---
_id: '64068'
abstract:
- lang: eng
text: "When do two irreducible polynomials with integer coefficients\r\n define
the same number field? One can define an action of\r\n $\\mathrm{GL}_2 \\times
\\mathrm{GL}_1$ on the space of polynomials of degree $n$ so that for any two\r\n
\ polynomials $f$ and $g$ in the same orbit, the roots of $f$ may be expressed\r\n
\ as rational linear transformations of the roots of $g$; thus, they generate\r\n
\ the same field. In this article, we show that almost all polynomials of\r\n
\ degree $n$ with size at most $X$ can only define the same number field as\r\n
\ another polynomial of degree $n$ with size at most $X$ if they lie in the\r\n
\ same orbit for this group action. (Here we measure the size of polynomials by\r\n
\ the greatest absolute value of their coefficients.)\r\n This improves on work
of Bhargava, Shankar, and Wang, who proved a similar\r\n statement for a positive
proportion of polynomials. Using this result, we\r\n prove that the number of
degree $n$ fields such that the smallest polynomial\r\n defining the field has
size at most $X$ is asymptotic to a constant times\r\n $X^{n+1}$ as long as $n\\geq
3$. For $n = 2$, we obtain a precise asymptotic of\r\n the form $\\frac{27}{π^2}
X^2$."
author:
- first_name: Santiago
full_name: Arango-Piñeros, Santiago
last_name: Arango-Piñeros
- first_name: Fabian
full_name: Gundlach, Fabian
id: '100450'
last_name: Gundlach
- first_name: Robert J.
full_name: Lemke Oliver, Robert J.
last_name: Lemke Oliver
- first_name: Kevin J.
full_name: McGown, Kevin J.
last_name: McGown
- first_name: Will
full_name: Sawin, Will
last_name: Sawin
- first_name: Allechar
full_name: Serrano López, Allechar
last_name: Serrano López
- first_name: Arul
full_name: Shankar, Arul
last_name: Shankar
- first_name: Ila
full_name: Varma, Ila
last_name: Varma
citation:
ama: Arango-Piñeros S, Gundlach F, Lemke Oliver RJ, et al. Counting number fields
of fixed degree by their smallest defining polynomial. arXiv:260206943.
Published online 2026.
apa: Arango-Piñeros, S., Gundlach, F., Lemke Oliver, R. J., McGown, K. J., Sawin,
W., Serrano López, A., Shankar, A., & Varma, I. (2026). Counting number fields
of fixed degree by their smallest defining polynomial. In arXiv:2602.06943.
bibtex: '@article{Arango-Piñeros_Gundlach_Lemke Oliver_McGown_Sawin_Serrano López_Shankar_Varma_2026,
title={Counting number fields of fixed degree by their smallest defining polynomial},
journal={arXiv:2602.06943}, author={Arango-Piñeros, Santiago and Gundlach, Fabian
and Lemke Oliver, Robert J. and McGown, Kevin J. and Sawin, Will and Serrano López,
Allechar and Shankar, Arul and Varma, Ila}, year={2026} }'
chicago: Arango-Piñeros, Santiago, Fabian Gundlach, Robert J. Lemke Oliver, Kevin
J. McGown, Will Sawin, Allechar Serrano López, Arul Shankar, and Ila Varma. “Counting
Number Fields of Fixed Degree by Their Smallest Defining Polynomial.” ArXiv:2602.06943,
2026.
ieee: S. Arango-Piñeros et al., “Counting number fields of fixed degree by
their smallest defining polynomial,” arXiv:2602.06943. 2026.
mla: Arango-Piñeros, Santiago, et al. “Counting Number Fields of Fixed Degree by
Their Smallest Defining Polynomial.” ArXiv:2602.06943, 2026.
short: S. Arango-Piñeros, F. Gundlach, R.J. Lemke Oliver, K.J. McGown, W. Sawin,
A. Serrano López, A. Shankar, I. Varma, ArXiv:2602.06943 (2026).
date_created: 2026-02-09T07:48:05Z
date_updated: 2026-02-09T07:49:17Z
external_id:
arxiv:
- '2602.06943'
language:
- iso: eng
publication: arXiv:2602.06943
status: public
title: Counting number fields of fixed degree by their smallest defining polynomial
type: preprint
user_id: '100450'
year: '2026'
...
