A Note on Geometric Degree of Finite Extensions of Mappings from a Smooth Variety
Volume 56 / 2008
Bulletin Polish Acad. Sci. Math. 56 (2008), 105-108
MSC: 14Rxx, 14R10.
DOI: 10.4064/ba56-2-2
Abstract
Let $f:V\rightarrow W$ be a finite polynomial mapping of algebraic subsets $V,W$ of $\mathbf{k}^{n}$ and $\mathbf{k}^{m},$ respectively, with $n\leq m.$ Kwieciński [J. Pure Appl. Algebra 76 (1991)] proved that there exists a finite polynomial mapping $F:\mathbf{k}^{n}\rightarrow \mathbf{k}^{m}$ such that $F|_{V}=f.$ In this note we prove that, if $V,W\subset \mathbf{k}^{n}$ are smooth of dimension $k$ with $3k+2\leq n,$ and $f:V\rightarrow W$ is finite, dominated and dominated on every component, then there exists a finite polynomial mapping $F: \mathbf{k}^{n}\rightarrow \mathbf{k}^{n}$ such that $F|_{V}=f$ and $\mathop{\rm gdeg}F\leq (\mathop{\rm gdeg}f)^{k+1}.$ This improves earlier results of the author.
