Related papers: Desingularization of Ore Operators
Desingularization is the problem of finding a left multiple of a given Ore operator in which some factor of the leading coefficient of the original operator is removed. An order-degree curve for a given Ore operator is a curve in the…
Ore operators form a common algebraic abstraction of linear ordinary differential and recurrence equations. Given an Ore operator $L$ with polynomial coefficients in $x$, it generates a left ideal $I$ in the Ore algebra over the field…
Ore operators with polynomial coefficients form a common algebraic abstraction for representing D-finite functions. They form the Ore ring $K(x)[D_x]$, where $K$ is the constant field. Suppose $K$ is the quotient field of some principal…
We give algorithms to construct the N\'eron Desingularization and the easy case from \cite{KK} of the General N\'eron Desingularization.
In this paper, we study the desingularization problem in the first $q$-Weyl algebra. We give an order bound for desingularized operators, and thus derive an algorithm for computing desingularized operators in the first $q$-Weyl algebra.…
Conventional ways to solve optimization problems on low-rank matrix sets which appear in great number of applications ignore its underlying structure of an algebraic variety and existence of singular points. This leads to appearance of…
Algorithmic approach to the problem of linearization by point transformation of ordinary differential equation of arbitrary order is presented. Test-linearization is purely algorithmic.
Linear recurrence operators in characteristic $p$ are classified by their $p$-curvature. For a recurrence operator $L$, denote by $\chi(L)$ the characteristic polynomial of its $p$-curvature. We can obtain information about the…
We give infinite triangularization and strict triangularization results for algebras of operators on infinite dimensional vector spaces. We introduce a class of algebras we call Ore-solvable algebras: these are similar to iterated Ore…
Over the last decade, implementations of several desingularization algorithms have appeared in various contexts. These differ as widely in their methods and in their practical efficiency as they differ in the situations in which they may be…
It is well known that for a first order system of linear difference equations with rational function coefficients, a solution that is holomorphic in some left half plane can be analytically continued to a meromorphic solution in the whole…
We present a Sage implementation of Ore algebras. The main features for the most common instances include basic arithmetic and actions; gcrd and lclm; D-finite closure properties; natural transformations between related algebras; guessing;…
We introduce basic aspects of new operator method, which is very suitable for practical solving differential equations of various types. The main advantage of the method is revealed in opportunity to find compact exact operator solutions of…
We obtain a new simple formula for the regularized traces of singular ordinary differential operators.
We introduce a search algorithm that utilises differential operator realisations to find polynomial Casimir operators of Lie algebras. To demonstrate the algorithm, we look at two classes of examples: (1) the model filiform Lie algebras and…
An algorithmic proof of the General N\'eron Desingularization theorem and its uniform version is given for morphisms with big smooth locus. This generalizes the results for the one-dimensional case.
After briefly recalling some computational aspects of blowing up and of representation of resolution data common to a wide range of desingularization algorithms (in the general case as well as in special cases like surfaces or binomial…
It is shown that, for any reduced algebraic variety in characteristic zero, one can resolve all but simple normal crossings (snc) singularities by a finite sequence of blowings-up with smooth centres which, at every step, avoids points…
It is well known that the affine matrix rank minimization problem is NP-hard and all known algorithms for exactly solving it are doubly exponential in theory and in practice due to the combinational nature of the rank function. In this…
In this set of papers we formulate a stand alone method to derive maximal number of linearizing transformations for nonlinear ordinary differential equations (ODEs) of any order including coupled ones from a knowledge of fewer number of…