---
_id: '65031'
abstract:
- lang: eng
text: 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:
- first_name: Fabian
full_name: Gundlach, Fabian
id: '100450'
last_name: Gundlach
- first_name: Beranger Fabrice
full_name: Seguin, Beranger Fabrice
id: '102487'
last_name: Seguin
orcid: 0000-0002-4800-4647
citation:
ama: Gundlach F, Seguin BF. Lifts of unramified twists and local-global principles.
arXiv:260315544. Published online 2026.
apa: Gundlach, F., & Seguin, B. F. (2026). Lifts of unramified twists and local-global
principles. In arXiv:2603.15544.
bibtex: '@article{Gundlach_Seguin_2026, title={Lifts of unramified twists and local-global
principles}, journal={arXiv:2603.15544}, author={Gundlach, Fabian and Seguin,
Beranger Fabrice}, year={2026} }'
chicago: Gundlach, Fabian, and Beranger Fabrice Seguin. “Lifts of Unramified Twists
and Local-Global Principles.” ArXiv:2603.15544, 2026.
ieee: F. Gundlach and B. F. Seguin, “Lifts of unramified twists and local-global
principles,” arXiv:2603.15544. 2026.
mla: Gundlach, Fabian, and Beranger Fabrice Seguin. “Lifts of Unramified Twists
and Local-Global Principles.” ArXiv:2603.15544, 2026.
short: F. Gundlach, B.F. Seguin, ArXiv:2603.15544 (2026).
date_created: 2026-03-17T12:17:42Z
date_updated: 2026-03-17T12:21:09Z
external_id:
arxiv:
- '2603.15544'
language:
- iso: eng
publication: arXiv:2603.15544
status: public
title: Lifts of unramified twists and local-global principles
type: preprint
user_id: '100450'
year: '2026'
...
---
_id: '66684'
abstract:
- lang: eng
text: Let $K$ be a global function field of characteristic~$p$ and let $G$ be a
finite abelian group of exponent $p^e$. We show that the multivariate generating
function counting sub-$G$-extensions of $K$ with respect to $e$ specific height
functions (encoding successive drops in the exponents of the higher ramification
groups) is rational.
author:
- first_name: Fabian
full_name: Gundlach, Fabian
id: '100450'
last_name: Gundlach
citation:
ama: Gundlach F. Multivariate counting of wild abelian extensions. arXiv:260727364.
Published online 2026.
apa: Gundlach, F. (2026). Multivariate counting of wild abelian extensions. In arXiv:2607.27364.
bibtex: '@article{Gundlach_2026, title={Multivariate counting of wild abelian extensions},
journal={arXiv:2607.27364}, author={Gundlach, Fabian}, year={2026} }'
chicago: Gundlach, Fabian. “Multivariate Counting of Wild Abelian Extensions.” ArXiv:2607.27364,
2026.
ieee: F. Gundlach, “Multivariate counting of wild abelian extensions,” arXiv:2607.27364.
2026.
mla: Gundlach, Fabian. “Multivariate Counting of Wild Abelian Extensions.” ArXiv:2607.27364,
2026.
short: F. Gundlach, ArXiv:2607.27364 (2026).
date_created: 2026-08-07T14:30:17Z
date_updated: 2026-08-07T14:30:51Z
external_id:
arxiv:
- '2607.27364'
language:
- iso: eng
publication: arXiv:2607.27364
status: public
title: Multivariate counting of wild abelian extensions
type: preprint
user_id: '100450'
year: '2026'
...
