ИСТИНА

Let F: Cn→ Cn be a polynomial mapping of a complex space into itself. When is it reversible? A necessary condition is local invertibility at each point. The famous Jacobian problem states that this condition is sufficient. For more than 20 years, until 1968, the Jacobian problem was considered solved for n = 2, since then new “evidence” has appeared every few months. The Jacobian problem is closely related to the Dixmier conjecture, the formulation of which for n=1 looks innocent: let P, Q be polynomials in x and (d/dx), and PQ– QP=1. Is it true that (d/dx) can be expressed in terms of P and Q. This statement has not yet been proven. Recently it was possible to prove the equivalence of this statement to the Jacobian problem for n=2. The stable equivalence of the Jacobian and Dixmier conjectures is proven in the work http://arxiv.org/abs/math/0512171. The proof uses an analogy between classical and quantum objects. It is intended to give an elementary explanation of this analogy and also discuss Kontsevich’s hypotheses. Another, close, statement is called the Abiencar–Moch theorem and looks like an Olympiad problem (which it is). Let P, Q, R be polynomials, and R(P(x),Q(x))=x. Prove that either the degree of P divides the degree of Q, or vice versa.