arXiv subjects

Angelot Behajaina

Publications and source records attributed to Angelot Behajaina.

At least 19 recordsLinked to original sources

arXiv2026

Monodromy groups of polynomials of composition length 2

We study the monodromy groups of compositions of two indecomposable polynomials. In particular, we show that such monodromy groups either fulfill a certain ``largeness" property, or belong to an explicit list of exceptions. Such largeness results are crucial for dealing with compositions of more than two polynomials, and consequently are expected to have a wide range of applications to problems concerning the arithmetic of polynomials and arithmetic dynamics. In particular, our main result is a key ingredient in the solution of a long-standing open problem due to Davenport, Lewis and Schinzel, achieved in a companion paper.

math.NT↗

arXiv2026

The Davenport-Lewis-Schinzel problem on the reducibility of $f(X)-g(Y)$

We solve the problem of Davenport--Lewis--Schinzel (DLS), originating in the 1950s, regarding the reducibility of $f(X)-g(Y)\in\mathbb C[X,Y]$. This yields an almost-complete solution to the Hilbert--Siegel problem: For a polynomial map $f$ whose composition factors avoid only very specific low-degree polynomials, we explicitly describe over which integers the fibers of $f$ are reducible. We further apply the solution to stability of iterates of $f$ in arithmetic dynamics, and to solving the functional equation $f(X)=g(Y)$ in $X,Y\in\mathbb{C}(z)$.

math.NT↗

arXiv2025

The intersection densities of transitive actions of $\operatorname{PSL}_{2}(q)$ with cyclic point stabilizers

Given a finite transitive group $G\leq \operatorname{Sym}{\Omega}$, the {intersection density} of $G$ is defined as the ratio between the size of the largest subsets of $G$ in which any two permutations agree on at least one element of $\Omega$, and the order of a point stabilizer of $G$. In this paper, we completely determine the intersection densities of the permutation groups $\operatorname{PSL}_{2}(q)$, where $q$ is a power of an odd prime $p$, acting transitively with point stabilizers conjugate to $\mathbb{Z}_p$. Our proof uses an auxiliary graph, which is a $\operatorname{PGL}_{2}{q}$-vertex-transitive graph, in which a clique corresponds to an intersecting set of $\operaotnrame{PSL}_{2}(q)$. For the transitive action of $\psl{2}{q}$ with point stabilizers conjugate to $\mathbb{Z}_r$, where $r\mid \frac{q-1}{2}$ is an odd prime, we show that the auxiliary graph is not regular, and we construct an intersecting set which is sometimes of maximum size.

math.CO↗

arXiv2025

The Combinatorial Nullstellensatz, Chevalley-Warning Theorem and weak Finitesatz in skew polynomial rings

We study zeros of polynomials in the multivariate skew polynomial ring $D[x_1,\ldots,x_n; \sigma]$, where $\sigma$ is an automorphism of a division ring $D$. We prove a generalization of Noga Alon's celebrated Combinatorial Nullstellensatz for such polynomials. In the case where $D$ is a finite field, we prove skew analogues of the Chevalley--Warning theorem, Ax's Lemma, and the weak case of Terjanian's Finitesatz.

math.AC↗

arXiv2025

Integrally Hilbertian rings and the polynomial Schinzel hypothesis

The classical Hilbert specialization property is a field-theoretic tool ensuring that polynomial irreducibility over a field is preserved under specialization of some of the variables. We develop an integral counterpart by introducing the notion of {integrally Hilbertian rings}, where specialization takes place inside a ring and irreducibility is required over the ring. A core part shows how new obstacles to irreducibility such as coefficient divisors or fixed divisors can be dealt with over Krull domains, a large class of rings including UFDs, Dedekind domains, etc. As a result, we obtain a general criterion for integral hilbertianity, along with many examples, \hbox{e.g.} all rings of integers of number fields. Polynomial rings over arbitrary domains are other examples. As an application, we prove a polynomial variant of the Schinzel Hypothesis on prime values of polynomials with integer coefficients: if $\mathcal{Z}$ is an integrally Hilbertian ring, the hypothesis becomes a true statement if the ring of integers ${\mathbb Z}$ is replaced by the polynomial ring $\mathcal{Z}[U]$ and ``prime'' by ``irreducible''. This result generalizes previous works and fits in a unified framework for Schinzel-type phenomena that we introduce. We further obtain an additional conclusion that has some noteworthy consequences for the classical Schinzel Hypothesis itself.

