@article{64180,
  author       = {{Gundlach, Fabian and Seguin, Beranger Fabrice}},
  issn         = {{1944-7833}},
  journal      = {{Algebra & Number Theory}},
  number       = {{2}},
  pages        = {{383--418}},
  publisher    = {{Mathematical Sciences Publishers}},
  title        = {{{Asymptotics of extensions of simple ℚ-algebras}}},
  doi          = {{10.2140/ant.2026.20.383}},
  volume       = {{20}},
  year         = {{2026}},
}

@article{64913,
  author       = {{Gundlach, Fabian and Seguin, Beranger Fabrice}},
  issn         = {{0021-8693}},
  journal      = {{Journal of Algebra}},
  publisher    = {{Elsevier BV}},
  title        = {{{On matrices commuting with their Frobenius}}},
  doi          = {{10.1016/j.jalgebra.2026.02.025}},
  year         = {{2026}},
}

@unpublished{65031,
  abstract     = {{We prove that two-step nilpotent $p$-extensions of rational global function fields of characteristic $p$ satisfy a quantitative local-global principle when they are counted according to their largest upper ramification break ("last jump"). We had previously shown this only for $p\neq2$. Compared to our previous proof, this proof is also more self-contained, and may apply to heights other than the last jump. As an application, we describe the distribution of last jumps of $D_4$-extensions of rational global function fields of characteristic $2$. We also exhibit a counterexample to the analogous local-global principle when counting by discriminants.}},
  author       = {{Gundlach, Fabian and Seguin, Beranger Fabrice}},
  booktitle    = {{arXiv:2603.15544}},
  title        = {{{Lifts of unramified twists and local-global principles}}},
  year         = {{2026}},
}

@unpublished{58852,
  abstract     = {{We study the asymptotic distribution of wildly ramified extensions of
function fields in characteristic $p > 2$, focusing on (certain) $p$-groups of
nilpotency class at most $2$. Rather than the discriminant, we count extensions
according to an invariant describing the last jump in the ramification
filtration at each place. We prove a local-global principle relating the
distribution of extensions over global function fields to their distribution
over local fields, leading to an asymptotic formula for the number of
extensions with a given global last-jump invariant. A key ingredient is
Abrashkin's nilpotent Artin-Schreier theory, which lets us parametrize
extensions and obtain bounds on the ramification of local extensions by
estimating the number of solutions to certain polynomial equations over finite
fields.}},
  author       = {{Gundlach, Fabian and Seguin, Beranger Fabrice}},
  booktitle    = {{arXiv:2502.18207}},
  title        = {{{Counting two-step nilpotent wildly ramified extensions of function  fields}}},
  year         = {{2025}},
}

@article{58187,
  abstract     = {{Let $K$ be a field of characteristic $0$ and $k \geq 2$ be an integer. We
prove that every $K$-linear bijection $f : K[X] \to K[X]$ strongly preserving
the set of $k$-free polynomials (or the set of polynomials with a $k$-fold root
in $K$) is a constant multiple of a $K$-algebra automorphism of $K[X]$, i.e.,
there are elements $a, c \in K^{\times}$, $b \in K$ such that $f(P)(X) = c P(a
X + b)$. When $K$ is a number field or $K=\mathbb{R}$, we prove that similar
statements hold when $f$ preserves the set of polynomials with a root in $K$.}},
  author       = {{Seguin, Beranger Fabrice}},
  journal      = {{Beiträge zur Algebra und Geometrie}},
  title        = {{{Symmetries of various sets of polynomials}}},
  doi          = {{10.1007/s13366-025-00800-2}},
  year         = {{2025}},
}

@article{63078,
  abstract     = {{For a finite group $G$, we describe the asymptotic growth of the number of
connected components of Hurwitz spaces of marked $G$-covers (of both the affine
and projective lines) whose monodromy classes are constrained in a certain way,
as the number of branch points grows to infinity. More precisely, we compute
both the exponent and (in many cases) the coefficient of the leading monomial
in the count of components containing covers whose monodromy group is a given
subgroup of $G$. By the work of Ellenberg, Tran, Venkatesh and Westerland, this
asymptotic behavior is related to the distribution of field extensions
of~$\mathbb{F}_q(T)$ with Galois group $G$.}},
  author       = {{Seguin, Beranger Fabrice}},
  issn         = {{0021-2172}},
  journal      = {{Israel Journal of Mathematics}},
  publisher    = {{Springer Science and Business Media LLC}},
  title        = {{{Counting Components of Hurwitz Spaces}}},
  doi          = {{10.1007/s11856-025-2848-5}},
  year         = {{2025}},
}

