_3\in \mathbb {F}_q[x,y,z]$
with
$\deg (R_i)=3$
such that
${\mathcal {L}_2=\langle R_0, R_1, R_2, R_3\rangle }$
. We claim that each
$\mathbb {F}_q$
-member of
$\mathcal {L}_2$
is geometrically irreducible except the member
$T\in \mathcal {L}_2$
which is a union of three lines conjugated by
$\operatorname {Gal}(\mathbb {F}_{q^3}/\mathbb {F}_q)$
. Indeed, we have a new coordinate system induced by the linear transformation:
$$ \begin{align*} x'&=\alpha x+\alpha^q y + \alpha^{q^2} z, \\[-2.2pt] y'&=\alpha^q x+\alpha^{q^2} y + \alpha z, \\[-2.2pt] z'&=\alpha^{q^2} x+\alpha y + \alpha^q z. \end{align*} $$
Applying Lemma 3.1 in the new coordinate system, we see that any geometrically reducible
$\mathbb {F}_q$
-member of
$\mathcal {L}_2$
, given by
$$ \begin{align*} a F + b G + c H + d T = 0, \end{align*} $$
satisfies
$abc=0$
. After applying the Frobenius map
$t\mapsto t^q$
twice and using the fact that T is defined over
$\mathbb {F}_q$
, we get two additional equations:
$$ \begin{align*} a G + b H + c F + d T &= 0, \\ a H + b F + c G + d T &= 0. \end{align*} $$
Since
$abc=0$
, at least one of
$a, b, c$
is zero. The three equations above and the linear independence of
$F, G, H, T$
imply
$a=b=c=0$
. Hence, the only geometrically reducible
$\mathbb {F}_q$
-member of
$\mathcal {L}_2$
is
$\{T=0\}$
. Note that
$\{T=0\}$
is irreducible over
$\mathbb {F}_q$
. Thus, the linear system
$\mathcal {L}_2$
satisfies the desired properties.
Appendix A Computational evidence for the conjecture
We verified Conjecture 1.4 for all
$q\leq 11$
using SageMath [6]. It suffices to randomly generate a cubic linear system
$\mathcal {L}=\langle F_0, F_1, F_2, F_3\rangle $
until all
$\mathbb {F}_q$
-members of
$\mathcal {L}$
are geometrically irreducible. Algorithm 1 formalises this procedure.
Table 1 lists the successful linear systems for
$q\in \{2,3,4,5,7,8,9,11\}$
.
Linear systems for
$q\le 11$
.
Interestingly, the linear system we found for
$\mathbb {F}_8$
has coefficients in
$\mathbb {F}_{2} = \{0, 1\}$
, which means that the table entry corresponding to
$q = 8$
also supports Conjecture 1.4 for
$q = 2, 4, 8$
. An intriguing question arises: how large are the values of k for which we can find a linear system over
$\mathbb {F}_{q}$
whose
$\mathbb {F}_{q^k}$
-members (not just
$\mathbb {F}_q$
-members) are geometrically irreducible? Such a result would provide an even stronger conclusion than Conjecture 1.4.
Acknowledgement
We thank Dino Lorenzini and the referee for valuable comments on the paper.
