## 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

### Citations:

Zbl 0249.12103; Zbl 0132.008
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### References:

  Halter-Koch, F., Narkiewicz, W., Finiteness properties of polynomial mappings, to appear. · Zbl 0779.12002  Lang, S., Fundamentals of Diophantine Geometry, Springer1983. · Zbl 0528.14013  Lang, S., Introduction to algebraic geometry, Interscience Publ.1958. · Zbl 0095.15301  Lewis, D.J., Invariant sets of morphisms in projective and affine number spaces, J. Algebra20 (1972), 419-434. · Zbl 0245.12003  Liardet, P., Sur les transformations polynomiales et rationnelles, Sém. Th. Nomb. Bordeaux exp. n° 29, 1971-1972. · Zbl 0273.12101  Liardet, P., Sur une conjecture de W. Narkiewicz, C. R. Acad. Sc. Paris274 (1972), 1836-1838. · Zbl 0249.12103  Narkiewicz, W., On transformations by polynomials in two variables, II, 13 (1964), 101-106. · Zbl 0132.00803  Serre, J.-P., Lectures on the Mordell-Weil-Theorem, Aspects of Mathematics, Braun schweig1989. · Zbl 0676.14005  Waerden, B. L. van der, Algebra, 2. Teil, Springer1967. · Zbl 0137.25403
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