math.NT↗

arXiv2024

Counting Problems for Orthogonal Sets and Sublattices in Function Fields

Let $\mathcal{K}=\mathbb{F}_q((x^{-1}))$. Analogous to orthogonality in the Euclidean space $\mathbb{R}^n$, there exists a well-studied notion of ultrametric orthogonality in $\mathcal{K}^n$. In this paper, we extend the work of Soffer-Aranov and Behajaina on counting problems related to orthogonality in $\mathcal{K}^n$. For example, we resolve an open question posed in Soffer-Aranov and Behajaina by bounding the size of the largest ``orthogonal sets'' in $\mathcal{K}^n$. Furthermore, using similar ideas and techniques, we investigate analogues of Hadamard matrices over $\mathcal{K}$. Finally, we also use ultrametric orthogonality to compute the number of sublattices of $\mathbb{F}_q[x]^n$ with a certain geometric structure, and to determine the number of orthogonal bases of a sublattice in $\mathcal{K}^n$. The resulting formulas depend crucially on successive minima.

math.CO↗

arXiv2024

On the maximum size of ultrametric orthogonal sets over discrete valued fields

Let $\mathcal{K}$ be a discrete valued field with finite residue field. In analogy with orthogonality in the Euclidean space $\mathbb{R}^n$, there is a well-studied notion of "ultrametric orthogonality" in $\mathcal{K}^n$. In this paper, motivated by a question of Erd{\H{o}}s in the real case, given integers $k \geq \ell \geq 2$, we investigate the maximum size of a subset $S \subseteq \mathcal{K}^n \setminus\{{\bf 0}\}$ satisfying the following property: for any $E \subseteq S$ of size $k$, there exists $F \subseteq E$ of size $\ell$ such that any two distinct vectors in $F$ are orthogonal. Other variants of this property are also studied.

math.NT↗

arXiv2024

On the stopping time of the Collatz map in $\mathbb{F}_2[x]$

We study the stopping time of the Collatz map for a polynomial $f \in \mathbb{F}_2[x]$, and bound it by $O({\rm deg} (f)^{1.5})$, improving upon the quadratic bound proven by Hicks, Mullen, Yucas and Zavislak. We also prove the existence arithmetic sequences of unbounded length in the stopping times of certain sequences of polynomials, a phenomenon observed in the classical Collatz map.

math.CO↗

arXiv2023

The Collatz map analogue in polynomial rings and in completions

We study an analogue of the Collatz map in the polynomial ring $R[x]$, where $R$ is an arbitrary commutative ring. We prove that if $R$ is of positive characteristic, then every polynomial in $R[x]$ is eventually periodic with respect to this map. This extends previous works of the authors and of Hicks, Mullen, Yucas and Zavislak, who studied the Collatz map on $\mathbb{F}_p[x]$ and $\mathbb{F}_2[x]$, respectively. We also consider the Collatz map on the ring of formal power series $R[[x]]$ when $R$ is finite: we characterize the eventually periodic series in this ring, and give formulas for the number of cycles induced by the Collatz map, of any given length. We provide similar formulas for the original Collatz map defined on the ring $\mathbb{Z}_2$ of $2$-adic integers, extending previous results of Lagarias.

