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Lang-Weil type bounds in finite difference fields
Multi angle
Authors : Hils, Martin (Author of the conference)
CIRM (Publisher )
03C13 - Finite structures [See also 68Q15, 68Q19]
03C20 - Ultraproducts and related constructions
03C60 - Model-theoretic algebra
11G25 - Varieties over finite and local fields
11U09 - Model theory, See also {03Cxx}
12L12 - Model theory
CIRM (Publisher )
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Abstract :
(joint work with Ehud Hrushovski, Jinhe Ye and Tingxiang Zou)
We prove Lang-Weil type bounds for the number of rational points of difference varieties over finite difference fields, in terms of the transformal dimension of the variety and assuming the existence of a smooth rational point. It follows that in (certain) non-principle ultraproducts of finite difference fields the course dimension of a quantifier free type equals its transformal tran-scendence degree.
The proof uses a strong form of the Lang-Weil estimates and, as key ingredi-ent to obtain equidimensional Frobenius specializations, the recent work of Dor and Hrushovski on the non-standard Frobenius acting on an algebraically closed non-trivially valued field, in particular the pure stable embeddedness of the residue difference field in this context.
Keywords : model theory; finite difference fields; Lang-Weil estimates; Frobenius specializations
MSC Codes : We prove Lang-Weil type bounds for the number of rational points of difference varieties over finite difference fields, in terms of the transformal dimension of the variety and assuming the existence of a smooth rational point. It follows that in (certain) non-principle ultraproducts of finite difference fields the course dimension of a quantifier free type equals its transformal tran-scendence degree.
The proof uses a strong form of the Lang-Weil estimates and, as key ingredi-ent to obtain equidimensional Frobenius specializations, the recent work of Dor and Hrushovski on the non-standard Frobenius acting on an algebraically closed non-trivially valued field, in particular the pure stable embeddedness of the residue difference field in this context.
Keywords : model theory; finite difference fields; Lang-Weil estimates; Frobenius specializations
03C13 - Finite structures [See also 68Q15, 68Q19]
03C20 - Ultraproducts and related constructions
03C60 - Model-theoretic algebra
11G25 - Varieties over finite and local fields
11U09 - Model theory, See also {03Cxx}
12L12 - Model theory
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Information on the Video
Film maker : Petit, JeanLanguage : English Available date : 23/06/2023 Conference Date : 30/05/2023 Subseries : Research talks arXiv category : Logic ; Commutative Algebra Mathematical Area(s) : Algebra ; Logic and Foundations Format : MP4 (.mp4) - HD Video Time : 01:07:10 Targeted Audience : Researchers ; Graduate Students ; Doctoral Students, Post-Doctoral Students Download : https://videos.cirm-math.fr/2023_05-30_Hils_1.mp4 
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Information on the Event
Event Title : Model theory of valued fields / Théorie des modèles des corps valuésEvent Organizers : Chatzidakis, Zoé ; Jahnke, Franziska ; Rideau-Kikuchi, Silvain Dates : 29/05/2023 - 02/06/2023 Event Year : 2023 Event URL : https://conferences.cirm-math.fr/2761.html
Citation Data
DOI : 10.24350/CIRM.V.20051803Cite this video as: Hils, Martin (2023). Lang-Weil type bounds in finite difference fields. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.20051803 URI : http://dx.doi.org/10.24350/CIRM.V.20051803 |
See Also
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Bibliography
- CAFURE, Antonio et MATERA, Guillermo. Improved explicit estimates on the number of solutions of equations over a finite field. Finite Fields and Their Applications, 2006, vol. 12, no 2, p. 155-185. - https://doi.org/10.1016/j.ffa.2005.03.003
- DOR, Yuval et HRUSHOVSKI, Ehud. Specialization of Difference Equations and High Frobenius Powers. arXiv preprint arXiv:2212.05366, 2022. - https://doi.org/10.48550/arXiv.2212.05366
- HRUSHOVSKI, Ehud. The elementary theory of the Frobenius automorphisms. arXiv preprint math/0406514, 2004. - https://doi.org/10.48550/arXiv.math/0406514
- ZOU, Tingxiang. Pseudofinite difference fields and counting dimensions. Journal of Mathematical Logic, 2021, vol. 21, no 01, p. 2050022. - https://doi.org/10.1142/S0219061320500221
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