×

Polynomial mappings defined by forms with a common factor. (English) Zbl 0778.12002

Let \(K\) be a field and let \(\widehat K\) be its algebraic closure. The authors consider the following finiteness property:
\((PO)\): Given \(m,n\in\mathbb{N}\), a \(K\)-homogeneous algebraic set \(V\subset\widehat K^ n\), a subset \({\mathcal X}\subset V\) and polynomial mappings \(F=(F_ 1,\ldots,F_ m):\widehat K^ n\to\widehat K^ m\), \(G=(G_ 1,\ldots,G_ m):\widehat K^ n\to\widehat K^ m\) defined by homogeneous polynomials \(F_ j\), \(G_ j\in K[X_ 1,\ldots,X_ n]\), with the following properties: (1) There exist homogeneous polynomials \(F_ 0\), \(F_ 1',\ldots,F_ m'\) defined on \(K\) such that \(F_ j=F_ 0F_ j'\) for each \(j\) and \(F_ 1',\ldots,F_ m'\) do not have a common nontrivial zero in \(V\); (2) \(\min\{\deg(F_ j);\;1\leq j\leq m\}>\max\{\deg(G_ j);\;1\leq j\leq m\}+2\deg(F_ 0);\) (3) \(\{[K(x):K];\;x\in{\mathcal X}\}\) is bounded: (4) \(F({\mathcal X})\supset G({\mathcal X})\); (5) \(G|_{\mathcal X}\) is injective. Then \({\mathcal X}\) is a finite set.
In this paper, it is proved in particular that global fields share the property \((PO)\) (see Theorem 1) and if every finite extension of a field \(K\) has the property \((PO)\) then (Theorem 2) every finitely generated extension field of \(K\) has the property \((PO)\). In the case \(F_ 0=1\) these results were established by the reviewer [Sur la stabilité rationnelle ou algébrique d’ensembles de nombres algébriques, Thesis, Univ. Aix-Marseille 2, 105 pp. (1975), see also C. R. Acad. Sci., Paris, Sér. A 274, 1836-1838 (1972; Zbl 0249.12103)]. The case \(F_ 0\neq 1\) was considered in a rather special case by the second author [Colloq. Math. 13, 101-106 (1964; Zbl 0132.008)]. The proofs use the theory of height functions on fields equipped with a family of absolute values satisfying the product formula.

MSC:

12E05 Polynomials in general fields (irreducibility, etc.)
11R09 Polynomials (irreducibility, etc.)
11C08 Polynomials in number theory
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[1] Halter-Koch, F., Narkiewicz, W., Finiteness properties of polynomial mappings, to appear. · Zbl 0779.12002
[2] Lang, S., Fundamentals of Diophantine Geometry, Springer1983. · Zbl 0528.14013
[3] Lang, S., Introduction to algebraic geometry, Interscience Publ.1958. · Zbl 0095.15301
[4] Lewis, D.J., Invariant sets of morphisms in projective and affine number spaces, J. Algebra20 (1972), 419-434. · Zbl 0245.12003
[5] Liardet, P., Sur les transformations polynomiales et rationnelles, Sém. Th. Nomb. Bordeaux exp. n° 29, 1971-1972. · Zbl 0273.12101
[6] Liardet, P., Sur une conjecture de W. Narkiewicz, C. R. Acad. Sc. Paris274 (1972), 1836-1838. · Zbl 0249.12103
[7] Narkiewicz, W., On transformations by polynomials in two variables, II, 13 (1964), 101-106. · Zbl 0132.00803
[8] Serre, J.-P., Lectures on the Mordell-Weil-Theorem, Aspects of Mathematics, Braun schweig1989. · Zbl 0676.14005
[9] Waerden, B. L. van der, Algebra, 2. Teil, Springer1967. · Zbl 0137.25403
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.