Estimation of the Number of the Critical Values at Infinity of a Polynomial Function $f:\mathbf C^2 → \mathbf C$

Abstract

is not necessarily a locally trivial fibration. In general, we have to exclude a finite values Z^ c C from the base space so that / : C f~ (L u Z^ ) — > C (Z u Z^ ) is a locally trivial fibration. We say that r e C is a regular value at infinity of the function / : C — > C if there exist positive numbers R and 8 so that the restriction of/, f:f-(D£(T))-B*^>D£(T),is a trivial fibration over the disc D£(r) where De(T) = {r]£C\\ri-i\<£} and B*=[(x,y);x 2 =\y <R}. Otherwise ris a called a critical value at infinity or an atypical value. We denote the set of the critical values at infinity by Z^. It is known that Z^ is finite ([V], [HI]). This fact also results from Theorem (1.4). The purpose of this note is to give an estimation on the number of critical values at infinity.