\r\n Let\r\n
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.
"}],"status":"public","_id":"64181","user_id":"100450","language":[{"iso":"eng"}]},{"status":"public","abstract":[{"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.","lang":"eng"}],"type":"preprint","publication":"arXiv:2502.18207","language":[{"iso":"eng"}],"user_id":"100450","_id":"58852","external_id":{"arxiv":["2502.18207"]},"citation":{"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} }","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).","ieee":"F. Gundlach and B. F. Seguin, “Counting two-step nilpotent wildly ramified extensions of function fields,” arXiv:2502.18207. 2025.","chicago":"Gundlach, Fabian, and Beranger Fabrice Seguin. “Counting Two-Step Nilpotent Wildly Ramified Extensions of Function Fields.” ArXiv:2502.18207, 2025.","ama":"Gundlach F, Seguin BF. Counting two-step nilpotent wildly ramified extensions of function fields. arXiv:250218207. Published online 2025."},"year":"2025","title":"Counting two-step nilpotent wildly ramified extensions of function fields","date_created":"2025-02-26T08:51:57Z","author":[{"last_name":"Gundlach","id":"100450","full_name":"Gundlach, Fabian","first_name":"Fabian"},{"first_name":"Beranger Fabrice","last_name":"Seguin","full_name":"Seguin, Beranger Fabrice","id":"102487"}],"date_updated":"2025-02-26T08:53:08Z"},{"status":"public","abstract":[{"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.","lang":"eng"}],"publication":"arXiv:2507.16138","type":"preprint","language":[{"iso":"eng"}],"user_id":"100450","external_id":{"arxiv":["2507.16138"]},"_id":"62308","citation":{"short":"T.C. Anderson, U.V. Asarhasa, A. Bertelli, F. Gundlach, E.M. O’Dorney, ArXiv:2507.16138 (2025).","mla":"Anderson, Theresa C., et al. “The Structure of the Double Discriminant.” ArXiv:2507.16138, 2025.","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} }","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.","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.","ama":"Anderson TC, Asarhasa UV, Bertelli A, Gundlach F, O’Dorney EM. The structure of the double discriminant. arXiv:250716138. Published online 2025."},"year":"2025","title":"The structure of the double discriminant","date_created":"2025-11-26T08:18:33Z","author":[{"last_name":"Anderson","full_name":"Anderson, Theresa C.","first_name":"Theresa C."},{"first_name":"Ufuoma V.","full_name":"Asarhasa, Ufuoma V.","last_name":"Asarhasa"},{"last_name":"Bertelli","full_name":"Bertelli, Adam","first_name":"Adam"},{"first_name":"Fabian","last_name":"Gundlach","id":"100450","full_name":"Gundlach, Fabian"},{"first_name":"Evan M.","full_name":"O'Dorney, Evan M.","last_name":"O'Dorney"}],"date_updated":"2025-11-26T08:21:27Z"},{"year":"2024","citation":{"apa":"Gundlach, F. (2024). Sampling cubic rings. In arXiv:2405.13734.","bibtex":"@article{Gundlach_2024, title={Sampling cubic rings}, journal={arXiv:2405.13734}, author={Gundlach, Fabian}, year={2024} }","mla":"Gundlach, Fabian. “Sampling Cubic Rings.” ArXiv:2405.13734, 2024.","short":"F. Gundlach, ArXiv:2405.13734 (2024).","ieee":"F. Gundlach, “Sampling cubic rings,” arXiv:2405.13734. 2024.","chicago":"Gundlach, Fabian. “Sampling Cubic Rings.” ArXiv:2405.13734, 2024.","ama":"Gundlach F. Sampling cubic rings. arXiv:240513734. Published online 2024."},"date_updated":"2024-05-24T09:14:58Z","author":[{"first_name":"Fabian","id":"100450","full_name":"Gundlach, Fabian","last_name":"Gundlach"}],"date_created":"2024-05-24T08:59:54Z","title":"Sampling cubic rings","publication":"arXiv:2405.13734","type":"preprint","abstract":[{"lang":"eng","text":"We explain how to construct a uniformly random cubic integral domain $S$ of\r\ngiven signature with $|\\text{disc}(S)| \\leq T$ in expected time $\\widetilde\r\nO(\\log T)$."