Summary: Problems to the Seminar : 15 january 2002
Vladimir ARNOLD #
CEREMADE ­ UMR C.N.R.S. 7534
Universit’e Paris 9 ­ Dauphine
Place du Mar’echal de Lattre de Tassigny
75775 Paris Cedex 16 ­ France
1. Let f : IR 2
# IR be a polynomial of degree D. Find the maximal
number of the connected components and the maximal number of closed
components of the parabolic curve of its graph (where f xx f yy = f 2
xy ) :
b 0 (Par(f)) = ?, b 1 (Par(f)) = ?
Even for D = 4, it is not known whether b 1 attaines the value 4, and the
constants C in the lower and upper bounds for large degrees D, b 1 # CD 2 ,
di#er 4 times :
(D - 1)(D - 2)/2 # b # (2D - 5)(D - 3) + 1 .
2. Let M # IRP 3 be a a smooth algebraic surface of degree D. Find the
maximal number of the connectec components of its parabolic line.
The constants C in the lower and in the upper bounds CD 3 di#er 20
times :

Source: Arnold, Vladimir Igorevich - Steklov Mathematical Institute

Collections: Mathematics