arXiv subjects
Publications and source records attributed to Dragos Ghioca.
A plane curve $C$ in $\mathbb{P}^2$ defined over $\mathbb{F}_q$ is called plane-filling if $C$ contains every $\mathbb{F}_q$-point of $\mathbb{P}^2$. Homma and Kim, building on the work of Tallini, proved that the minimum degree of a smooth plane-filling curve is $q+2$. We study smooth plane-filling curves of degree $q+3$ and higher.
Let $G$ be a semiabelian variety defined over 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 recent years, many useful applications of the polynomial method have emerged in finite geometry. Indeed, algebraic curves, especially those defined by Rédei-type polynomials, are powerful in studying blocking sets. In this paper, we reverse the engine and study when blocking sets can arise from rational points on plane curves over finite fields. We show that irreducible curves of low degree cannot provide blocking sets and prove more refined results for cubic and quartic curves. On the other hand, using tools from number theory, we construct smooth plane curves defined over $\mathbb{F}_p$ of degree at most $4p^{3/4}+1$ whose points form blocking sets.
We study plane curves over finite fields whose tangent lines at smooth $\mathbb{F}_q$-points together cover all the points of $\mathbb{P}^2(\mathbb{F}_q)$.
We complete the proof of a Siegel type statement for finitely generated $Φ$-submodules of $\mathbb{G}_a$ under the action of a Drinfeld module $Φ$.
Let $K$ be a finitely generated field. We construct an $n$-dimensional linear system $\mathcal{L}$ of hypersurfaces of degree $d$ in $\mathbb{P}^n$ defined over $K$ such that each member of $\mathcal{L}$ defined over $K$ is smooth, under the hypothesis that the characteristic $p$ does not divide $\gcd(d, n+1)$ (in particular, there is no restriction when $K$ has characteristic $0$). Moreover, we exhibit a counterexample when $p$ divides $\gcd(d, n+1)$.
Let $G$ be a semiabelian variety defined over an algebraically closed field $K$, endowed with a rational self-map $Φ$. Let $α\in G(K)$ and let $Γ\subseteq G(K)$ be a finitely generated subgroup. We show that the set $\{n\in\mathbb{N}\colon Φ^n(α)\in Γ\}$ is a union of finitely many arithmetic progressions along with a set of Banach density equal to $0$. In addition, assuming $Φ$ is regular, we prove that the set $S$ must be finite.
Let $G$ be a semiabelian variety defined over an algebraically closed field $K$ of characteristic $0$. Let $Φ\colon G\dashrightarrow G$ be a dominant rational self-map. Assume that an iterate $Φ^m \colon G \to G$ is regular for some $m \geqslant 1$ and that there exists no non-constant homomorphism $τ: G\to G_0$ of semiabelian varieties such that $τ\circ Φ^{m k}=τ$ for some $k \geqslant 1$. We show that under these assumptions $Φ$ itself must be a regular. We also prove a variant of this assertion in prime characteristic and present examples showing that our results are sharp.
We study an open question at the interplay between the classical and the dynamical Mordell-Lang conjectures in positive characteristic. Let $K$ be an algebraically closed field of positive characteristic, let $G$ be a finitely generated subgroup of the multiplicative group of $K$, and let $X$ be a (irreducible) quasiprojective variety defined over $K$. We consider $K$-valued sequences of the form $a_n:=f(φ^n(x_0))$, where $φ\colon X\rightarrow X$ and $f\colon X\rightarrow\mathbb{P}^1$ are rational maps defined over $K$ and $x_0\in X$ is a point whose forward orbit avoids the indeterminacy loci of $φ$ and $f$. We show that the set of $n$ for which $a_n\in G$ is a finite union of arithmetic progressions along with a set of upper Banach density zero. In addition, we show that if $a_n\in G$ for every $n$ and the $φ$ orbit of $x$ is Zariski dense in $X$ then {there is} a multiplicative torus $\mathbb{G}_m^d$ and maps $Ψ:\mathbb{G}_m^d \to \mathbb{G}_m^d$ and $g:\mathbb{G}_m^d \to \mathbb{G}_m$ such that $a_n = g\circ Ψ^n(y)$ for some $y\in \mathbb{G}_m^d$. We then describe various applications of our results.
