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arXiv2010

Multiplication of sparse Laurent polynomials and Poisson series on modern hardware architectures

In this paper we present two algorithms for the multiplication of sparse Laurent polynomials and Poisson series (the latter being algebraic structures commonly arising in Celestial Mechanics from the application of perturbation theories). Both algorithms first employ the Kronecker substitution technique to reduce multivariate multiplication to univariate multiplication, and then use the schoolbook method to perform the univariate multiplication. The first algorithm, suitable for moderately-sparse multiplication, uses the exponents of the monomials resulting from the univariate multiplication as trivial hash values in a one dimensional lookup array of coefficients. The second algorithm, suitable for highly-sparse multiplication, uses a cache-optimised hash table which stores the coefficient-exponent pairs resulting from the multiplication using the exponents as keys. Both algorithms have been implemented with attention to modern computer hardware architectures. Particular care has been devoted to the efficient exploitation of contemporary memory hierarchies through cache-blocking techniques and cache-friendly term ordering. The first algorithm has been parallelised for shared-memory multicore architectures, whereas the second algorithm is in the process of being parallelised. We present benchmarks comparing our algorithms to the routines of other computer algebra systems, both in sequential and parallel mode.

cs.SC↗
arXiv2010

Chunky and Equal-Spaced Polynomial Multiplication

Finding the product of two polynomials is an essential and basic problem in computer algebra. While most previous results have focused on the worst-case complexity, we instead employ the technique of adaptive analysis to give an improvement in many "easy" cases. We present two adaptive measures and methods for polynomial multiplication, and also show how to effectively combine them to gain both advantages. One useful feature of these algorithms is that they essentially provide a gradient between existing "sparse" and "dense" methods. We prove that these approaches provide significant improvements in many cases but in the worst case are still comparable to the fastest existing algorithms.

cs.SC↗
arXiv2010

Formal Proof of SCHUR Conjugate Function

The main goal of our work is to formally prove the correctness of the key commands of the SCHUR software, an interactive program for calculating with characters of Lie groups and symmetric functions. The core of the computations relies on enumeration and manipulation of combinatorial structures. As a first "proof of concept", we present a formal proof of the conjugate function, written in C. This function computes the conjugate of an integer partition. To formally prove this program, we use the Frama-C software. It allows us to annotate C functions and to generate proof obligations, which are proved using several automated theorem provers. In this paper, we also draw on methodology, discussing on how to formally prove this kind of program.

cs.LO↗
arXiv2010

Some Results on the Functional Decomposition of Polynomials

If g and h are functions over some field, we can consider their composition f = g(h). The inverse problem is decomposition: given f, determine the ex- istence of such functions g and h. In this thesis we consider functional decom- positions of univariate and multivariate polynomials, and rational functions over a field F of characteristic p. In the polynomial case, "wild" behaviour occurs in both the mathematical and computational theory of the problem if p divides the degree of g. We consider the wild case in some depth, and deal with those polynomials whose decompositions are in some sense the "wildest": the additive polynomials. We determine the maximum number of decompositions and show some polynomial time algorithms for certain classes of polynomials with wild decompositions. For the rational function case we present a definition of the problem, a normalised version of the problem to which the general problem reduces, and an exponential time solution to the normal problem.

cs.SC↗
arXiv2010

Symbolic Domain Decomposition

Decomposing the domain of a function into parts has many uses in mathematics. A domain may naturally be a union of pieces, a function may be defined by cases, or different boundary conditions may hold on different regions. For any particular problem the domain can be given explicitly, but when dealing with a family of problems given in terms of symbolic parameters, matters become more difficult. This article shows how hybrid sets, that is multisets allowing negative multiplicity, may be used to express symbolic domain decompositions in an efficient, elegant and uniform way, simplifying both computation and reasoning. We apply this theory to the arithmetic of piecewise functions and symbolic matrices and show how certain operations may be reduced from exponential to linear complexity.

cs.SC↗
arXiv2010

When can we decide that a P-finite sequence is positive?