}],"status":"public","external_id":{"arxiv":["2405.13734"]},"_id":"54442","user_id":"100450","language":[{"iso":"eng"}]},{"abstract":[{"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.","lang":"eng"}],"status":"public","publication":"arXiv:2407.08438","type":"preprint","language":[{"iso":"eng"}],"external_id":{"arxiv":["2407.08438"]},"_id":"55192","user_id":"82981","year":"2024","citation":{"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} }","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).","apa":"Gundlach, F., & Klüners, J. (2024). Symmetries of power-free integers in number fields and their shift spaces. In arXiv:2407.08438.","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.","ama":"Gundlach F, Klüners J. Symmetries of power-free integers in number fields and their shift spaces. arXiv:240708438. Published online 2024."},"title":"Symmetries of power-free integers in number fields and their shift spaces","date_updated":"2024-07-12T08:19:11Z","author":[{"first_name":"Fabian","last_name":"Gundlach","id":"100450","full_name":"Gundlach, Fabian"},{"last_name":"Klüners","full_name":"Klüners, Jürgen","id":"21202","first_name":"Jürgen"}],"date_created":"2024-07-12T08:16:37Z"},{"_id":"53421","external_id":{"arxiv":["2211.16698"]},"user_id":"100450","extern":"1","language":[{"iso":"eng"}],"type":"preprint","publication":"arXiv:2211.16698","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 $\\operatorname{inv}_1,\\dots,\\operatorname{inv}_m$."}],"status":"public","date_updated":"2024-04-11T12:50:44Z","author":[{"first_name":"Fabian","full_name":"Gundlach, Fabian","id":"100450","last_name":"Gundlach"}],"date_created":"2024-04-11T12:43:14Z","title":"Malle's conjecture with multiple invariants","year":"2022","citation":{"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} }","mla":"Gundlach, Fabian. “Malle’s Conjecture with Multiple Invariants.” ArXiv:2211.16698, 2022.","short":"F. Gundlach, ArXiv:2211.16698 (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.","ama":"Gundlach F. Malle’s conjecture with multiple invariants. arXiv:221116698. Published online 2022."}},{"date_created":"2024-04-11T12:43:04Z","author":[{"first_name":"Fabian","last_name":"Gundlach","full_name":"Gundlach, Fabian","id":"100450"}],"date_updated":"2024-04-11T12:50:48Z","title":"Polynomials vanishing at lattice points in a convex set","citation":{"apa":"Gundlach, F. (2021). Polynomials vanishing at lattice points in a convex set. In arXiv:2107.05353.","mla":"Gundlach, Fabian. “Polynomials Vanishing at Lattice Points in a Convex Set.” ArXiv:2107.05353, 2021.","short":"F. Gundlach, ArXiv:2107.05353 (2021).","bibtex":"@article{Gundlach_2021, title={Polynomials vanishing at lattice points in a convex set}, journal={arXiv:2107.05353}, author={Gundlach, Fabian}, year={2021} }","ama":"Gundlach F. Polynomials vanishing at lattice points in a convex set. arXiv:210705353. Published online 2021.","ieee":"F. Gundlach, “Polynomials vanishing at lattice points in a convex set,” arXiv:2107.05353. 2021.","chicago":"Gundlach, Fabian. “Polynomials Vanishing at Lattice Points in a Convex Set.” ArXiv:2107.05353, 2021."},"year":"2021","user_id":"100450","external_id":{"arxiv":["2107.05353"]},"_id":"53420","language":[{"iso":"eng"}],"extern":"1","type":"preprint","publication":"arXiv:2107.05353","status":"public","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."}]},{"language":[{"iso":"eng"}],"extern":"1","_id":"53424","user_id":"100450","status":"public","type":"dissertation","title":"Parametrizing extensions with fixed Galois group","date_updated":"2024-04-11T12:46:20Z","author":[{"full_name":"Gundlach, Fabian","id":"100450","last_name":"Gundlach","first_name":"Fabian"}],"date_created":"2024-04-11T12:45:34Z","year":"2019","citation":{"chicago":"Gundlach, Fabian. Parametrizing Extensions with Fixed Galois Group, 2019.","ieee":"F. Gundlach, Parametrizing extensions with fixed Galois group. 2019.","ama":"Gundlach F. Parametrizing Extensions with Fixed Galois Group.; 2019.","short":"F. Gundlach, Parametrizing Extensions with Fixed Galois Group, 2019.","mla":"Gundlach, Fabian. Parametrizing Extensions with Fixed Galois Group. 2019.","bibtex":"@book{Gundlach_2019, title={Parametrizing extensions with fixed Galois group}, author={Gundlach, Fabian}, year={2019} }","apa":"Gundlach, F. (2019). Parametrizing extensions with fixed Galois group."