We study pencils of hypersurfaces over finite fields $\mathbb{F}_q$ such that each of the $q+1$ members defined over $\mathbb{F}_q$ is smooth.
We prove the Zariski dense orbit conjecture in positive characteristic for endomorphisms of $\mathbb{G}_a^N$ defined over $\overline{\mathbb{F}_p}$.
We formulate a strengthening of the Zariski dense orbit conjecture for birational maps of dynamical degree one. So, given a quasiprojective variety $X$ defined over an algebraically closed field $K$ of characteristic $0$, endowed with a birational self-map $ϕ$ of dynamical degree $1$, we expect that either there exists a non-constant rational function $f:X\dashrightarrow \mathbb{P}^1$ such that $f\circ ϕ=f$, or there exists a proper subvariety $Y\subset X$ with the property that for any invariant proper subvariety $Z\subset X$, we have that $Z\subseteq Y$. We prove our conjecture for automorphisms $ϕ$ of dynamical degree $1$ of semiabelian varieties $X$. Also, we prove a related result for regular dominant self-maps $ϕ$ of semiabelian varieties $X$: assuming $ϕ$ does not preserve a non-constant rational function, we have that the dynamical degree of $ϕ$ is larger than $1$ if and only if the union of all $ϕ$-invariant proper subvarieties of $X$ is Zariski dense. We give applications of our results to representation theoretic questions about twisted homogeneous coordinate rings associated to abelian varieties.
We prove the Zariski dense orbit conjecture in positive characteristic for regular self-maps of split semiabelian varieties.
Let $q$ be a power of the prime number $p$, let $K={\mathbb F}_q(t)$, and let $r\ge 2$ be an integer. For points ${\mathbf a}, {\mathbf b}\in K$ which are $\mathbb{F}_q$-linearly independent, we show that there exist positive constants $N_0$ and $c_0$ such that for each integer $\ell\ge N_0$ and for each generator $τ$ of ${\mathbb F}_{q^\ell}/{\mathbb F}_q$, we have that for all except $N_0$ values $λ\in{\overline{\mathbb{F}_q}}$, the corresponding specializations ${\mathbf a}, {\mathbf b}(τ)$ and ${\mathbf b}(τ)$ cannot have orders of degrees less than $c_0\log\log\ell$ as torsion points for the Drinfeld module $Φ^{(τ,λ)}:\mathbb{F}_q[T] {\longrightarrow} {\mathrm{End}}_{\overline{\mathbb{F}_q}}({\mathbb G}_a)$ (where ${\mathbb G}_a$ is the additive group scheme), given by $Φ^{(τ,λ)}_T(x)=τx+λx^q + x^{q^r}$.
Given an integer $g$ and also some given integers $m$ (sufficiently large) and $c_1,\dots, c_m$, we show that the number of all non-negative integers $n\le M$ with the property that there exist non-negative integers $k_1,\dots, k_m$ such that $$n^2=\sum_{i=1}^m c_i g^{k_i}$$ is $o\left(\left(\log M \right)^{m-1/2}\right)$. We also obtain a similar bound when dealing with more general inequalities $$\left|Q(n)-\sum_{i=1}^m c_iλ^{k_i}\right|\le B,$$ where $Q\in {\mathbb C}[X]$ and also $λ\in {\mathbb C}$ (while $B$ is a real number).
We prove a quantitative partial result in support of the Dynamical Mordell-Lang Conjecture (also known as the DML conjecture) in positive characteristic. More precisely, we show the following: given a field $K$ of characteristic $p$, given a semiabelian variety $X$ defined over a finite subfield of $K$ and endowed with a regular self-map $Φ:X \longrightarrow X$ defined over $K$, given a point $α\in X(K)$ and a subvariety $V\subseteq X$, then the set of all non-negative integers $n$ such that $Φ^n(α)\in V(K)$ is a union of finitely many arithmetic progressions along with a subset $S$ with the property that there exists a positive real number $A$ (depending only on $N$, $Φ$, $α$, $V$) such that for each positive integer $M$, we have $$\#\left\{n\in S\colon~ n\le M\right\}\le A\cdot \left(1+\log M\right)^{\dim V}.$$
We study the question of finding smooth hyperplane sections to a pencil of hypersurfaces over finite fields.
We formulate a general question regarding the size of the iterated Galois groups associated to an algebraic dynamical system and then we discuss some special cases of our question.