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.


12E05 Polynomials in general fields (irreducibility, etc.)
11R09 Polynomials (irreducibility, etc.)
11C08 Polynomials in number theory
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