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Dragos Ghioca

Publications and source records attributed to Dragos Ghioca.

At least 19 recordsLinked to original sources
arXiv2026

Collision Parameters in Higher-Dimensional Families of Polynomials

Let $d\geq 3$ and consider the generic family of monic normalized polynomials $f_{\mathbf{t}}(z)=z^d+t_1z^{d-2}+t_2z^{d-3}+\cdots+t_{d-2}z+t_{d-1}$, where $\mathbf{t}=(t_1,\ldots,t_{d-1})$. Given pairwise distinct starting points $α_1,\ldots,α_d\in\bar{\mathbb{Q}}$ and a target point $β\in\bar{\mathbb{Q}}$, we show that the set of parameters $\boldsymbolλ\in\mathbb{A}^{d-1}(\bar{\mathbb{Q}})$ for which $β$ belongs to the forward orbit of every $α_i$ under $f_{\boldsymbolλ}$ is not Zariski dense in $\mathbb{A}^{d-1}$.

math.NT↗
arXiv2026

Covering plane curves over finite fields with lines

We prove that, for every fixed $d\geq 2$ and $\varepsilon>0$, the $\mathbb{F}_q$-points of a geometrically irreducible reflexive plane curve of degree $d$ over $\mathbb{F}_q$ can be covered by at most $\left(\frac{1}{d}+\varepsilon\right)q$ many $\mathbb{F}_q$-lines, provided that $q$ is sufficiently large. The same conclusion holds for $\mathbb{F}_q$-irreducible curves that are not geometrically irreducible, since they have only $O(d^2)$ $\mathbb{F}_q$-points. In particular, for sufficiently large $q$, the bound applies to every $\mathbb{F}_q$-irreducible plane curve of degree $d$ when $\operatorname{char}(\mathbb{F}_q)>d$. Our bound is asymptotically sharp. The proof combines Entin's Chebotarev density theorem for varieties over finite fields with the Pippenger--Spencer theorem on covers of nearly regular hypergraphs.

math.AG↗
arXiv2026

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↗
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 $κ$ with no non-trivial automorphisms for all $2\leκ\le \aleph_1$.

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 $f_λ(z):=z^d+\sum_{i=0}^{d-2} c_i(λ)\cdot z^i$ parameterized by $λ\in\bar{\mathbb{Q}}$; 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 $α_1(t),α_2(t),β(t)\in\bar{\mathbb{Q}}[t]$, where $α_i(t)$ 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 $λ\in\bar{\mathbb{Q}}$ with the property that for some $m,n\in\mathbb{N}$ (depending on $λ$), we have that $f_λ^m(α_1(λ))=f_λ^n(α_2(λ))=β(λ)$.

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

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↗