Endomorphisms of quantized Weyl algebras

Belov-Kanel and Kontsevich conjectured that the group of automorphisms of the n'th Weyl algebra and the group of polynomial symplectomorphisms of C^2 are canonically isomorphic. We discuss how this conjecture can be approached by means of (second) quantized Weyl algebras at roots of unity

A proof of the equivalence of the Dixmier, Jacobian and Poisson Conjectures

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Eulerian operators and the Jacobian conjecture III

AbstractLet k be a field of characteristic zero and F:kn→kn a polynomial map with det JFϵk∗ and F(0)=0. Using the Euler operator it is shown that if the k-subalgebra of Mn(k[k1,…,xn]) generated by the homogeneous components of the matrices JF and (JF)-1 is finite-dimensional over k and such that each element in it is a Jacobian matrix, then F is invertible. This implies a result of Connell and Zweibel. Furthermore, it is shown that the Jacobian Conjecture is equivalent with the statement that for every F with det JF ϵ k∗ and F(0)=0, the shifted Euler operator 1+ΣFi(∂∂Fi) is Eulerian

A new inversion formula for a polynomial map in two variables

AbstractBy computing one resultant, the inverse of an invertible polynomial map in two variables is given

Generalized Hermite Reduction, Creative Telescoping and Definite Integration of D-Finite Functions

International audienceHermite reduction is a classical algorithmic tool in symbolic integration. It is used to decompose a given rational function as a sum of a function with simple poles and the derivative of another rational function. We extend Hermite reduction to arbitrary linear differential operators instead of the pure derivative, and develop efficient algorithms for this reduction. We then apply the generalized Hermite reduction to the computation of linear operators satisfied by single definite integrals of D-finite functions of several continuous or discrete parameters. The resulting algorithm is a generalization of reduction-based methods for creative telescoping