---
_id: '62308'
abstract:
- lang: eng
text: '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:
- first_name: Theresa C.
full_name: Anderson, Theresa C.
last_name: Anderson
- first_name: Ufuoma V.
full_name: Asarhasa, Ufuoma V.
last_name: Asarhasa
- first_name: Adam
full_name: Bertelli, Adam
last_name: Bertelli
- first_name: Fabian
full_name: Gundlach, Fabian
id: '100450'
last_name: Gundlach
- first_name: Evan M.
full_name: O'Dorney, Evan M.
last_name: O'Dorney
citation:
ama: Anderson TC, Asarhasa UV, Bertelli A, Gundlach F, O’Dorney EM. The structure
of the double discriminant. arXiv:250716138. Published online 2025.
apa: Anderson, T. C., Asarhasa, U. V., Bertelli, A., Gundlach, F., & O’Dorney,
E. M. (2025). The structure of the double discriminant. In arXiv:2507.16138.
bibtex: '@article{Anderson_Asarhasa_Bertelli_Gundlach_O’Dorney_2025, title={The
structure of the double discriminant}, journal={arXiv:2507.16138}, author={Anderson,
Theresa C. and Asarhasa, Ufuoma V. and Bertelli, Adam and Gundlach, Fabian and
O’Dorney, Evan M.}, year={2025} }'
chicago: Anderson, Theresa C., Ufuoma V. Asarhasa, Adam Bertelli, Fabian Gundlach,
and Evan M. O’Dorney. “The Structure of the Double Discriminant.” ArXiv:2507.16138,
2025.
ieee: T. C. Anderson, U. V. Asarhasa, A. Bertelli, F. Gundlach, and E. M. O’Dorney,
“The structure of the double discriminant,” arXiv:2507.16138. 2025.
mla: Anderson, Theresa C., et al. “The Structure of the Double Discriminant.” ArXiv:2507.16138,
2025.
short: T.C. Anderson, U.V. Asarhasa, A. Bertelli, F. Gundlach, E.M. O’Dorney, ArXiv:2507.16138
(2025).
date_created: 2025-11-26T08:18:33Z
date_updated: 2025-11-26T08:21:27Z
external_id:
arxiv:
- '2507.16138'
language:
- iso: eng
publication: arXiv:2507.16138
status: public
title: The structure of the double discriminant
type: preprint
user_id: '100450'
year: '2025'
...
---
_id: '58852'
abstract:
- lang: eng
text: "We study the asymptotic distribution of wildly ramified extensions of\r\nfunction
fields in characteristic $p > 2$, focusing on (certain) $p$-groups of\r\nnilpotency
class at most $2$. Rather than the discriminant, we count extensions\r\naccording
to an invariant describing the last jump in the ramification\r\nfiltration at
each place. We prove a local-global principle relating the\r\ndistribution of
extensions over global function fields to their distribution\r\nover local fields,
leading to an asymptotic formula for the number of\r\nextensions with a given
global last-jump invariant. A key ingredient is\r\nAbrashkin's nilpotent Artin-Schreier
theory, which lets us parametrize\r\nextensions and obtain bounds on the ramification
of local extensions by\r\nestimating the number of solutions to certain polynomial
equations over finite\r\nfields."
author:
- first_name: Fabian
full_name: Gundlach, Fabian
id: '100450'
last_name: Gundlach
- first_name: Beranger Fabrice
full_name: Seguin, Beranger Fabrice
id: '102487'
last_name: Seguin
citation:
ama: Gundlach F, Seguin BF. Counting two-step nilpotent wildly ramified extensions
of function fields. arXiv:250218207. Published online 2025.
apa: Gundlach, F., & Seguin, B. F. (2025). Counting two-step nilpotent wildly
ramified extensions of function fields. In arXiv:2502.18207.
bibtex: '@article{Gundlach_Seguin_2025, title={Counting two-step nilpotent wildly
ramified extensions of function fields}, journal={arXiv:2502.18207}, author={Gundlach,
Fabian and Seguin, Beranger Fabrice}, year={2025} }'
chicago: Gundlach, Fabian, and Beranger Fabrice Seguin. “Counting Two-Step Nilpotent
Wildly Ramified Extensions of Function Fields.” ArXiv:2502.18207, 2025.
ieee: F. Gundlach and B. F. Seguin, “Counting two-step nilpotent wildly ramified
extensions of function fields,” arXiv:2502.18207. 2025.
