CLASSIFICATION OF POLYNOMIALS FROM C

CLASSIFICATION OF POLYNOMIALS FROM C 2 TO C WITH ONE CRITICAL VALUE ARNAUD BODIN 1. Introdu tion Let f : C 2 ! C be a polynomial map. The bifur ation set B is the minimal set of points of C su h that f : C 2 n f 1 (B) ! C n B is a lo ally trivial bration. We an des ribe B as follows: let B a = f(x; y) j grad f (x; y) = (0; 0) be the set of aÆne riti al values. The set B a is a subset of B but is not ne essarily equal to B. The value 2 C is regular at innity if there exists a disk D entered at and a ompa t set K of C 2 with a lo ally trivial bration f : f 1 (D) nK ! D. There is only a nite number of non-regular values at innity: the riti al values at innity olle ted in B 1 . The bifur ation set B is now: B = B a [ B 1 : For 2 C , we denote the ber f 1 ( ) by F .

• redu ed polynomial


• no riti


• polynomial map


• has genus


• indu ed


• topologi al las


• map oming


• situation has