In this paper, in addition to the earlier introduced involutive divisions, we consider a new class of divisions induced by admissible monomial orderings. We prove that these divisions are noetherian and constructive. Thereby each of them allows one to compute an involutive Gröbner basis of a polynomial ideal by sequentially examining multiplicative reductions of nonmultiplicative prolongations....

A three-by-three matrix spectral problem for AKNS soliton hierarchy is proposed and the corresponding Bargmann symmetry constraint involved in Lax pairs and adjoint Lax pairs is discussed. The resulting nonlinearized Lax systems possess classical Hamiltonian structures, in which the nonlinearized spatial system is intimately related to stationary AKNS flows. These nonlinearized Lax systems also...

For a specific exactly solvable 2×2 matrix model with a PT -symmetric Hamiltonian possessing a real spectrum, we construct all the eligible physical metrics Θ > 0 and show that none of them admits a factorization Θ = CP in terms of an involutive charge operator C. Alternative ways of restricting the physical metric to a unique form are briefly discussed. PACS: 03.65.-w; 03.65.Ca; 03.65.Ta

The Drinfeld-Sokolov reduction method has been used to associate with gln extensions of the matrix r-KdV system. Reductions of these systems to the fixed point sets of involutive Poisson maps, implementing reduction of gln to classical Lie algebras of type B, C, D, are here presented. Modifications corresponding, in the first place to factorisation of the Lax operator, and then to Wakimoto real...

A three-by-three matrix spectral problem for AKNS soliton hierarchy is proposed and the corresponding Bargmann symmetry constraint involved in Lax pairs and adjoint Lax pairs is discussed. The resulting nonlinearized Lax systems possess classical Hamiltonian structures, in which the nonlinearized spatial system is intimately related to stationary AKNS ows. These nonlin-earized Lax systems also ...

We compute the involutive Heegaard Floer homology of the family of threemanifolds obtained by plumbings along almost-rational graphs. (This includes all Seifert fibered homology spheres.) We also study the involutive Heegaard Floer homology of connected sums of such three-manifolds, and explicitly determine the involutive correction terms in the case that all of the summands have the same orien...

Let J be an involutive Hermitian matrix with signature (t, n− t), 0 ≤ t ≤ n, that is, with t positive and n− t negative eigenvalues. The Krein space numerical range of a complex matrix A of size n is the collection of complex numbers of the form ξ ∗JAξ ξ∗Jξ , with ξ ∈ Cn and ξ∗Jξ = 0. In this note, a class of tridiagonal matrices with hyperbolical numerical range is investigated. A Matlab progr...

In this paper we present results of our computer experiments to study effectiveness of involutive criteria for avoiding useless prolongations in construction of polynomial Janet bases. These bases are typical representative of involutive bases. Though involutive bases are usually redundant as Gröbner ones, they can be used as well as the reduced Gröbner bases in whose theory Volker Weispfenning...

Buchberger’s algorithm for computing a Gröbner basis solves the ideal membership problem over commutative polynomial rings. In the early 1990’s, an alternative to this algorithm was found, namely the involutive basis algorithm, which provides a Gröbner basis with extra combinatorial properties. Buchberger’s work was generalised to noncommutative polynomial rings by Bergman and Mora during the 1...

In this paper we generalize the involutive methods and algorithms devised for polynomial ideals to differential ones generated by a finite set of linear differential polynomials in the differential polynomial ring over a zero characteristic differential field. Given a ranking of derivative terms and an involutive division, we formulate the involutivity conditions which form a basis of involutiv...