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Dragos Ghioca

Publications and source records attributed to Dragos Ghioca.

At least 19 recordsLinked to original sources
arXiv2026

Finding suitably generic points on curves with an application to the construction of rigid real closed fields

Let $K$ be an algebraically closed field of characteristic 0 and transcendence degree at least 2. Let $C\subset K^2$ be an irreducible curve defined over $K$ but not defined over the algebraic closure of $\mathbb Q$. There is $(x ,y)$ a $K$-point of $C$ such that $x$ and $y$ are algebraically independent. Moreover, if $C_0$ and $C_1$ are two such curves and there is a finite-to-finite algebraic correspondence between them defined over $K$, then there are corresponding $K$-points $(x_0,y_0)\in C_0$ and $(x_1,y_1)\in C_1$ such that $x_0$ and $y_0$ are algebraically independent and $x_1$ and $y_1$ are algebraically independent. We use the latter result to construct non-Archimedean real closed fields of transcendence degree $\kappa$ with no non-trivial automorphisms for all $2\le\kappa\le \aleph_1$.

math.LO↗
arXiv2026

Collision of Orbits for Families of Polynomials Defined over Number Fields

Let $d\ge 2$ be an integer and let $c_0(t),\dots, c_{d-2}(t)\in\bar{\mathbb{Q}}[t]$. We consider the family of normalized polynomials $f_\lambda(z):=z^d+\sum_{i=0}^{d-2} c_i(\lambda)\cdot z^i$ parameterized by $\lambda\in\bar{\mathbb{Q}}$; the generic element of our family of polynomials is $f_t(z):=z^d+\sum_{i=0}^{d-2}c_i(t)\cdot z^i\in \bar{\mathbb{Q}}[t][z]$. Also, let $\alpha_1(t),\alpha_2(t),\beta(t)\in\bar{\mathbb{Q}}[t]$, where $\alpha_i(t)$ is not preperiodic under the action of $f_t(z)$ for each $i=1,2$. Under some natural hypotheses, we obtain precise necessary and sufficient conditions for which there exist infinitely many $\lambda\in\bar{\mathbb{Q}}$ with the property that for some $m,n\in\mathbb{N}$ (depending on $\lambda$), we have that $f_\lambda^m(\alpha_1(\lambda))=f_\lambda^n(\alpha_2(\lambda))=\beta(\lambda)$.

math.NT↗
arXiv2026

Collision of Orbits on an Elliptic Surface

Let $C$ be a smooth projective curve defined over $\Qbar$, let $π:\mathcal{E}\lra C$ be an elliptic surface and let $σ_{P_1},σ_{P_2},σ_{Q}$ be sections of $π$ (corresponding to points $P_1,P_2, Q$ of the generic fiber $E$ of $\mathcal{E}$). We obtain a precise characterization, expressed solely in terms of the dynamical relations between the points $P_1,P_2,Q$ with respect to the endomorphism ring of $E$, so that there exist infinitely many $ł\in C(\Qbar)$ with the property that for some nonzero integers $m_{1,ł},m_{2,ł}$, we have that $[m_{i,ł}](σ_{P_{i}}(ł))=σ_{Q}(ł)$ (for $i=1,2$) on the smooth fiber $E_ł$ of $\mathcal{E}$.

math.NT↗
arXiv2026

Collision of orbits for families of polynomials defined over fields of positive characteristic

Let $L$ be a field of positive characteristic $p$ with a fixed algebraic closure $\overline{L}$, and let $α_1,α_2,β\in L$. For an integer $d\ge 2$, we consider the family of polynomials $f_λ(z) := z^d+λ$, parameterized by $λ\in\overline{L}$. Define $C(α_1,α_2;β)$ to be the set of all $λ\in\overline{L}$ for which there exist $m,n\in\mathbb{N}$ such that $f_λ^m(α_1)=f_λ^n(α_2)=β$. In other words, $C(α_1,α_2;β)$ consists of all $λ\in\overline{L}$ with the property that the orbit of $α_1$ collides with the orbit of $α_2$ under the same polynomial $f_λ$ precisely at the point $β$. Assuming $α_1,α_2,β$ are not all contained in a finite subfield of $L$, we provide explicit necessary and sufficient conditions under which $C(α_1,α_2;β)$ is infinite. We also discuss the remaining case where $α_1,α_2,β\in \overline{\mathbb F}_p$ and provide ample computational data that suggest a somewhat surprising conjecture. Our problem fits into a long series of questions in the area of unlikely intersections in arithmetic dynamics, which have been primarily studied over fields of characteristic $0$. Working in characteristic $p$ adds significant difficulties, but also reveals the subtlety of our problem, especially when some of the points lie in a finite field or when $d$ is a power of $p$.

