Abstract
In this short note, we survey some degree and height bound results for arithmetic Nullstellensatz from the literature. We also introduce the notion of height functions, ultraproducts and nonstandard extensions. As our main remark, we find height bounds for polynomial rings over integral domains via nonstandard methods.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
M. Aschenbrenner, Ideal membership in polynomial rings over the integers, J. Amer. Math. Soc. 17 (2004), 407–441.
E. Bombieri, W. Gubler, Heights in Diophantine Geometry, Cambridge University Press; 1 edition (September 24, 2007).
L. van den Dries and K. Schmidt, Bounds in the theory of polynomial rings over fields. A nonstandard approach, Inventiones Math. 76 (1984), 77–91.
M. Giusti, J. Heintz, J. Sabia, On the efficiency of effective Nullstellensätze, Comput. Complexity 3 (1993), 56–95.
R. Goldblatt, Lectures on the Hyperreals, A Introduction to Nonstandard Analysis Springer-Verlag, New York, 1998.
H. Göral, Model Theory of Fields and Heights, Ph.D thesis, Lyon, 2015.
H. Göral, Height Bounds, Nullstellensatz and Primality, to appear in Comm. in Algebra, 46 (2018), no. 10, 4463–4472.
C. W. Henson, Foundation of Nonstandard Analysis, A Gentle Introduction to Nonstandard Extensions, Lecture Notes.
G. Hermann Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Math. Ann. 95 (1926), 736–788.
M. Hindry, J.H. Silverman, Diophantine Geometry, An Introduction, Springer-Verlag, 2000.
J. Kollár, Sharp Effective Nullstellensatz, J. Amer. Math. Soc. Volume 1, Number 4, 1988.
T. Krick, L.M. Pardo, M. Sombra Sharp estimates for the arithmetic Nullstellensatz, Duke Math. J. 109 (2001), no. 3, 521–598.
D. Marker, Model Theory: An Introduction, Springer-Verlag, New York, 2002.
P. Philippon, Dénominateurs dans le théorème des zeros de Hilbert, Acta Arith. 58 (1990), 1–25.
A. Robinson, Théorie métamathématiques des idéaux, collection de logique mathématiques, Dactyl-offset. Gauthier-Villars, E. Nauwelaerts, Ser A, Paris-Louvain, 1955.
A. Robinson, Some problems of definability in the lower predicate calculus, J. Symbolic Logic, Volume 25, Issue 2 (1960), 171.
A. Seidenberg, Constructions in algebra, Trans. AMS 197, 273–313 (1974).
M. Sombra, Sparse Effective Nullstellensatz, Advances in Applied Mathematics, Volume 22, Issue 2, February 1999, Pages 271–295.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Göral, H. (2019). Nullstellensatz via Nonstandard Methods. In: Inam, I., Büyükaşık, E. (eds) Notes from the International Autumn School on Computational Number Theory. Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-12558-5_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-12558-5_5
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-12557-8
Online ISBN: 978-3-030-12558-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)