Order zeta functions of number rings and resolution of singularities

• Applicants: Professorin Dr. Anne Frühbis-Krüger; Professor Dr. Christopher Voll
• Subject Area: Mathematics
• Term: from 2017 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 373111162

• Final Report Year: 2022

Final Report Abstract

Based on our original project we did indeed increase our understanding of the complexity of cone integrals describing subring zeta functions of arithmetically motivated rings. Even though our initially envisaged final goal—the computation of the zeta function of the ring Zn for all n—proved unattainable for the time being, our efforts towards solving it paid off in several ways: the methods we devised allowed us to solve related, simpler counting problems and inspired related work on a number of topics, including parallel computing, resolution of singularities for binomial ideals, and the combinatorics of hyperplane arrangements. Covid 19 had a massive negative impact on the project, curtailing mutual visits and inhibiting research stays at conferences, workshops etc.

Publications

• Invariance of Hironaka’s characteristic polyhedron. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 113(4), 4145-4169.
  Cossart, Vincent; Jannsen, Uwe & Schober, Bernd
  
• Towards Massively Parallel Computations in Algebraic Geometry. Foundations of Computational Mathematics, 21(3), 767-806.
  Böhm, Janko; Decker, Wolfram; Frühbis-Krüger, Anne; Pfreundt, Franz-Josef; Rahn, Mirko & Ristau, Lukas
  
• Embedded desingularization for arithmetic surfaces – toward a parallel implementation. Mathematics of Computation, 90(330), 1957-1997.
  Frühbis-Krüger, Anne; Ristau, Lukas & Schober, Bernd
  
• Algorithmic local monomialization of a binomial: A comparison of different approaches. International Journal of Algebra and Computation, 33(01), 161-195.
  Gaube, Sabrina Alexandra & Schober, Bernd
  
• On the geometry of flag Hilbert–Poincaré series for matroids. Algebraic Combinatorics, 6(3), 623-638.
  Kühne, Lukas & Maglione, Joshua
  
• Flag Hilbert–Poincaré series and Igusa zeta functions of hyperplane arrangements. Israel Journal of Mathematics, 264(1), 177-233.
  Maglione, Joshua & Voll, Christopher