ИСТИНА

The talk is devoted to the famous Jacobian conjecture: JCnJCn: Has any polynomial mapping of Cn→CnCn→Cn with constant Jacobian a polynomial inverse? Diximer conjecture (DCn)(DCn): End(Wn)=Aut(Wn)End(Wn)=Aut(Wn), where Wn=C[x1,…,xn,∂x1,…,∂xn]Wn=C[x1,…,xn,∂x1,…,∂xn]. It was well known that DCnDCn implies JCnJCn. Recently, together with Kontzevich, the author proved that JC2nJC2n implies DCnDCn. This is related to Kontzevich conjecture, saying that Aut(Wn)Aut(Wn) is isomorphic to the group of polynomial symplectomorphisms of C2nC2n. These questions are related to describing aut(aut(Wn))aut(aut(Wn)). Recently author proved that the group of algebraic Aut(Aut(Cn))Aut(Aut(Cn)) contains only inner automorphisms.