Journal of Symbolic Computation

Consider a card guessing game with complete feedback in which a deck of n cards ordered 1 , … , n is riffle-shuffled once. With the goal to maximize the number of correct guesses, a player guesses cards from the top of the deck one at ...

We apply Kovacic's algorithm, a tool that is developed from differential Galois theory, to discuss the existence of Liouvillian solutions of Whittaker-Ince equation, ellipsoidal wave equation and the Picard-Fuchs equation of a K3 ...

It is well known that every non-degenerate quadratic form admits a decomposition into an orthogonal sum of its anisotropic part and a hyperbolic form. This decomposition is unique up to isometry. In this paper we present an algorithm ...

Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The fundamental theorem ...

Arbitrary-precision integer libraries such as GMP are a critical building block of computer algebra systems. GMP provides state-of-the-art algorithms that are intricate enough to justify formal verification. In this paper, we present a ...

In 1977, Strassen invented a famous baby-step/giant-step algorithm that computes the factorial N! in arithmetic complexity quasi-linear in N. In 1988, the Chudnovsky brothers generalized Strassen's algorithm to the computation of the N-...

We make explicit the exponential bound on the degrees of the polynomials appearing in the Effective Quillen-Suslin Theorem, and apply it jointly with the Hilbert-Burch Theorem to show that the syzygy module of a sequence of m ...

The condition-based complexity analysis framework is one of the gems of modern numerical algebraic geometry and theoretical computer science. Among the challenges that it poses is to expand the currently limited range of random ...

An interpolation problem is defined by a set of linear forms on the (multivariate) polynomial ring and values to be achieved by an interpolant. For Lagrange interpolation the linear forms consist of evaluations at some nodes, while ...

With this paper we present an extension of our recent ISSAC paper about computations of Gröbner(-Shirshov) bases over free associative algebras Z 〈 X 〉. We present all the needed proofs in details, add a part on the direct treatment of ...

In this paper we provide a new method to certify that a nearby polynomial system has a singular isolated root and we compute its multiplicity structure. More precisely, given a polynomial system f = ( f 1 , … , f N ) ∈ C [ x 1 , … , x ...

The first extended greatest common right divisor (GCRD) algorithm for parametric univariate polynomial matrices is presented. The starting point of this GCRD algorithm is the free property of submodules over univariate polynomial ...

This paper is concerned with the factorization and equivalence problems of multivariate polynomial matrices. We present some new criteria for the existence of matrix factorizations for a class of multivariate polynomial matrices, and ...

We present a non-commutative algorithm for the multiplication of a 2 × 2 block-matrix by its adjoint, defined by a matrix ring anti-homomorphism. This algorithm uses 5 block products (3 recursive calls and 2 general products). It works ...

Tropical Newton-Puiseux polynomials, defined as piece-wise linear functions with rational coefficients of the variables, play a role as tropical algebraic functions. We provide explicit formulas for tropical Newton-Puiseux polynomials ...

In this paper we introduce methods and algorithms that will help us solve connectivity queries of parameterized semi-algebraic sets. Answering these connectivity queries is applied in the design of robotic structures having similar ...

The second-order cone (SOC) is a class of simple convex cones and optimizing over them can be done more efficiently than with semidefinite programming. It is interesting both in theory and in practice to investigate which convex cones ...

We discuss the parallelization of algorithms for solving polynomial systems by way of triangular decomposition. The Triangularize algorithm proceeds through incremental intersections of polynomials to produce different components (...

Parameterized systems of polynomial equations arise in many applications in science and engineering with the real solutions describing, for example, equilibria of a dynamical system, linkages satisfying design constraints, and scene ...

We consider the problem of computing the topology and describing the geometry of a parametric curve in R n. We present an algorithm, PTOPO, that constructs an abstract graph that is isotopic to the curve in the embedding space. Our ...

We apply numerical algebraic geometry to the invariant-theoretic problem of detecting symmetries between two plane algebraic curves. We describe an efficient equality test which determines, with “probability-one”, whether or not two ...

We show that standard machine learning algorithms may be trained to predict certain invariants of low genus arithmetic curves. Using datasets of size around 10⁵, we demonstrate the utility of machine learning in classification problems ...

This paper introduces an algorithmic approach to the analysis of bifurcation of limit cycles from the centers of nonlinear continuous differential systems via the averaging method. We develop three algorithms to implement the averaging ...