math.NT↗
arXiv2025

Blocking sets from a union of plane curves

Motivated by a question of Erdős on blocking sets in a projective plane that intersect every line only a few times, several authors have used unions of algebraic curves to construct such sets in $\mathbb{P}^2(\mathbb{F}_q)$. In this paper, we provide new constructions of blocking sets in $\mathbb{P}^2(\mathbb{F}_q)$ from a union of geometrically irreducible curves of a fixed degree $d$. We also establish lower bounds on the number of such curves required to form a blocking set. Our proofs combine tools from arithmetic geometry and combinatorics.

math.AG↗
arXiv2025

Arboreal Galois groups of postcritically finite quadratic polynomials: The strictly preperiodic case

In a previous paper, we provided an explicit description of the arboreal Galois group of the postcritically finite polynomial $f(z) = z^2 +c$ in the special case when the critical point $0$ is periodic under the action of $f(z)$. In the current paper, we complete the picture for all postcritically finite polynomials by addressing the cases when $0$ is strictly preperiodic for the polynomial $f(z)$.

math.NT↗
arXiv2025

Proportion of blocking curves in a pencil

Let $\mathcal{L}$ be a pencil of plane curves defined over $\mathbb{F}_q$ with no $\mathbb{F}_q$-points in its base locus. We investigate the number of curves in $\mathcal{L}$ whose $\mathbb{F}_q$-points form a blocking set. When the degree of the pencil is allowed to grow with respect to $q$, we show that the geometric problem can be translated into a purely combinatorial problem about disjoint blocking sets. We also study the same problem when the degree of the pencil is fixed.

math.AG↗
arXiv2024

Linear system of geometrically irreducible plane cubics over finite fields

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.

math.AG↗
arXiv2024

Effective isotrivial Mordell-Lang in positive characteristic

The isotrivial Mordell-Lang theorem of Moosa and Scanlon describes the set $X\capΓ$ when $X$ is a subvariety of a semiabelian variety $G$ over a finite field $\mathbb{F}_q$ and $Γ$ is a finitely generated subgroup of $G$ that is invariant under the $q$-power Frobenius endomorphism $F$. That description is here made effective, and extended to arbitrary commutative algebraic groups $G$ and arbitrary finitely generated $\mathbb{Z}[F]$-submodules $Γ$. The approach is to use finite automata to give a concrete description of $X\cap Γ$. These methods and results have new applications even when specialised to the case when $G$ is an abelian variety over a finite field, $X\subseteq G$ a subvariety defined over a function field $K$, and $Γ=G(K)$. As an application of the automata-theoretic approach, a dichotomy theorem is established for the growth of the number of points in $X(K)$ of bounded height. As an application of the effective description of $X\capΓ$, decision procedures are given for the following three diophantine problems: Is $X(K)$ nonempty? Is it infinite? Does it contain an infinite coset?

math.NT↗
arXiv2024

Intersection of orbits for polynomials in characteristic $p$

In [GTZ08, GTZ12], the following result was established: given polynomials $f,g\in\mathbb{C}[x]$ of degrees larger than $1$, if there exist $α,β\in\mathbb{C}$ such that their corresponding orbits $\mathcal{O}_f(α)$ and $\mathcal{O}_g(β)$ (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.

math.NT↗
arXiv2024

Linear system of hypersurfaces passing through a Galois orbit

Let $d$ and $n$ be positive integers, and $E/F$ be a separable field extension of degree $m=\binom{n+d}{n}$. 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.

math.AG↗
arXiv2024

Simultaneously preperiodic points for a family of polynomials in positive characteristic

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 $f_λ(x):=x^d+λ$ (parameterized by $λ\in\mathbb{C}$), given two starting points $a$ and $b$ in $\mathbb{C}$, if there exist infinitely many $λ\in\mathbb{C}$ such that both $a$ and $b$ are preperiodic under the action of $f_λ$, 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 $\Fpbar$; (ii) $d$ is a power of $p$; and (iii) not both $a$ and $b$ live in $\Fpbar$, 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 $λ\in\Fpbar$, both $a$ and $b$ are preperiodic under the action of $f_λ$, while in case~(ii), one obtains that \emph{also} whenever $a-b\in\Fpbar$, then for each parameter $λ$, we have that $a$ is preperiodic under the action of $f_λ$ if and only if $b$ is preperiodic under the action of $f_λ$.

math.NT↗
arXiv2024

Most plane curves over finite fields are not blocking

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 $k=1, 2, \ldots, q^2+q+1$.

math.AG↗
arXiv2023

Existence of pencils with nonblocking hypersurfaces

We prove that there is a pencil of hypersurfaces in $\mathbb{P}^n$ of any given degree over a finite field $\mathbb{F}_q$ such that every $\mathbb{F}_q$-member of the pencil is not blocking with respect to $\mathbb{F}_q$-lines.

math.AG↗
arXiv2023

Plane-filling curves of small degree over finite fields

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.

math.AG↗