We consider two algorithms which can be used for proving positivity of sequences that are defined by a linear recurrence equation with polynomial coefficients (P-finite sequences). Both algorithms have in common that while they do succeed on a great many examples, there is no guarantee for them to terminate, and they do in fact not terminate for every input. For some restricted classes of P-finite recurrence equations of order up to three we provide a priori criteria that assert the termination of the algorithms.

cs.SC↗
arXiv2010

Partial Denominator Bounds for Partial Linear Difference Equations

We investigate which polynomials can possibly occur as factors in the denominators of rational solutions of a given partial linear difference equation (PLDE). Two kinds of polynomials are to be distinguished, we call them /periodic/ and /aperiodic/. The main result is a generalization of a well-known denominator bounding technique for univariate equations to PLDEs. This generalization is able to find all the aperiodic factors of the denominators for a given PLDE.

cs.SC↗
arXiv2010

Integrating multiple sources to answer questions in Algebraic Topology

We present in this paper an evolution of a tool from a user interface for a concrete Computer Algebra system for Algebraic Topology (the Kenzo system), to a front-end allowing the interoperability among different sources for computation and deduction. The architecture allows the system not only to interface several systems, but also to make them cooperate in shared calculations.

cs.SC↗
arXiv2010

Polynomial integration on regions defined by a triangle and a conic

We present an efficient solution to the following problem, of relevance in a numerical optimization scheme: calculation of integrals of the type \[\iint_{T \cap \{f\ge0\}} ϕ_1ϕ_2 \, dx\,dy\] for quadratic polynomials $f,ϕ_1,ϕ_2$ on a plane triangle $T$. The naive approach would involve consideration of the many possible shapes of $T\cap\{f\geq0\}$ (possibly after a convenient transformation) and parameterizing its border, in order to integrate the variables separately. Our solution involves partitioning the triangle into smaller triangles on which integration is much simpler.

cs.SC↗
arXiv2010

Composition collisions and projective polynomials

The functional decomposition of polynomials has been a topic of great interest and importance in pure and computer algebra and their applications. The structure of compositions of (suitably normalized) polynomials f=g(h) over finite fields is well understood in many cases, but quite poorly when the degrees of both components are divisible by the characteristic p. This work investigates the decomposition of polynomials whose degree is a power of p.

math.AC↗
arXiv2010

How to correctly prune tropical trees

We present tropical games, a generalization of combinatorial min-max games based on tropical algebras. Our model breaks the traditional symmetry of rational zero-sum games where players have exactly opposed goals (min vs. max), is more widely applicable than min-max and also supports a form of pruning, despite it being less effective than alpha-beta. Actually, min-max games may be seen as particular cases where both the game and its dual are tropical: when the dual of a tropical game is also tropical, the power of alpha-beta is completely recovered. We formally develop the model and prove that the tropical pruning strategy is correct, then conclude by showing how the problem of approximated parsing can be modeled as a tropical game, profiting from pruning.

cs.AI↗
arXiv2010

Random polynomials and expected complexity of bisection methods for real solving

Our probabilistic analysis sheds light to the following questions: Why do random polynomials seem to have few, and well separated real roots, on the average? Why do exact algorithms for real root isolation may perform comparatively well or even better than numerical ones? We exploit results by Kac, and by Edelman and Kostlan in order to estimate the real root separation of degree $d$ polynomials with i.i.d.\ coefficients that follow two zero-mean normal distributions: for SO(2) polynomials, the $i$-th coefficient has variance ${d \choose i}$, whereas for Weyl polynomials its variance is ${1/i!}$. By applying results from statistical physics, we obtain the expected (bit) complexity of \func{sturm} solver, $\sOB(r d^2 τ)$, where $r$ is the number of real roots and $τ$ the maximum coefficient bitsize. Our bounds are two orders of magnitude tighter than the record worst case ones. We also derive an output-sensitive bound in the worst case. The second part of the paper shows that the expected number of real roots of a degree $d$ polynomial in the Bernstein basis is $\sqrt{2d}\pm\OO(1)$, when the coefficients are i.i.d.\ variables with moderate standard deviation. Our paper concludes with experimental results which corroborate our analysis.

