A Note on Geometric Degree of Finite Extensions of Mappings from a Smooth Variety

Marek Karaś ¹

¹ Instytut Matematyki Uniwersytet Jagielloński Reymonta 4 30-059 Kraków, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 56 (2008) no. 2, pp. 105-108

Résumé

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.

Détail

DOI : 10.4064/ba56-2-2

Keywords: rightarrow finite polynomial mapping algebraic subsets mathbf mathbf respectively leq kwieci ski pure appl algebra proved there exists finite polynomial mapping mathbf rightarrow mathbf note prove subset mathbf smooth dimension leq rightarrow finite dominated dominated every component there exists finite polynomial mapping mathbf rightarrow mathbf mathop gdeg leq mathop gdeg improves earlier results author