The Groebner basis of a polynomial system

We compute the Groebner basis of a system of polynomial equations related to the Jacobian conjecture using a recursive formula for the Catalan numbers.Comment: From the Magister Thesis of Marco Solorzan

Excision in bivariant periodic cyclic cohomology: a categorical approach

We extend Cuntz-Quillen's excision theorem for algebras and pro-algebras in arbitrary Q-linear categories with tensor product.The excision theorems for the bivariant periodic cyclic cohomology of discrete,topological and bornological algebras and pro-algebras follow from this.Comment: 32 pages, Amslatex, uses xypic. Final version to appear in K-theor

The starred Dixmier conjecture for A1A_1

Let $A_1(K)=K \langle x,y | yx-xy= 1 \rangle$ be the first Weyl algebra over a characteristic zero field $K$ and let $\alpha$ be the exchange involution on $A_1(K)$ given by $\alpha(x)= y$ and $\alpha(y)= x$. The Dixmier conjecture of Dixmier (1968) asks: Is every algebra endomorphism of the Weyl algebra $A_1(K)$ an automorphism? The aim of this paper is to prove that each $\alpha$-endomorphism of $A_1(K)$ is an automorphism. Here an $\alpha$-endomorphism of $A_1(K)$ is an endomorphism which preserves the involution $\alpha$. We also prove an analogue result for the Jacobian conjecture in dimension 2, called $\alpha-JC_2$.Comment: arXiv admin note: text overlap with arXiv:1310.756

A short and elementary proof of Jung's theorem

We give a short and elementary proof of Jung's theorem, which states that for a field K of characteristic zero the automorphisms of K[x,y] are generated by elementary automorphisms and linear automorphisms.Comment: 1 figur

Solutions of the Braid Equation with set-type square

For a family of height one orders $(X,\le)$ and each non-degenerate solution $r_0\colon X\times X\longrightarrow X\times X$ of the set-theoretic braid equation on $X$ satisfying suitable conditions, we obtain all the non-degenerate solutions of the braid equation on the incidence coalgebra of $(X,\le)$ that extend $r_0$.Comment: 19 pages, 1 tabl

The two-dimensional Jacobian conjecture and the lower side of the Newton polygon

We prove that if the Jacobian Conjecture in two variables is false and (P,Q) is a standard minimal pair, then the Newton polygon HH(P) of P must satisfy several restrictions that had not been found previously. This allows us to discard some of the corners found in [GGV, Remark 7.14] for HH(P), together with some of the infinite families found in [H, Theorem~2.25]Comment: 17 pages. We add Remark 3.9 and generalize Proposition 3.12 to be valid in L^{(l). Moreover we improve the presentation of the pape

Twisted planes

Let k be a commutative ring. We find and characterize a new family of twisted planes (i. e. associative unitary k-algebra structures on the k-module k[X,Y], having k[X] and and k[Y] as subalgebras).Similar results are obtained for the k-module of two variables power series k[[X,Y]].Comment: 33 page

The Dixmier conjecture and the shape of possible counterexamples II

We continue with the investigation began in "The Dixmier conjecture and the shape of possible counterexamples". In that paper we introduced the notion of an irreducible pair (P,Q) as the image of the pair (X,Y) of the canonical generators of W via an endomorphism which is not an automorphism, such that it cannot be made "smaller", we let B denote the minimum of the greatest common divisor of the total degrees of P and Q, where (P,Q) runs on the irreducible pairs and we prove that $\ge$. In the present work we improve this lower bound by proving that B\ge 15. In order to do this we need to show the the main results of our previous paper remain valid for a family of algebras (W^{(l)})_{l\in \mathds{N}} that extend W.Comment: 44 pages: 3 figure

A system of polynomial equations related to the Jacobian Conjecture

We prove that the Jacobian conjecture is false if and only if there exists a solution to a certain system of polynomial equations. We analyse the solution set of this system. In particular we prove that it is zero dimensional.Comment: 26 pages, Proposition 4.3 is ne

On the shape of possible counterexamples to the Jacobian Conjecture

We improve the algebraic methods of Abhyankar for the Jacobian Conjecture in dimension two and describe the shape of possible counterexamples. We give an elementary proof of the result of Heitmann, which states that gcd(deg(P),deg(Q)) is greater than or equal to 16 for any counterexample (P,Q). We also prove that gcd(deg(P),deg(Q)) \ne 2p for any prime p and analyze thoroughly the case 16, adapting a reduction of degree technique introduced by Moh.Comment: 71 pages, 8 figures. We improve the presentation and fix some minor mistake