Computer Science

We consider the formal reduction of a system of linear differential equations and show that, if the system can be block-diagonalised through transformation with a ramified Shearing-transformation and following application of the Splitting Lemma, and if the spectra of the leading block matrices of the ramified system satisfy a symmetry condition, this block-diagonalisation can also be achieved through an unramified transformation. Combined with classical results by Turritin and Wasow as well as work by Balser, this yields a constructive and simple proof of the existence of an unramified block-diagonal form from which formal invariants such as the Newton polygon can be read directly. Our result is particularly useful for designing efficient algorithms for the formal reduction of the system.

Signature-based algorithms have become a standard approach for Gröbner basis computations for polynomial systems over fields, but how to extend these techniques to coefficients in general rings is not yet as well understood. In this paper, we present a proof-of-concept signature-based algorithm for computing Gröbner bases over commutative integral domains. It is adapted from a general version of Möller's algorithm (1988) which considers reductions by multiple polynomials at each step. This algorithm performs reductions with non-decreasing signatures, and in particular, signature drops do not occur. When the coefficients are from a principal ideal domain (e.g. the ring of integers or the ring of univariate polynomials over a field), we prove correctness and termination of the algorithm, and we show how to use signature properties to implement classic signature-based criteria to eliminate some redundant reductions. In particular, if the input is a regular sequence, the algorithm operates without any reduction to 0. We have written a toy implementation of the algorithm in Magma. Early experimental results suggest that the algorithm might even be correct and terminate in a more general setting, for polynomials over a unique factorization domain (e.g. the ring of multivariate polynomials over a field or a PID).

We introduce a method for computing some pseudo-elliptic integrals in terms of elementary functions. The method is simple and fast in comparison to the algebraic case of the Risch-Trager-Bronstein algorithm. This method can quickly solve many pseudo-elliptic integrals, which other well-known computer algebra systems either fail, return an answer in terms of special functions, or require more than 20 seconds of computing time. Randomised tests showed our method solved 73.4% of the integrals that could be solved with the best implementation of the Risch-Trager-Bronstein algorithm. Unlike the symbolic integration algorithms of Risch, Davenport, Trager, Bronstein and Miller; our method is not a decision process. The implementation of this method is less than 200 lines of Mathematica code and can be easily ported to other CAS that can solve systems of polynomial equations.

We present a simple and fast algorithm for computing the N -th term of a given linearly recurrent sequence. Our new algorithm uses O(M(d)logN) arithmetic operations, where d is the order of the recurrence, and M(d) denotes the number of arithmetic operations for computing the product of two polynomials of degree d . The state-of-the-art algorithm, due to Charles Fiduccia (1985), has the same arithmetic complexity up to a constant factor. Our algorithm is simpler, faster and obtained by a totally different method. We also discuss several algorithmic applications, notably to polynomial modular exponentiation, powering of matrices and high-order lifting.

A special homotopy continuation method, as a combination of the polyhedral homotopy and the linear product homotopy, is proposed for computing all the isolated solutions to a special class of polynomial systems. The root number bound of this method is between the total degree bound and the mixed volume bound and can be easily computed. The new algorithm has been implemented as a program called LPH using C++. Our experiments show its efficiency compared to the polyhedral or other homotopies on such systems. As an application, the algorithm can be used to find witness points on each connected component of a real variety.

In this paper, a new triangular decomposition algorithm is proposed for ordinary differential polynomial systems, which has triple exponential computational complexity. The key idea is to eliminate one algebraic variable from a set of polynomials in one step using the theory of multivariate resultant. This seems to be the first differential triangular decomposition algorithm with elementary computation complexity.

Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gr{ö}bner bases taking into account the valuation of K. While generalizing the classical theory of Gr{ö}bner bases, it is not clear how modern algorithms for computing Gr{ö}bner bases can be adapted to the tropical case. Among them, one of the most efficient is the celebrated F5 Algorithm of Faug{è}re. In this article, we prove that, for homogeneous ideals, it can be adapted to the tropical case. We prove termination and correctness. Because of the use of the valuation, the theory of tropical Gr{ö}b-ner bases is promising for stable computations over polynomial rings over a p-adic field. We provide numerical examples to illustrate time-complexity and p-adic stability of this tropical F5 algorithm.

Cylindrical algebraic decomposition (CAD) is a key tool for problems in real algebraic geometry and beyond. When using CAD there is often a choice over the variable ordering to use, with some problems infeasible in one ordering but simple in another. Here we discuss a recent experiment comparing three heuristics for making this choice on thousands of examples.

Let f 1 ,…, f m be elements in a quotient R n /N which has finite dimension as a K -vector space, where R=K[ X 1 ,…, X r ] and N is an R -submodule of R n . We address the problem of computing a Gröbner basis of the module of syzygies of ( f 1 ,…, f m ) , that is, of vectors ( p 1 ,…, p m )∈ R m such that p 1 f 1 +⋯+ p m f m =0 . An iterative algorithm for this problem was given by Marinari, Möller, and Mora (1993) using a dual representation of R n /N as the kernel of a collection of linear functionals. Following this viewpoint, we design a divide-and-conquer algorithm, which can be interpreted as a generalization to several variables of Beckermann and Labahn's recursive approach for matrix Padé and rational interpolation problems. To highlight the interest of this method, we focus on the specific case of bivariate Padé approximation and show that it improves upon the best known complexity bounds.

We discuss theoretical and algorithmic questions related to the p -curvature of differential operators in characteristic p . Given such an operator L , and denoting by $\Chi(L)$ the characteristic polynomial of its p -curvature, we first prove a new, alternative, description of $\Chi(L)$. This description turns out to be particularly well suited to the fast computation of $\Chi(L)$ when p is large: based on it, we design a new algorithm for computing $\Chi(L)$, whose cost with respect to p is $\softO(p^{0.5})$ operations in the ground field. This is remarkable since, prior to this work, the fastest algorithms for this task, and even for the subtask of deciding nilpotency of the p -curvature, had merely slightly subquadratic complexity $\softO(p^{1.79})$.