mla: Gundlach, Fabian, and Beranger Fabrice Seguin. “Counting Two-Step Nilpotent
Wildly Ramified Extensions of Function Fields.” ArXiv:2502.18207, 2025.
short: F. Gundlach, B.F. Seguin, ArXiv:2502.18207 (2025).
date_created: 2025-02-26T08:51:57Z
date_updated: 2025-02-26T08:53:08Z
external_id:
arxiv:
- '2502.18207'
language:
- iso: eng
publication: arXiv:2502.18207
status: public
title: Counting two-step nilpotent wildly ramified extensions of function fields
type: preprint
user_id: '100450'
year: '2025'
...
---
_id: '55192'
abstract:
- lang: eng
text: "We describe the group of $\\mathbb Z$-linear automorphisms of the ring of\r\nintegers
of a number field $K$ that preserve the set $V_{K,k}$ of $k$th\r\npower-free integers:
every such map is the composition of a field automorphism\r\nand the multiplication
by a unit.\r\n We show that those maps together with translations generate the
extended\r\nsymmetry group of the shift space $\\mathbb D_{K,k}$ associated to
$V_{K,k}$.\r\nMoreover, we show that no two such dynamical systems $\\mathbb D_{K,k}$
and\r\n$\\mathbb D_{L,l}$ are topologically conjugate and no one is a factor system
of\r\nanother.\r\n We generalize the concept of $k$th power-free integers to
sieves and study\r\nthe resulting admissible shift spaces."
author:
- first_name: Fabian
full_name: Gundlach, Fabian
id: '100450'
last_name: Gundlach
- first_name: Jürgen
full_name: Klüners, Jürgen
id: '21202'
last_name: Klüners
citation:
ama: Gundlach F, Klüners J. Symmetries of power-free integers in number fields and
their shift spaces. arXiv:240708438. Published online 2024.
apa: Gundlach, F., & Klüners, J. (2024). Symmetries of power-free integers in
number fields and their shift spaces. In arXiv:2407.08438.
bibtex: '@article{Gundlach_Klüners_2024, title={Symmetries of power-free integers
in number fields and their shift spaces}, journal={arXiv:2407.08438}, author={Gundlach,
Fabian and Klüners, Jürgen}, year={2024} }'
chicago: Gundlach, Fabian, and Jürgen Klüners. “Symmetries of Power-Free Integers
in Number Fields and Their Shift Spaces.” ArXiv:2407.08438, 2024.
ieee: F. Gundlach and J. Klüners, “Symmetries of power-free integers in number fields
and their shift spaces,” arXiv:2407.08438. 2024.
mla: Gundlach, Fabian, and Jürgen Klüners. “Symmetries of Power-Free Integers in
Number Fields and Their Shift Spaces.” ArXiv:2407.08438, 2024.
short: F. Gundlach, J. Klüners, ArXiv:2407.08438 (2024).
date_created: 2024-07-12T08:16:37Z
date_updated: 2024-07-12T08:19:11Z
external_id:
arxiv:
- '2407.08438'
language:
- iso: eng
publication: arXiv:2407.08438
status: public
title: Symmetries of power-free integers in number fields and their shift spaces
type: preprint
user_id: '82981'
year: '2024'
...
---
_id: '53421'
abstract:
- lang: eng
text: "We define invariants $\\operatorname{inv}_1,\\dots,\\operatorname{inv}_m$
of\r\nGalois extensions of number fields with a fixed Galois group. Then, we propose\r\na
heuristic in the spirit of Malle's conjecture which asymptotically predicts\r\nthe
number of extensions that satisfy $\\operatorname{inv}_i\\leq X_i$ for all\r\n$X_i$.
The resulting conjecture is proved for abelian Galois groups. We also\r\ndescribe
refined Artin conductors that carry essentially the same information\r\nas the
invariants $\\operatorname{inv}_1,\\dots,\\operatorname{inv}_m$."
author:
- first_name: Fabian
full_name: Gundlach, Fabian
id: '100450'
last_name: Gundlach
citation:
ama: Gundlach F. Malle’s conjecture with multiple invariants. arXiv:221116698.
