Geometric degree of finite extensions of mappings from a smooth variety

Let f:V→Wf:V→W be a finite polynomial mapping of algebraic subsets V,WV,W of kn and km, respectively, with n≤mn≤m. It is known that ff can be extended to a finite polynomial mapping F:kn→km. Moreover, it is known that, if V,WV,W are smooth of dimension k,4k+2≤n=mk,4k+2≤n=m, and ff is dominated on every component (without vertical components) then there exists a finite polynomial extension F:kn→kn such that gdegF≤(gdegf)2k+1, where gdegh means the number of points in the generic fiber of hh. In this note we improve this result. Namely we show that there exists a finite polynomial extension F:kn→kn such that gdegF≤(gdegf)k+1.