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The growth of regular functions on algebraic sets. (English) Zbl 0755.32022
Let \(V\subset\mathbb{C}^ N\) be an algebraic set of positive dimension and let \(f\in\mathbb{C}[V]\). Define \[ M(V,f)=\{B\geq 0:\;\exists A\geq 0,\quad\text{ such that } | f(z)|\leq A(1+| z|)^ B\quad\text{ for } z\in V\}, \]
\[ B(V,f)=\inf M(V,f)\quad\text{ and }B_ V=\{B(V,f):f\in\mathbb{C}[V]\}. \] The quantity \(B(V,f)\) is called the growth exponent of \(f\). The aim of this paper is to study the dependence of the structure of the set \(B_ V\) on the geometric properties of \(V\), in particular when \(V\) is a curve in \(\mathbb{C}^ N\) (section 5), and when \(V\) is a hypersurface (section 6). Before that, it is shown that neither the smallest number of generators, nor the number of denominators of irreductible ratios belonging to \(B_ V\) are invariants of biregular mappings of \(\mathbb{C}^ N\). However, if \(V\) has a one-dimensional polynomial parametrization, one can calculate \(B_ V\).

MSC:
32H30 Value distribution theory in higher dimensions
14N99 Projective and enumerative algebraic geometry
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
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