Proof of Two Dimensional Jacobian Conjecture

Polynomial Retracts and the Jacobian Conjecture

Let $ K[x, y]$ be the polynomial algebra in two variables over a field $K$ of characteristic $0$. A subalgebra $R$ of $K[x, y]$ is called a retract if there is an idempotent homomorphism (a {\it retraction}, or {\it projection}) $φ: K[x, y] \to K[x, y]$ such that $φ(K[x, y]) = R$. The presence of other, equivalent, definitions of retracts provides several different methods of studying them, and brings together ideas from combinatorial algebra, homological algebra, and algebraic geometry. In this paper, we characterize all the retracts of $ K[x, y]$ up to an automorphism, and give several appli…

On the equivalence of the Jacobian, Dixmier and Poisson Conjectures in any characteristic

Thanks to recent results on ring homomorphisms of Azumaya algebras and to the following ones about endomorphisms of canonical Poisson algebras and Dirac quantum algebras, and about the reformulation in positive characteristic of these conjectures in characteristic zero, we prove the equivalence of these three conjectures in any characteristic, giving also by this way a new proof of the equivalence of the complex version of the two first conjectures recently proved by Y. Tsuchimoto.

截断多项式自同构的逆极限及其与 Jacobian 猜想相关的应用

Inverse limits of automorphisms of truncated polynomials and applications related to Jacobian conjecture

本文利用特征 $p$ 多项式环与逆极限技术，证明满足 Jacobian 条件且保持极大理想的多项式环同态必为自同构，为 Jacobian 猜想提供新结论。

Involutions and the Jacobian conjecture

The famous Jacobian conjecture asks if an endomorphism $f$ of $K[x,y]$ ($K$ is a characteristic zero field) having a non-zero scalar Jacobian is invertible. Let $α$ be the exchange involution on $K[x,y]$: $α(x)= y$ and $α(y)= x$. An $α$-endomorphism $f$ of $K[x,y]$ is an endomorphism of $K[x,y]$ that preserves the involution $α$: $f α= αf$. It was shown that if $f$ is an $α$-endomorphism of $K[x,y]$ having a non-zero scalar Jacobian, then $f$ is invertible. Based on this, we bring more results that imply that a given endomorphism $f$ having a non-zero scalar Jacobian and additional conditions …

The Jacobian Conjecture as a Problem of Perturbative Quantum Field Theory

The Jacobian conjecture is an old unsolved problem in mathematics, which has been unsuccessfully attacked from many different angles. We add here another point of view pertaining to the so called formal inverse approach, that of perturbative quantum field theory.

A differential equation for polynomials related to the Jacobian conjecture

We analyze a possible minimal counterexample to the Jacobian Conjecture $P,Q$ with $\gcd(deg(P),deg(Q))=16$ and show that its existence depends only on the existence of solutions for a certain Abel differential equation of the second kind.

Special cases of the Jacobian conjecture

The famous Jacobian conjecture asks if a morphism $f:K[x,y]\to K[x,y]$ having an invertible Jacobian is invertible ($K$ is a characteristic zero field). We show that if one of the following three equivalent conditions is satisfied, then $f$ is invertible: $K[f(x),f(y)][x+y]$ is normal; $K[x,y]$ is flat over $K[f(x),f(y)][x+y]$; $K[f(x),f(y)][x+y]$ is separable over $K[f(x),f(y)]$.

The Jacobian map, the Jacobian group and the group of automorphisms of the Grassmann algebra

There are nontrivial dualities and parallels between polynomial algebras and the Grassmann algebras. This paper is an attempt to look at the Grassmann algebras at the angle of the Jacobian conjecture for polynomial algebras (which is the question/conjecture about the $ $ {\em Jacobian set} -- the set of all algebra endomorphisms of a polynomial algebra with the Jacobian 1 -- the Jacobian conjecture claims that the Jacobian set is a {\em group}). In this paper, we study in detail the Jacobian set for the Grassmann algebra which turns out to be a {\em group} -- the {\em Jacobian group} $Σ$ -- a …

Chow groups of intersections of quadrics via homological projective duality and (Jacobians of) noncommutative motives

The Beilinson-Bloch type conjectures predict that the low degree rational Chow groups of intersections of quadrics are one dimensional. This conjecture was proved by Otwinowska. Making use of homological projective duality and the recent theory of (Jacobians of) noncommutative Chow motives, we give an alternative proof of this conjecture in the case of a complete intersection of either two quadrics or three odd-dimensional quadrics. Moreover, without the use of the powerful Lefschetz theorem, we prove that in these cases the unique non-trivial algebraic Jacobian is the middle one. As an applic…