Nonstandard arithmetic of Hilbert subsets
type: Item
instance of: scholarly article
title: Nonstandard arithmetic of Hilbert subsets
zbMATH Open document ID: 0733.03045
DOI: 10.1016/0168-0072(91)90046-O
author: Masahiro Yasumoto
published in: Annals of Pure and Applied Logic
publication date: 1991-01-01T00:00:00Z
review text: For an irreducible polynomial f(X,Y) in R[X,Y], J(f) denotes the set of those r in R for which f(r,Y) is reducible. Thus, in the case of \(R={\mathbb{Z}}\), J(f) is the complement of the Hilbert subset of \({\mathbb{Z}}\) corresponding to f. Hilbert's irreducibility theorem says that all Hilbert subsets of \({\mathbb{Z}}\) are infinite. The paper discusses the problem of the size of J(f). In the case of \(R={\mathbb{Z}}\), a sufficient condition for finiteness of J(f) is given, the condition also yields an upper bound for J(f) in terms of the coefficients and the degree of f. In the case of \(R=K[t]\) and an irreducible f in K[X,Y,t], where K is a field and t is transcendental over K, an analogous condition is used to obtain an upper bound for the degree of those \(\phi\) (t) in K[t] for which f(\(\phi\) (t),Y,t) is reducible. Both results are proved by Robinson's methods of nonstandard arithmetic.
Mathematics Subject Classification ID: 03H15
Mathematics Subject Classification ID: 11U10
Mathematics Subject Classification ID: 12L15
zbMATH DE Number: 4212039
zbMATH Keywords: Hilbert sets
zbMATH Keywords: irreducible polynomial
zbMATH Keywords: Hilbert's irreducibility theorem
zbMATH Keywords: nonstandard arithmetic
reviewed by: Roman Kossak
MaRDI profile type: Publication
cites work: The Distribution of Galois Groups and Hilbert's Irreducibility Theorem
cites work: On the finiteness theorem of Siegel and Mahler concerning diophantine equations
cites work: Q4075557
cites work: Nonstandard arithmetic of function fields over H-convex subfields of *Q.
cites work: Nonstandard Arithmetic of Polynomial Rings
cites work: Algebraic extensions in nonstandard models and Hilbert's irreducibility theorem
Wikidata QID: Q127706188
description: scientific article; zbMATH DE number 4212039