Let
This is in contrast with the generating functions for the ordinary conductor (from class field theory) and the discriminant, which in general have no meromorphic continuation to the entire complex plane.
}}, author = {{Gundlach, Fabian}}, issn = {{1088-6826}}, journal = {{Proceedings of the American Mathematical Society}}, publisher = {{American Mathematical Society (AMS)}}, title = {{{Counting abelian extensions by Artin–Schreier conductor}}}, doi = {{10.1090/proc/17440}}, year = {{2025}}, } @unpublished{58852, abstract = {{We study the asymptotic distribution of wildly ramified extensions of function fields in characteristic $p 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title = {{{Counting two-step nilpotent wildly ramified extensions of function fields}}}, year = {{2025}}, } @unpublished{62308, abstract = {{For a polynomial $f(x) = \sum_{i=0}^n a_i x^i$, we study the double discriminant $DD_{n,k} = \operatorname{disc}_{a_k} \operatorname{disc}_x f(x)$. This object has been well studied in algebraic geometry, but has been brought to recent prominence in number theory by its key role in the proof of the Bhargava--van der Waerden theorem. We bridge the knowledge gap for this object by proving an explicit factorization: $DD_{n,k}$ is the product of a square, a cube, and possibly a linear monomial. Our proof is entirely algebraic. We also investigate other aspects of this factorization.}}, author = {{Anderson, Theresa C. and Asarhasa, Ufuoma V. and Bertelli, Adam and Gundlach, Fabian and O'Dorney, Evan M.}}, booktitle = {{arXiv:2507.16138}}, title = {{{The structure of the double discriminant}}}, year = {{2025}}, } @unpublished{54442, abstract = {{We explain how to construct a uniformly random cubic integral domain $S$ of given signature with $|\text{disc}(S)| \leq T$ in expected time $\widetilde O(\log T)$.}}, author = {{Gundlach, Fabian}}, booktitle = {{arXiv:2405.13734}}, title = {{{Sampling cubic rings}}}, year = {{2024}}, } @unpublished{55192, abstract = {{We describe the group of $\mathbb Z$-linear automorphisms of the ring of integers of a number field $K$ that preserve the set $V_{K,k}$ of $k$th power-free integers: every such map is the composition of a field automorphism and the multiplication by a unit. We show that those maps together with translations generate the extended symmetry group of the shift space $\mathbb D_{K,k}$ associated to $V_{K,k}$. Moreover, we show that no two such dynamical systems $\mathbb D_{K,k}$ and $\mathbb D_{L,l}$ are topologically conjugate and no one is a factor system of another. We generalize the concept of $k$th power-free integers to sieves and study the resulting admissible shift spaces.}}, author = {{Gundlach, Fabian and Klüners, Jürgen}}, booktitle = {{arXiv:2407.08438}}, title = {{{Symmetries of power-free integers in number fields and their shift spaces}}}, year = {{2024}}, } @unpublished{53421, abstract = {{We define invariants $\operatorname{inv}_1,\dots,\operatorname{inv}_m$ of Galois extensions of number fields with a fixed Galois group. Then, we propose a heuristic in the spirit of Malle's conjecture which asymptotically predicts the number of extensions that satisfy $\operatorname{inv}_i\leq X_i$ for all $X_i$. The resulting conjecture is proved for abelian Galois groups. We also describe refined Artin conductors that carry essentially the same information as the invariants $\operatorname{inv}_1,\dots,\operatorname{inv}_m$.}}, author = {{Gundlach, Fabian}}, booktitle = {{arXiv:2211.16698}}, title = {{{Malle's conjecture with multiple invariants}}}, year = {{2022}}, } @unpublished{53420, abstract = {{Let $P$ be a bounded convex subset of $\mathbb R^n$ of positive volume. Denote the smallest degree of a polynomial $p(X_1,\dots,X_n)$ vanishing on $P\cap\mathbb Z^n$ by $r_P$ and denote the smallest number $u\geq0$ such that every function on $P\cap\mathbb Z^n$ can be interpolated by a polynomial of degree at most $u$ by $s_P$. We show that the values $(r_{d\cdot P}-1)/d$ and $s_{d\cdot P}/d$ for dilates $d\cdot P$ converge from below to some numbers $v_P,w_P>0$ as $d$ goes to infinity. The limits satisfy $v_P^{n-1}w_P \leq n!\cdot\operatorname{vol}(P)$. When $P$ is a triangle in the plane, we show equality: $v_Pw_P = 2\operatorname{vol}(P)$. These results are obtained by looking at the set of standard monomials of the vanishing ideal of $d\cdot P\cap\mathbb Z^n$ and by applying the Bernstein--Kushnirenko theorem. Finally, we study irreducible Laurent polynomials that vanish with large multiplicity at a point. This work is inspired by questions about Seshadri constants.}}, author = {{Gundlach, Fabian}}, booktitle = {{arXiv:2107.05353}}, title = {{{Polynomials vanishing at lattice points in a convex set}}}, year = {{2021}}, } @phdthesis{53424, author = {{Gundlach, Fabian}}, title = {{{Parametrizing extensions with fixed Galois group}}}, year = {{2019}}, } @misc{53423, author = {{Gundlach, Fabian}}, title = {{{Del Pezzo Surface Fibrations of Degree 4}}}, year = {{2014}}, } @article{53419, author = {{Gundlach, Fabian}}, issn = {{0024-6107}}, journal = {{Journal of the London Mathematical Society}}, keywords = {{General Mathematics}}, number = {{2}}, pages = {{599--618}}, publisher = {{Wiley}}, title = {{{Integral Brauer-Manin obstructions for sums of two squares and a power}}}, doi = {{10.1112/jlms/jdt042}}, volume = {{88}}, year = {{2013}}, } @misc{53422, author = {{Gundlach, Fabian}}, title = {{{Brauer-Manin obstructions for sums of two squares and a power}}}, year = {{2012}}, }