cs.SC↗
arXiv2010

Constructive $D$-module Theory with \textsc{Singular}

We overview numerous algorithms in computational $D$-module theory together with the theoretical background as well as the implementation in the computer algebra system \textsc{Singular}. We discuss new approaches to the computation of Bernstein operators, of logarithmic annihilator of a polynomial, of annihilators of rational functions as well as complex powers of polynomials. We analyze algorithms for local Bernstein-Sato polynomials and also algorithms, recovering any kind of Bernstein-Sato polynomial from partial knowledge of its roots. We address a novel way to compute the Bernstein-Sato polynomial for an affine variety algorithmically. All the carefully selected nontrivial examples, which we present, have been computed with our implementation. We address such applications as the computation of a zeta-function for certain integrals and revealing the algebraic dependence between pairwise commuting elements.

math.AG↗
arXiv2010

An Algebraic Approach for Computing Equilibria of a Subclass of Finite Normal Form Games

A Nash equilibrium has become important solution concept for analyzing the decision making in Game theory. In this paper, we consider the problem of computing Nash equilibria of a subclass of generic finite normal form games. We define "rational payoff irrational equilibria games" to be the games with all rational payoffs and all irrational equilibria. We present a purely algebraic method for computing all Nash equilibria of these games that uses knowledge of Galois groups. Some results, showing properties of the class of games, and an example to show working of the method concludes the paper.

cs.GT↗
arXiv2010

The DMM bound: multivariate (aggregate) separation bounds

In this paper we derive aggregate separation bounds, named after Davenport-Mahler-Mignotte (\dmm), on the isolated roots of polynomial systems, specifically on the minimum distance between any two such roots. The bounds exploit the structure of the system and the height of the sparse (or toric) resultant by means of mixed volume, as well as recent advances on aggregate root bounds for univariate polynomials, and are applicable to arbitrary positive dimensional systems. We improve upon Canny's gap theorem \cite{c-crmp-87} by a factor of $\OO(d^{n-1})$, where $d$ bounds the degree of the polynomials, and $n$ is the number of variables. One application is to the bitsize of the eigenvalues and eigenvectors of an integer matrix, which also yields a new proof that the problem is polynomial. We also compare against recent lower bounds on the absolute value of the root coordinates by Brownawell and Yap \cite{by-issac-2009}, obtained under the hypothesis there is a 0-dimensional projection. Our bounds are in general comparable, but exploit sparseness; they are also tighter when bounding the value of a positive polynomial over the simplex. For this problem, we also improve upon the bounds in \cite{bsr-arxix-2009,jp-arxiv-2009}. Our analysis provides a precise asymptotic upper bound on the number of steps that subdivision-based algorithms perform in order to isolate all real roots of a polynomial system. This leads to the first complexity bound of Milne's algorithm \cite{Miln92} in 2D.

cs.SC↗
arXiv2010

Parallel versions of the symbolic manipulation system FORM

The symbolic manipulation program FORM is specialized to handle very large algebraic expressions. Some specific features of its internal structure make FORM very well suited for parallelization. We have now two parallel versions of FORM, one is based on POSIX threads and is optimal for modern multicore computers while another one uses MPI and can be used to parallelize FORM on clusters and Massive Parallel Processing systems. Most existing FORM programs will be able to take advantage of the parallel execution without the need for modifications.

hep-ph↗
arXiv2010

The F5 Algorithm in Buchberger's Style

The famous F5 algorithm for computing \gr basis was presented by Faugère in 2002. The original version of F5 is given in programming codes, so it is a bit difficult to understand. In this paper, the F5 algorithm is simplified as F5B in a Buchberger's style such that it is easy to understand and implement. In order to describe F5B, we introduce F5-reduction, which keeps the signature of labeled polynomials unchanged after reduction. The equivalence between F5 and F5B is also shown. At last, some versions of the F5 algorithm are illustrated.

cs.SC↗