Symbolic Computation
Let $a_n$ denote the number of ways in which a chess rook can move from a corner cell to the opposite corner cell of an $n \times n \times n$ three-dimensional chessboard, assuming that the piece moves closer to the goal cell at each step.…
We introduce a new algorithm denoted DSC2 to isolate the real roots of a univariate square-free polynomial f with integer coefficients. The algorithm iteratively subdivides an initial interval which is known to contain all real roots of f.…
We study the complexity of some fundamental operations for triangular sets in dimension zero. Using Las-Vegas algorithms, we prove that one can perform such operations as change of order, equiprojectable decomposition, or quasi-inverse…
The paper reports on a computer algebra program LSSS (Linear Selective Systems Solver) for solving linear algebraic systems with rational coefficients. The program is especially efficient for very large sparse systems that have a solution…
The purpose of this paper is twofold. An immediate practical use of the presented algorithm is its applicability to the parametric solution of underdetermined linear ordinary differential equations (ODEs) with coefficients that are…
One of the most famous conjectures in computer algebra is that matrix multiplication might be feasible in not much more than quadratic time. The best known exponent is 2.376, due to Coppersmith and Winograd. Many attempts to solve this…
Ritt-Wu's algorithm of characteristic sets is the most representative for triangularizing sets of multivariate polynomials. Pseudo-division is the main operation used in this algorithm. In this paper we present a new algorithmic scheme for…
Signature-based algorithms are a popular kind of algorithms for computing Groebner basis, including the famous F5 algorithm, F5C, extended F5, G2V and the GVW algorithm. In this paper, an efficient method is proposed to solve the…
We consider algebras over a field K, generated by two variables x and y subject to the single relation yx = qxy + ax + by + c for q in K^* and a, b, c in K. We prove, that among such algebras there are precisely five isomorphism classes.…
In this paper, we consider an extended concept of invariant for polynomial dynamical system (PDS) with domain and initial condition, and establish a sound and complete criterion for checking semi-algebraic invariants (SAI) for such PDSs.…
In this paper we explore the possibility of using computational algebraic methods to analyze a class of consensus protocols. We state some necessary conditions for convergence under consensus protocols that are polynomials.
In order to appreciate how well off we mathematicians and scientists are today, with extremely fast hardware and lots and lots of memory, as well as with powerful software, both for numeric and symbolic computation, it may be a good idea to…
In this note we reinvestigate the task of computing creative telescoping relations in differential-difference operator algebras. Our approach is based on an ansatz that explicitly includes the denominators of the delta parts. We contribute…
A generalized criterion for signature-based algorithms to compute Gr\"obner bases is proposed in this paper. This criterion is named by "generalized criterion", because it can be specialized to almost all existing criteria for…
We consider the problem of isolating the real roots of a square-free polynomial with integer coefficients using (variants of) the continued fraction algorithm (CF). We introduce a novel way to compute a lower bound on the positive real…
The problem of the determination of the Hilbert-space metric which renders a given Hamiltonian $H$ self-adjoint is addressed from the point of view of applicability of computer-assisted algebraic manipulations. An exactly solvable example…
We present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial in $B_{\alpha} \in L[y]$, where $L=\QQ(\alpha)$ is a simple algebraic extension of the rational numbers. We…
We consider the problem of isolating the real roots of a square-free polynomial with integer coefficients using (variants of) the continued fraction algorithm (CF). We introduce a novel way to compute a lower bound on the positive real…
An algorithm for the symbolic computation of recursion operators for systems of nonlinear differential-difference equations (DDEs) is presented. Recursion operators allow one to generate an infinite sequence of generalized symmetries. The…
An overview of the solution methods for ordinary differential equations in the Mathematica function DSolve is presented.