Published online 2022.
apa: Gundlach, F. (2022). Malle’s conjecture with multiple invariants. In arXiv:2211.16698.
bibtex: '@article{Gundlach_2022, title={Malle’s conjecture with multiple invariants},
journal={arXiv:2211.16698}, author={Gundlach, Fabian}, year={2022} }'
chicago: Gundlach, Fabian. “Malle’s Conjecture with Multiple Invariants.” ArXiv:2211.16698,
2022.
ieee: F. Gundlach, “Malle’s conjecture with multiple invariants,” arXiv:2211.16698.
2022.
mla: Gundlach, Fabian. “Malle’s Conjecture with Multiple Invariants.” ArXiv:2211.16698,
2022.
short: F. Gundlach, ArXiv:2211.16698 (2022).
date_created: 2024-04-11T12:43:14Z
date_updated: 2024-04-11T12:50:44Z
extern: '1'
external_id:
arxiv:
- '2211.16698'
language:
- iso: eng
publication: arXiv:2211.16698
status: public
title: Malle's conjecture with multiple invariants
type: preprint
user_id: '100450'
year: '2022'
...
---
_id: '53420'
abstract:
- lang: eng
text: "Let $P$ be a bounded convex subset of $\\mathbb R^n$ of positive volume.\r\nDenote
the smallest degree of a polynomial $p(X_1,\\dots,X_n)$ vanishing on\r\n$P\\cap\\mathbb
Z^n$ by $r_P$ and denote the smallest number $u\\geq0$ such that\r\nevery function
on $P\\cap\\mathbb Z^n$ can be interpolated by a polynomial of\r\ndegree at most
$u$ by $s_P$. We show that the values $(r_{d\\cdot P}-1)/d$ and\r\n$s_{d\\cdot
P}/d$ for dilates $d\\cdot P$ converge from below to some numbers\r\n$v_P,w_P>0$
as $d$ goes to infinity. The limits satisfy $v_P^{n-1}w_P \\leq\r\nn!\\cdot\\operatorname{vol}(P)$.
When $P$ is a triangle in the plane, we show\r\nequality: $v_Pw_P = 2\\operatorname{vol}(P)$.
These results are obtained by\r\nlooking at the set of standard monomials of the
vanishing ideal of $d\\cdot\r\nP\\cap\\mathbb Z^n$ and by applying the Bernstein--Kushnirenko
theorem. Finally,\r\nwe study irreducible Laurent polynomials that vanish with
large multiplicity at\r\na point. This work is inspired by questions about Seshadri
constants."
author:
- first_name: Fabian
full_name: Gundlach, Fabian
id: '100450'
last_name: Gundlach
citation:
ama: Gundlach F. Polynomials vanishing at lattice points in a convex set. arXiv:210705353.
Published online 2021.
apa: Gundlach, F. (2021). Polynomials vanishing at lattice points in a convex set.
In arXiv:2107.05353.
bibtex: '@article{Gundlach_2021, title={Polynomials vanishing at lattice points
in a convex set}, journal={arXiv:2107.05353}, author={Gundlach, Fabian}, year={2021}
}'
chicago: Gundlach, Fabian. “Polynomials Vanishing at Lattice Points in a Convex
Set.” ArXiv:2107.05353, 2021.
ieee: F. Gundlach, “Polynomials vanishing at lattice points in a convex set,” arXiv:2107.05353.
2021.
mla: Gundlach, Fabian. “Polynomials Vanishing at Lattice Points in a Convex Set.”
ArXiv:2107.05353, 2021.
short: F. Gundlach, ArXiv:2107.05353 (2021).
date_created: 2024-04-11T12:43:04Z
date_updated: 2024-04-11T12:50:48Z
extern: '1'
external_id:
arxiv:
- '2107.05353'
language:
- iso: eng
publication: arXiv:2107.05353
status: public
title: Polynomials vanishing at lattice points in a convex set
type: preprint
user_id: '100450'
year: '2021'
...