arXiv subjects
Publications and source records attributed to Christian Valqui.
We translate the results of Yansong Xu into the language of~\cite{GGV1}, obtaining nearly the same formulas for the intersection number of Jacobian pairs, but with an inequality instead of an equality.
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 $r_0$.
We compute the isomorphism class in $\mathfrak{KK}^{alg}$ of all noncommutative generalized Weyl algebras $A=\CC[h](σ, P)$, where $σ(h)=qh+h_0$ is an automorphism of $\CC[h]$, except when $q\neq 1$ is a root of unity. In particular, we compute the isomorphism class in $\mathfrak{KK}^{alg}$ of the quantum Weyl algebra, the primitive factors $B_λ$ of $U(\mathfrak{sl}_2)$ and the quantum weighted projective lines $\mathcal{O}(\mathbb{WP}_q(k, l))$.
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]
We describe an algorithm that computes possible corners of hypothetical counterexamples to the Jacobian Conjecture up to a given bound. Using this algorithm we compute the possible families corresponding to $\gcd(deg(P),deg(Q))\le 35$, and all the pairs $(deg(P),deg(Q))$ with $\max(deg(P),deg(Q))\le 150$ for any hypothetical counterexample.
We introduce the notion of a non-degenerate solution of the braid equation on the incidence coalgebra of a locally finite order. Each one of these solutions induce by restriction a non-degenerate set theoretic solution over the underlying set. So, it makes sense to ask if a non-degenerate set theoretic solution on the underlying set of a locally finite order extends to a non-degenerate solution on its incidence coalgebra. In this paper we begin the study of this problem.
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.
Let k be a commutative algebra with the field of the rational numbers included in k and let (E,p,i) be a cleft extension of A. We obtain a new mixed complex, simpler than the canonical one, giving the Hochschild and cyclic homologies of E relative to ker(p). This complex resembles the canonical reduced mixed complex of an augmented algebra. We begin the study of our complex showing that it has a harmonic decomposition like to the one considered by Cuntz and Quillen for the normalized mixed complex of an algebra.
We compute the Groebner basis of a system of polynomial equations related to the Jacobian conjecture using a recursive formula for the Catalan numbers.
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.
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.
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 $α$ be the exchange involution on $A_1(K)$ given by $α(x)= y$ and $α(y)= x$. 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 $α$-endomorphism of $A_1(K)$ is an automorphism. Here an $α$-endomorphism of $A_1(K)$ is an endomorphism which preserves the involution $α$. We also prove an analogue result for the Jacobian conjecture in dimension 2, called $α-JC_2$.
We establish a lower bound for the size of possible counterexamples of the Dixmier Conjecture. We prove that $B>15$, where $B$ is the minimum of the greatest common divisor of the total degrees of $P$ and $Q$, where $(P,Q)$ runs over the counterexamples of the Dixmier Conjecture.
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.
We introduce the notion of non commutative truncated polynomial extension of an algebra A. We study two families of these extensions. For the first one we obtain a complete classification and for the second one, which we call upper triangular, we find that the obstructions to inductively construct them, lie in the Hochschild homology of A, with coefficients in a suitable A-bimodule.
Let k be a field, A a unitary associative k-algebra and V a k-vector space endowed with a distinguished element 1_V. We obtain a mixed complex, simpler that the canonical one, that gives the Hochschild, cyclic, negative and periodic homology of a crossed product E:=A#_f V, in the sense of Brzezinski. We actually work in the more general context of relative cyclic homology. Specifically, we consider a subalgebra K of A that satisfies suitable hypothesis and we find a mixed complex computing the Hochschild, cyclic, negative and periodic homology of E relative to K. Then, when E is a cleft braided Hopf crossed product, we obtain a simpler mixed complex, that also gives the Hochschild, cyclic, negative and periodic homology of E.
We study formal deformations of a crossed product $S(V)#_f G$, of a polynomial algebra with a group, induced from a universal deformation formula introduced by Witherspoon. These deformations arise from braided actions of Hopf algebras generated by automorphisms and skew derivations. We show that they are nontrivial in the characteristic free context, even if $G$ is infinite, by showing that their infinitesimals are not coboundaries. For this we construct a new complex which computes the Hochschild cohomology of $S(V)#_f G$.
Let P,Q be elements of the Weyl algebra W. We prove that if [Q,P]=1, then the centralizer of P is the polynomial algebra k[P].