Fraction-free algorithm for the computation of diagonal forms matrices over Ore domains using Gr{\"o}bner bases

This paper is a sequel to "Computing diagonal form and Jacobson normal form of a matrix using Groebner bases", J. of Symb. Computation, 46 (5), 2011. We present a new fraction-free algorithm for the computation of a diagonal form of a matrix over a certain non-commutative Euclidean domain over a computable field with the help of Gr\"obner bases. This algorithm is formulated in a general constructive framework of non-commutative Ore localizations of $G$-algebras (OLGAs). We split the computation of a normal form of a matrix into the diagonalization and the normalization processes. Both of them can be made fraction-free. For a matrix $M$ over an OLGA we provide a diagonalization algorithm to compute $U,V$ and $D$ with fraction-free entries such that $UMV=D$ holds and $D$ is diagonal. The fraction-free approach gives us more information on the system of linear functional equations and its solutions, than the classical setup of an operator algebra with rational functions coefficients. In particular, one can handle distributional solutions together with, say, meromorphic ones. We investigate Ore localizations of common operator algebras over $K[x]$ and use them in the unimodularity analysis of transformation matrices $U,V$. In turn, this allows to lift the isomorphism of modules over an OLGA Euclidean domain to a polynomial subring of it. We discuss the relation of this lifting with the solutions of the original system of equations. Moreover, we prove some new results concerning normal forms of matrices over non-simple domains. Our implementation in the computer algebra system {\sc Singular:Plural} follows the fraction-free strategy and shows impressive performance, compared with methods which directly use fractions. Since we experience moderate swell of coefficients and obtain simple transformation matrices, the method we propose is well suited for solving nontrivial practical problems.Comment: 25 pages, to appear in Journal of Symbolic Computatio

Computing diagonal form and Jacobson normal form of a matrix using Gr\"obner bases

In this paper we present two algorithms for the computation of a diagonal form of a matrix over non-commutative Euclidean domain over a field with the help of Gr\"obner bases. This can be viewed as the pre-processing for the computation of Jacobson normal form and also used for the computation of Smith normal form in the commutative case. We propose a general framework for handling, among other, operator algebras with rational coefficients. We employ special "polynomial" strategy in Ore localizations of non-commutative $G$-algebras and show its merits. In particular, for a given matrix $M$ we provide an algorithm to compute $U,V$ and $D$ with fraction-free entries such that $UMV=D$ holds. The polynomial approach allows one to obtain more precise information, than the rational one e. g. about singularities of the system. Our implementation of polynomial strategy shows very impressive performance, compared with methods, which directly use fractions. In particular, we experience quite moderate swell of coefficients and obtain uncomplicated transformation matrices. This shows that this method is well suitable for solving nontrivial practical problems. We present an implementation of algorithms in SINGULAR:PLURAL and compare it with other available systems. We leave questions on the algorithmic complexity of this algorithm open, but we stress the practical applicability of the proposed method to a bigger class of non-commutative algebras

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

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

Computing images of Galois representations attached to elliptic curves

Let E be an elliptic curve without complex multiplication (CM) over a number field K, and let G_E(ell) be the image of the Galois representation induced by the action of the absolute Galois group of K on the ell-torsion subgroup of E. We present two probabilistic algorithms to simultaneously determine G_E(ell) up to local conjugacy for all primes ell by sampling images of Frobenius elements; one is of Las Vegas type and the other is a Monte Carlo algorithm. They determine G_E(ell) up to one of at most two isomorphic conjugacy classes of subgroups of GL_2(Z/ell Z) that have the same semisimplification, each of which occurs for an elliptic curve isogenous to E. Under the GRH, their running times are polynomial in the bit-size n of an integral Weierstrass equation for E, and for our Monte Carlo algorithm, quasi-linear in n. We have applied our algorithms to the non-CM elliptic curves in Cremona's tables and the Stein--Watkins database, some 140 million curves of conductor up to 10^10, thereby obtaining a conjecturally complete list of 63 exceptional Galois images G_E(ell) that arise for E/Q without CM. Under this conjecture we determine a complete list of 160 exceptional Galois images G_E(ell) the arise for non-CM elliptic curves over quadratic fields with rational j-invariants. We also give examples of exceptional Galois images that arise for non-CM elliptic curves over quadratic fields only when the j-invariant is irrational.Comment: minor edits, 47 pages, to appear in Forum of Mathematics, Sigm

On the Generation of Positivstellensatz Witnesses in Degenerate Cases

One can reduce the problem of proving that a polynomial is nonnegative, or more generally of proving that a system of polynomial inequalities has no solutions, to finding polynomials that are sums of squares of polynomials and satisfy some linear equality (Positivstellensatz). This produces a witness for the desired property, from which it is reasonably easy to obtain a formal proof of the property suitable for a proof assistant such as Coq. The problem of finding a witness reduces to a feasibility problem in semidefinite programming, for which there exist numerical solvers. Unfortunately, this problem is in general not strictly feasible, meaning the solution can be a convex set with empty interior, in which case the numerical optimization method fails. Previously published methods thus assumed strict feasibility; we propose a workaround for this difficulty. We implemented our method and illustrate its use with examples, including extractions of proofs to Coq.Comment: To appear in ITP 201