Improved Polynomial Remainder Sequences for Ore Polynomials

Polynomial remainder sequences contain the intermediate results of the Euclidean algorithm when applied to (non-)commutative polynomials. The running time of the algorithm is dependent on the size of the coefficients of the remainders. Different ways have been studied to make these as small as possible. The subresultant sequence of two polynomials is a polynomial remainder sequence in which the size of the coefficients is optimal in the generic case, but when taking the input from applications, the coefficients are often larger than necessary. We generalize two improvements of the subresultant sequence to Ore polynomials and derive a new bound for the minimal coefficient size. Our approach also yields a new proof for the results in the commutative case, providing a new point of view on the origin of the extraneous factors of the coefficients

Fast Computation of Common Left Multiples of Linear Ordinary Differential Operators

We study tight bounds and fast algorithms for LCLMs of several linear differential operators with polynomial coefficients. We analyze the arithmetic complexity of existing algorithms for LCLMs, as well as the size of their outputs. We propose a new algorithm that recasts the LCLM computation in a linear algebra problem on a polynomial matrix. This algorithm yields sharp bounds on the coefficient degrees of the LCLM, improving by one order of magnitude the best bounds obtained using previous algorithms. The complexity of the new algorithm is almost optimal, in the sense that it nearly matches the arithmetic size of the output.Comment: The final version will appear in Proceedings of ISSAC 201

On the complexity of skew arithmetic

13 pagesIn this paper, we study the complexity of several basic operations on linear differential operators with polynomial coefficients. As in the case of ordinary polynomials, we show that these complexities can be expressed in terms of the cost of multiplication

Computing greatest common divisor of several parametric univariate polynomials via generalized subresultant polynomials

In this paper, we tackle the following problem: compute the gcd for several univariate polynomials with parametric coefficients. It amounts to partitioning the parameter space into ``cells'' so that the gcd has a uniform expression over each cell and constructing a uniform expression of gcd in each cell. We tackle the problem as follows. We begin by making a natural and obvious extension of subresultant polynomials of two polynomials to several polynomials. Then we develop the following structural theories about them. 1. We generalize Sylvester's theory to several polynomials, in order to obtain an elegant relationship between generalized subresultant polynomials and the gcd of several polynomials, yielding an elegant algorithm. 2. We generalize Habicht's theory to several polynomials, in order to obtain a systematic relationship between generalized subresultant polynomials and pseudo-remainders, yielding an efficient algorithm. Using the generalized theories, we present a simple (structurally elegant) algorithm which is significantly more efficient (both in the output size and computing time) than algorithms based on previous approaches

Resultant-based Elimination in Ore Algebra

We consider resultant-based methods for elimination of indeterminates of Ore polynomial systems in Ore algebra. We start with defining the concept of resultant for bivariate Ore polynomials then compute it by the Dieudonne determinant of the polynomial coefficients. Additionally, we apply noncommutative versions of evaluation and interpolation techniques to the computation process to improve the efficiency of the method. The implementation of the algorithms will be performed in Maple to evaluate the performance of the approaches.Comment: An updated (and shorter) version published in the SYNASC '21 proceedings (IEEE CS) with the title "Resultant-based Elimination for Skew Polynomials

A first approach to the Burchnall-Chaundy theory for quadratic algebras having PBW bases

In this paper, we present a first approach toward a Burchnall-Chaundy theory for the skew Ore polynomials of higher order generated by quadratic relations defined by Golovashkin and Maksimov \cite{GolovashkinMaksimov1998}.Comment: 24 page

Decomposition of ordinary difference polynomials

AbstractIn this paper, we present an algorithm to decompose ordinary non-linear difference polynomials with rational functions as coefficients. The algorithm provides an effective reduction of the decomposition of difference polynomials to the decomposition of linear difference polynomials over the same coefficient field. The algorithm is implemented in Maple for the constant coefficient case. Experimental results show that the algorithm is quite effective and can be used to decompose difference polynomials with thousands of terms