ИСТИНА

Let $F: \mathbb{C}^n\to \mathbb{C}^n$~- be a polynomial mapping of a complex space into itself. When is it invertible? The necessary condition is local invertibility at every point. The famous {\em Jacobian conjecture} states that this condition is sufficient. For more than 20 years, until 1968, the Jacobian conjecture was considered solved for $n=2$, since then new ``proofs'' appear every few months. The {\em Jacobian conjecture} is closely related to Dixmier's 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 claim has not yet been proven. Recently it was possible to prove that this assertion is equivalent to the Jacobian problem for $n=2$. The stable equivalence of the Jacobian and Dixmier conjectures was proved by Tsuchimoto and Belov and Kontsevich in http://arxiv.org/abs/math/0512171. The proof uses an analogy between classical and quantum objects. It is supposed to give an elementary explanation of this analogy and also to discuss Kontsevich's hypotheses. Another, close, statement is called the Abhyankar –Moh theorem and looks like an Olympiad problem. 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. The {\em Abhyankar problem} states that all embeddings of an affine line in $\mathbb{C}^3$ are isomorphic. Over $\mathbb{R}$ the answer is no -- there are polynomial knots, so Abyankar's problem can be viewed as a way to define a knot in an abstract algebraic way. Significant advances in the problem of the Jacobian were made by the remarkable mathematician A.Yagzhev (a student of L.A. Bokut) by linking the topic with universal algebra, the review of Alexei Belov, Leonid Bokut, Louis Rowen, Jie-Tai Yu, ``The Jacobian Conjecture, Together with Specht and Burnside-Type Problems'', Automorphisms in Birational and Affine Geometry (Bellavista Relax Hotel, Levico Terme –Trento, October 29th – November 3rd, 2012, Italy), Springer Proceedings in Mathematics & Statistics, 79, Springer Verlag, 2014, 249– 285 http://link.springer.com/chapter/10.1007/978-3-319-05681-4_15, arXiv: 1308.0674 Join our Cloud HD Video Meeting https://us02web.zoom.us The Jacobian Conjecture is stably equivalent to the Dixmier Conjecture http://arxiv.org