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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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