@unpublished{63077,
  abstract     = {{We study $n$-flimsy spaces, which are the topological spaces that remain connected when removing fewer than $n$ points but become disconnected when removing exactly $n$ points. We show that no such space exists for $n \geq 3$, and that the compact $2$-flimsy spaces are precisely the dense and order-complete cyclically ordered sets equipped with their order topology. Furthermore, we examine variants of the definition obtained by replacing connectedness by path-connectedness, where paths are either parametrized by $[0,1]$ or by arbitrary compact linear continua.}},
  author       = {{Khanfir, Robin and Seguin, Beranger Fabrice}},
  booktitle    = {{arXiv:2511.17745}},
  title        = {{{Flimsy Spaces}}},
  year         = {{2025}},
}

@inproceedings{58183,
  author       = {{Seguin, Beranger Fabrice}},
  booktitle    = {{MFO–RIMS Tandem Workshop: Arithmetic Homotopy and Galois Theory}},
  issn         = {{1660-8933}},
  number       = {{3}},
  pages        = {{2377--2488}},
  publisher    = {{European Mathematical Society - EMS - Publishing House GmbH}},
  title        = {{{Covers of ℙ¹ and their moduli: where arithmetic, geometry and combinatorics meet}}},
  doi          = {{10.4171/owr/2023/42}},
  volume       = {{20}},
  year         = {{2024}},
}

@article{58182,
  abstract     = {{We study a weak divisibility property for noncommutative rings: a nontrivial ring is fadelian if for all nonzero a and x there exist b, c such that x=ab+ca. We prove properties of fadelian rings and construct examples thereof which are not division rings, as well as non-Noetherian and non-Ore examples.}},
  author       = {{Khanfir, Robin and Seguin, Beranger Fabrice}},
  issn         = {{0219-4988}},
  journal      = {{Journal of Algebra and Its Applications}},
  publisher    = {{World Scientific Pub Co Pte Ltd}},
  title        = {{{Study of a division-like property}}},
  doi          = {{10.1142/s0219498825502214}},
  year         = {{2024}},
}

@phdthesis{58189,
  abstract     = {{Hurwitz spaces are moduli spaces that classify ramified covers of the projective line on
which a fixed group G acts. Their geometric and arithmetic properties are related to
number theoretical questions, particularly the inverse Galois problem. In this thesis, we
study the connected components of these spaces. Firstly, we prove results concerning
the asymptotic behavior of the count of connected components of Hurwitz spaces as
the number of branch points of the covers they classify grows. Secondly, we establish
stability results for fields of definitions of connected components of Hurwitz spaces
under the gluing operation. These results relate topological and arithmetical properties
of covers. Three expository chapters, devoid of original statements, present the various
objects. In an appendix, we summarize the thesis for the general public.}},
  author       = {{Seguin, Beranger Fabrice}},
  title        = {{{Geometry and arithmetic of components of Hurwitz spaces}}},
  year         = {{2023}},
}

@unpublished{58184,
  abstract     = {{For a fixed finite group $G$, we study the fields of definition of
geometrically irreducible components of Hurwitz moduli schemes of marked
branched $G$-covers of the projective line. The main focus is on determining
whether components obtained by "gluing" two other components, both defined over
a number field $K$, are also defined over $K$. The article presents a list of
situations in which a positive answer is obtained. As an application, when $G$
is a semi-direct product of symmetric groups or the Mathieu group $M_{23}$,
components defined over $\mathbb{Q}$ of small dimension ($6$ and $4$,
respectively) are shown to exist.}},
  author       = {{Seguin, Beranger Fabrice}},
  booktitle    = {{arXiv:2303.05903}},
  title        = {{{Fields of Definition of Components of Hurwitz Spaces}}},
  year         = {{2023}},
}

@unpublished{58185,
  abstract     = {{We consider a variant of the ring of components of Hurwitz spaces introduced
by Ellenberg, Venkatesh and Westerland. By focusing on Hurwitz spaces
classifying covers of the projective line, the resulting ring of components is
commutative, which lets us study it from the point of view of algebraic
geometry and relate its geometric properties to numerical invariants involved
in our previously obtained asymptotic counts. Specifically, we describe a
stratification of the prime spectrum of the ring of components, and we compute
the dimensions and degrees of the strata. Using the stratification, we give a
complete description of the spectrum in some cases.}},
  author       = {{Seguin, Beranger Fabrice}},
  booktitle    = {{arXiv:2210.12793}},
  title        = {{{The Geometry of Rings of Components of Hurwitz Spaces}}},
  year         = {{2022}},
}

@misc{58190,
  author       = {{Seguin, Beranger Fabrice}},
  title        = {{{Les Déformations des Représentations Galoisiennes}}},
  year         = {{2019}},
}