A Survey of Local–Global Methods for Hilbert’s Tenth Problem

Sylvy Anscombe; Valentijn Karemaker; Zeynep Kisakürek; Vlerë Mehmeti; Margherita Pagano; Laura Paladino

doi:10.1007/978-3-031-52163-8_2

A Survey of Local–Global Methods for Hilbert’s Tenth Problem

Sylvy Anscombe^*, Valentijn Karemaker, Zeynep Kisakürek, Vlerë Mehmeti, Margherita Pagano, Laura Paladino

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Chapter › Academic › peer-review

Abstract

Hilbert’s Tenth Problem (H10) for a ring R asks for an algorithm to decide correctly, for each f∈ℤ[X₁,…,X_n], whether the diophantine equation f(X₁,…,X_n)=0 has a solution in R. The celebrated ‘Davis–Putnam–Robinson–Matiyasevich theorem’ shows that H10 for ℤ is unsolvable, i.e., there is no such algorithm. Since then, Hilbert’s Tenth Problem has been studied in a wide range of rings and fields. Most importantly, for number fields and in particular for ℚ, H10 is still an unsolved problem. A recent work of Eisenträger, Poonen, Koenigsmann, Park, Dittmann, Daans, and others has dramatically pushed forward what is known in this area and has made essential use of local–global principles for quadratic forms and for central simple algebras. We give a concise survey and introduction to this particular rich area of interaction between logic and number theory, without assuming a detailed background of either subject. We also sketch two further directions of future research, one inspired by model theory and one by arithmetic geometry.

Original language	English
Title of host publication	Women in Numbers Europe IV
Publisher	Springer
Pages	29-61
Number of pages	33
ISBN (Electronic)	978-3-031-52163-8
ISBN (Print)	978-3-031-52162-1
DOIs	https://doi.org/10.1007/978-3-031-52163-8_2
Publication status	Published - 6 Mar 2024

Publication series

Name	Association for Women in Mathematics Series
Volume	32
ISSN (Print)	2364-5733
ISSN (Electronic)	2364-5741

Bibliographical note

Publisher Copyright:
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.

Funding

This collaboration started at the workshop Women in Numbers Europe 4 which took place in Utrecht, the Netherlands in August 2022. S. A. was supported by GeoMod AAPG2019 (ANR-DFG). V. K. was supported by the Dutch Research Council (NWO) through grant VI.Veni.192.038. Z. K. was partially supported by the research training group \u2018GRK 2240: Algebro-Geometric Methods in Algebra, Arithmetic and Topology\u2019, funded by the DFG. L. P. is a member of INdAM-GNSAGA. We thank the referees for helpful comments. Acknowledgments This collaboration started at the workshop Women in Numbers Europe 4 which took place in Utrecht, the Netherlands in August 2022. S. A. was supported by GeoMod AAPG2019 (ANR-DFG). V. K. was supported by the Dutch Research Council (NWO) through grant VI.Veni.192.038. Z. K. was partially supported by the research training group \u2018GRK 2240: Algebro-Geometric Methods in Algebra, Arithmetic and Topology\u2019, funded by the DFG. L. P. is a member of INdAM-GNSAGA. We thank the referees for helpful comments.

Funders	Funder number
ANR-DFG
Deutsche Forschungsgemeinschaft
INdAM-GNSAGA
Nederlandse Organisatie voor Wetenschappelijk Onderzoek	VI.Veni.192.038

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978-3-031-52163-8_2Final published version, 987 KBLicence: Taverne

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A Survey of Local–Global Methods for Hilbert’s Tenth Problem

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