math.CO↗

arXiv2023

On the intersection spectrum of $\operatorname{PSL}_2(q)$

Given a group $G$ and a subgroup $H \leq G$, a set $\mathcal{F}\subset G$ is called $H$\emph{-intersecting} if for any $g,g' \in \mathcal{F}$, there exists $xH \in G/H$ such that $gxH=g'xH$. The \emph{intersection density} of the action of $G$ on $G/H$ by (left) multiplication is the rational number $\rho(G,H)$, equal to the maximum ratio $\frac{|\mathcal{F}|}{|H|}$, where $\mathcal{F} \subset G$ runs through all $H$-intersecting sets of $G$. The \emph{intersection spectrum} of the group $G$ is then defined to be the set $$ \sigma(G) := \left\{ \rho(G,H) : H\leq G \right\}. $$ It was shown by Bardestani and Mallahi-Karai [{\it J. Algebraic Combin.}, 42(1):111-128, 2015] that if $\sigma(G) = \{1\}$, then $G$ is necessarily solvable. The natural question that arises is, therefore, which rational numbers larger than $1$ belong to $\sigma(G)$, whenever $G$ is non-solvable. In this paper, we study the intersection spectrum of the linear group $\operatorname{PSL}_2(q)$. It is shown that $2 \in \sigma\left(\operatorname{PSL}_2(q)\right)$, for any prime power $q\equiv 3 \pmod 4$. Moreover, when $q\equiv 1 \pmod 4$, it is proved that $\rho(\operatorname{PSL}_2(q),H)=1$, for any odd index subgroup $H$ (containing $\mathbb{F}_q$) of the Borel subgroup (isomorphic to $\mathbb{F}_q\rtimes \mathbb{Z}_{\frac{q-1}{2}}$) consisting of all upper triangular matrices.

math.CO↗

arXiv2022

Twisted skew $G$-codes

In this paper we investigate left ideals as codes in twisted skew group rings. The considered rings, which are often algebras over a finite field, allows us to detect many of the well-known codes. The presentation, given here, unifies the concept of group codes, twisted group codes and skew group codes.

cs.IT↗

arXiv2022

Intersection density of imprimitive groups of degree $pq$

A subset $\mathcal{F}$ of a finite transitive group $G\leq \operatorname{Sym}(\Omega)$ is \emph{intersecting} if any two elements of $\mathcal{F}$ agree on an element of $\Omega$. The \emph{intersection density} of $G$ is the number $$\rho(G) = \max\left\{ \mathcal{|F|}/|G_\omega| \mid \mathcal{F}\subset G \mbox{ is intersecting} \right\},$$ where $\omega \in\Omega$ and $G_\omega$ is the stabilizer of $\omega$ in $G$. It is known that if $G\leq \operatorname{Sym}(\Omega)$ is an imprimitive group of degree a product of two odd primes $p>q$ admitting a block of size $p$ or two complete block systems, whose blocks are of size $q$, then $\rho(G) = 1$. In this paper, we analyse the intersection density of imprimitive groups of degree $pq$ with a unique block system with blocks of size $q$ based on the kernel of the induced action on blocks. For those whose kernels are non-trivial, it is proved that the intersection density is larger than $1$ whenever there exists a cyclic code $C$ with parameters $[p,k]_q$ such that any codeword of $C$ has weight at most $p-1$, and under some additional conditions on the cyclic code, it is a proper rational number. For those that are quasiprimitive, we reduce the cases to almost simple groups containing $\operatorname{Alt}(5)$ or a projective special linear group. We give some examples where the latter has intersection density equal to $1$, under some restrictions on $p$ and $q$.

math.CO↗

arXiv2022

On integral mixed Cayley graphs over non-abelian finite groups admitting an abelian subgroup of index 2

Recently, several works by a number of authors have provided characterizations of integral undirected Cayley graphs over generalized dihedral groups and generalized dicyclic groups. We generalize and unify these results in two different ways. Firstly, we work over arbitrary non-abelian finite groups admitting an abelian subgroup of index 2. Secondly, our main result actually characterizes integral mixed Cayley graphs over such finite groups, in the spirit of a very recent result of Kadyan--Bhattarcharjya in the abelian case.

math.CO↗

arXiv2022

On Cayley graphs over generalized dicyclic groups

Recently, several works by a number of authors have studied integrality, distance integrality, and distance powers of Cayley graphs over some finite groups, such as dicyclic groups and (generalized) dihedral groups. Our aim is to generalize and/or to give analogues of these results for generalized dicyclic groups. For example, we give a necessary and sufficient condition for a Cayley graph over a generalized dicyclic group to be integral (i.e., all eigenvalues of its adjacency matrix are in $\mathbb{Z}$). We also obtain sufficient conditions for the integrality of all distance powers of a Cayley graph over a given generalized dicyclic group. These results extend works on dicyclic groups by Cheng--Feng--Huang and Cheng--Feng--Liu--Lu--Stevanovic, respectively.

