Dragos Ghioca’s research while affiliated with University of British Columbia and other places

Let G be a semiabelian variety defined over a finite subfield of an algebraically closed field K of prime characteristic. We describe the intersection of a subvariety X of G with a finitely generated subgroup of G(K) .

In the goundbreaking paper [BD11] (which opened a wide avenue of research regarding unlikely intersections in arithmetic dynamics), Baker and DeMarco prove that for the family of polynomials $f_\lambda (x):=x^d+\lambda$ (parameterized by $\lambda \in \mathbb {C}$), given two starting points a and b in $\mathbb {C}$, if there exist infinitely many $\lambda \in \mathbb {C}$ such that both a and b are preperiodic under the action of $f_\lambda$, then $a^d=b^d$. In this paper, we study the same question, this time working in a field of characteristic $p>0$. The answer in positive characteristic is more nuanced, as there are three distinct cases: (i) both starting points a and b live in ${\overline {\mathbb F}_p}$; (ii) d is a power of p; and (iii) not both a and b live in ${\overline {\mathbb F}_p}$, while d is not a power of p. Only in case (iii), one derives the same conclusion as in characteristic 0 (i.e., that $a^d=b^d$). In case (i), one has that for each $\lambda \in {\overline {\mathbb F}_p}$, both a and b are preperiodic under the action of $f_\lambda$, while in case (ii), one obtains that also whenever $a-b\in {\overline {\mathbb F}_p}$, then for each parameter $\lambda$, we have that a is preperiodic under the action of $f_\lambda$ if and only if b is preperiodic under the action of $f_\lambda$.

We provide an explicit construction of the arboreal Galois group for the postcritically finite polynomial $f(z) = z^2 +c$, where c belongs to some arbitrary field of characteristic not equal to 2. In this first of two papers, we consider the case that the critical point is periodic.

Let d and n be positive integers, and E/F be a separable field extension of degree $m=\left( {\begin{array}{c}n+d\\ n\end{array}}\right)$. We show that if $|F| > 2$, then there exists a point $P\in \mathbb {P}^n(E)$ which does not lie on any degree d hypersurface defined over F. In other words, the m Galois conjugates of P impose independent conditions on the m-dimensional F-vector space of degree d forms in $x_0, x_1, \ldots , x_n$. As an application, we determine the maximal dimensions of linear systems $\mathcal {L}_1$ and $\mathcal {L}_2$ of hypersurfaces in $\mathbb P^n$ over a finite field F, where every F-member of $\mathcal {L}_1$ is reducible and every F-member of $\mathcal {L}_2$ is irreducible.

We prove a nonabelian variant of the classical Mordell–Lang conjecture in the context of finite- dimensional central simple algebras. We obtain the following result as a particular case of a more general statement. Let K be an algebraically closed field of characteristic zero, let $B_1,\dots ,B_r\in \mathrm {GL}_m(K)$ be matrices with multiplicatively independent eigenvalues and let V be a closed subvariety of $\mathrm {GL}_m(K)$ not passing through zero. Then there exist only finitely many elements of $\mathrm {GL}_m(K)$ of the form $B_1^{n_1}\cdots B_r^{n_r}$ (as we vary $n_1,\dots ,n_r$ in $\mathbb {Z}$ ) lying on the subvariety V .

In [GTZ08, GTZ12], the following result was established: given polynomials $f,g\in\mathbb{C}[x]$ of degrees larger than 1, if there exist $\alpha,\beta\in\mathbb{C}$ such that their corresponding orbits $\mathcal{O}_f(\alpha)$ and $\mathcal{O}_g(\beta)$ (under the action of f, respectively of g) intersect in infinitely many points, then f and g must share a common iterate, i.e., $f^m=g^n$ for some $m,n\in\mathbb{N}$. If one replaces $\mathbb{C}$ with a field K of characteristic p, then the conclusion fails; we provide numerous examples showing the complexity of the problem over a field of positive characteristic. We advance a modified conjecture regarding polynomials f and g which admit two orbits with infinite intersection over a field of characteristic p. Then we present various partial results, along with connections with another deep conjecture in the area, the dynamical Mordell-Lang conjecture.

We examine the maximum dimension of a linear system of plane cubic curves whose $\mathbb{F}_q$-members are all geometrically irreducible. Computational evidence suggests that such a system has a maximum (projective) dimension of 3. As a step towards the conjecture, we prove that there exists a 3-dimensional linear system $\mathcal{L}$ with at most one geometrically reducible $\mathbb{F}_q$-member.

A plane curve $C\subset\mathbb{P}^2$ of degree d is called \emph{blocking} if every $\mathbb{F}_q$-line in the plane meets C at some $\mathbb{F}_q$-point. We prove that the proportion of blocking curves among those of degree d is o(1) when $d\geq 2q-1$ and $q \to \infty$. We also show that the same conclusion holds for smooth curves under the somewhat weaker condition $d\geq 3p$ and $d, q \to \infty$. Moreover, the two events in which a random plane curve is smooth and respectively blocking are shown to be asymptotically independent. Extending a classical result on the number of $\mathbb{F}_q$-roots of random polynomials, we find that the limiting distribution of the number of $\mathbb{F}_q$-points in the intersection of a random plane curve and a fixed $\mathbb{F}_q$-line is Poisson with mean 1. We also present an explicit formula for the proportion of blocking curves involving statistics on the number of $\mathbb{F}_q$-points contained in a union of k lines for $k=1, 2, \ldots, q^2+q+1$.