}},{"year":"2014","citation":{"bibtex":"@book{Gundlach_2014, title={Del Pezzo Surface Fibrations of Degree 4}, author={Gundlach, Fabian}, year={2014} }","mla":"Gundlach, Fabian. Del Pezzo Surface Fibrations of Degree 4. 2014.","short":"F. Gundlach, Del Pezzo Surface Fibrations of Degree 4, 2014.","apa":"Gundlach, F. (2014). Del Pezzo Surface Fibrations of Degree 4.","chicago":"Gundlach, Fabian. Del Pezzo Surface Fibrations of Degree 4, 2014.","ieee":"F. Gundlach, Del Pezzo Surface Fibrations of Degree 4. 2014.","ama":"Gundlach F. Del Pezzo Surface Fibrations of Degree 4.; 2014."},"date_updated":"2024-04-11T12:46:21Z","date_created":"2024-04-11T12:45:07Z","author":[{"last_name":"Gundlach","full_name":"Gundlach, Fabian","id":"100450","first_name":"Fabian"}],"title":"Del Pezzo Surface Fibrations of Degree 4","type":"mastersthesis","status":"public","_id":"53423","user_id":"100450","extern":"1","language":[{"iso":"eng"}]},{"publisher":"Wiley","date_created":"2024-04-11T12:41:31Z","title":"Integral Brauer-Manin obstructions for sums of two squares and a power","issue":"2","year":"2013","keyword":["General Mathematics"],"language":[{"iso":"eng"}],"publication":"Journal of the London Mathematical Society","date_updated":"2024-04-11T12:50:38Z","author":[{"first_name":"Fabian","full_name":"Gundlach, Fabian","id":"100450","last_name":"Gundlach"}],"volume":88,"doi":"10.1112/jlms/jdt042","publication_status":"published","publication_identifier":{"issn":["0024-6107"]},"citation":{"apa":"Gundlach, F. (2013). Integral Brauer-Manin obstructions for sums of two squares and a power. Journal of the London Mathematical Society, 88(2), 599–618. https://doi.org/10.1112/jlms/jdt042","bibtex":"@article{Gundlach_2013, title={Integral Brauer-Manin obstructions for sums of two squares and a power}, volume={88}, DOI={10.1112/jlms/jdt042}, number={2}, journal={Journal of the London Mathematical Society}, publisher={Wiley}, author={Gundlach, Fabian}, year={2013}, pages={599–618} }","mla":"Gundlach, Fabian. “Integral Brauer-Manin Obstructions for Sums of Two Squares and a Power.” Journal of the London Mathematical Society, vol. 88, no. 2, Wiley, 2013, pp. 599–618, doi:10.1112/jlms/jdt042.","short":"F. Gundlach, Journal of the London Mathematical Society 88 (2013) 599–618.","ama":"Gundlach F. Integral Brauer-Manin obstructions for sums of two squares and a power. Journal of the London Mathematical Society. 2013;88(2):599-618. doi:10.1112/jlms/jdt042","ieee":"F. Gundlach, “Integral Brauer-Manin obstructions for sums of two squares and a power,” Journal of the London Mathematical Society, vol. 88, no. 2, pp. 599–618, 2013, doi: 10.1112/jlms/jdt042.","chicago":"Gundlach, Fabian. “Integral Brauer-Manin Obstructions for Sums of Two Squares and a Power.” Journal of the London Mathematical Society 88, no. 2 (2013): 599–618. https://doi.org/10.1112/jlms/jdt042."},"intvolume":" 88","page":"599-618","_id":"53419","user_id":"100450","extern":"1","type":"journal_article","status":"public"},{"user_id":"100450","_id":"53422","language":[{"iso":"eng"}],"extern":"1","type":"bachelorsthesis","status":"public","date_created":"2024-04-11T12:44:37Z","author":[{"first_name":"Fabian","id":"100450","full_name":"Gundlach, Fabian","last_name":"Gundlach"}],"date_updated":"2024-04-11T12:46:22Z","title":"Brauer-Manin obstructions for sums of two squares and a power","citation":{"apa":"Gundlach, F. (2012). Brauer-Manin obstructions for sums of two squares and a power.","mla":"Gundlach, Fabian. Brauer-Manin Obstructions for Sums of Two Squares and a Power. 2012.","bibtex":"@book{Gundlach_2012, title={Brauer-Manin obstructions for sums of two squares and a power}, author={Gundlach, Fabian}, year={2012} }","short":"F. Gundlach, Brauer-Manin Obstructions for Sums of Two Squares and a Power, 2012.","ama":"Gundlach F. Brauer-Manin Obstructions for Sums of Two Squares and a Power.; 2012.","chicago":"Gundlach, Fabian. Brauer-Manin Obstructions for Sums of Two Squares and a Power, 2012.","ieee":"F. Gundlach, Brauer-Manin obstructions for sums of two squares and a power. 2012."},"year":"2012"}]