math.CO↗

arXiv2022

On the intersection density of the symmetric group acting on uniform subsets of small size

Given a finite transitive group $G\leq \operatorname{Sym}(Ω)$, a subset $\mathcal{F}$ of $G$ is \emph{intersecting} if any two elements of $\mathcal{F}$ agree on some element of $Ω$. The \emph{intersection density} of $G$, denoted by $ρ(G)$, is the maximum of the rational number $|\mathcal{F}|\left(\frac{|G|}{|Ω|}\right)^{-1}$ when $\mathcal{F}$ runs through all intersecting sets in $G$. In this paper, we prove that if $G$ is the group $\operatorname{Sym}(n)$ or $\operatorname{Alt}(n)$ acting on the $k$-subsets of $\{1,2,3\ldots,n\}$, for $k\in \{3,4,5\}$, then $ρ(G)=1$. Our proof relies on the representation theory of the symmetric group and the ratio bound.

math.CO↗

arXiv2021

On non-normal subgroup perfect codes

Let $X = (V,E)$ be a graph. A subset $C \subseteq V(X)$ is a \emph{perfect code} of $X$ if $C$ is a coclique of $X$ with the property that any vertex in $V(X)\setminus C$ is adjacent to exactly one vertex in $C$. Given a finite group $G$ with identity element $e$ and $H\leq G$, $H$ is a \emph{subgroup perfect code} of $G$ if there exists an inverse-closed subset $S \subseteq G\setminus \{e\}$ such that $H$ is a perfect code of the Cayley graph $\operatorname{Cay}(G,S)$ of $G$ with connection set $S$. In this short note, we give an infinite family of finite groups $G$ admitting a non-normal subgroup perfect code $H$ such that there exists $ g\in G$ with $g^2\in H$ but $(gh)^2 \neq e$, for all $h \in H$; thus, answering a question raised by Wang, Xia, and Zhou in [Perfect sets in Cayley graphs. {\it arXiv preprint} arXiv:2006.05100, 2020].

math.CO↗

arXiv2021

$3$-setwise intersecting families of the symmetric group

Given two positive integers $n\geq 3$ and $t\leq n$, the permutations $σ,π\in \operatorname{Sym}(n)$ are $t$-setwise intersecting if they agree (setwise) on a $t$-subset of $\{1,2,\ldots,n\}$. A family $\mathcal{F} \subset \operatorname{Sym}(n)$ is $t$-setwise intersecting if any two permutations of $\mathcal{F}$ are $t$-setwise intersecting. Ellis [Journal of Combinatorial Theory, Series A, 119(4), 825--849, 2012] conjectured that if $t\leq n$ and $\mathcal{F} \subset \operatorname{Sym}(n)$ is a $t$-setwise intersecting family, then $|\mathcal{F}|\leq t!(n-t)!$ and equality holds only if $\mathcal{F}$ is a coset of a setwise stablizer of a $t$-subset of $\{1,2,\ldots,n\}$. In this paper, we prove that if $n\geq 11$ and $\mathcal{F}$ is $3$-setwise intersecting, then $|\mathcal{F}|\leq 6(n-3)!$. Moreover, we prove that the characteristic vector of a $3$-setwise intersecting family of maximum size lies in the sum of the eigenspaces induced by the permutation module of $\operatorname{Sym}(n)$ acting on the $3$-subsets of $\{1,2,\ldots,n\}$.

math.CO↗

arXiv2021

Problèmes de plongement finis sur les corps non commutatifs

We extend finite embedding problems over fields, a central notion in inverse Galois theory, to the situation of a skew field $H$ of finite dimension over its center $h$. First, we show that solving a finite embedding problem over $H$ is equivalent to finding a solution to some finite embedding problem over $h$ fulfilling a polynomial constraint. Next, we show that every constant finite split embedding problem over the skew field of fractions $H(t)$ with central indeterminate $t$ has a solution, if $h$ is an ample field. This is a non-commutative analogue of a deep result of Pop. More generally, we solve such finite embedding problems over the skew field of fractions $H(t, σ)$ of the twisted polynomial ring $H[t, σ]$, for some automorphisms $σ$ of $H$ of finite order. Our results extend previous works on the inverse Galois problem over skew fields.

math.NT↗