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At least 235 records · Page 13Linked to original sources
arXiv2010

A new conception for computing gröbner basis and its applications

This paper presents a conception for computing gröbner basis. We convert some of gröbner-computing algorithms, e.g., F5, extended F5 and GWV algorithms into a special type of algorithm. The new algorithm's finite termination problem can be described by equivalent conditions, so all the above algorithms can be determined when they terminate finitely. At last, a new criterion is presented. It is an improvement for the Rewritten and Signature Criterion.

cs.SC↗
arXiv2011

FORM development

I give an overview of FORM development based on a few pilot projects, explaining how they have influenced the FORM capabilities. Next I explain what is happnening right now in the field of Open Sourcing and the FORM Forum.

hep-ph↗
arXiv2011

A Refined Denominator Bounding Algorithm for Multivariate Linear Difference Equations

We continue to investigate which polynomials can possibly occur as factors in the denominators of rational solutions of a given partial linear difference equation. In an earlier article we had introduced the distinction between periodic and aperiodic factors in the denominator, and we gave an algorithm for predicting the aperiodic ones. Now we extend this technique towards the periodic case and present a refined algorithm which also finds most of the periodic factors.

cs.SC↗
arXiv2011

Deflation and Certified Isolation of Singular Zeros of Polynomial Systems

We develop a new symbolic-numeric algorithm for the certification of singular isolated points, using their associated local ring structure and certified numerical computations. An improvement of an existing method to compute inverse systems is presented, which avoids redundant computation and reduces the size of the intermediate linear systems to solve. We derive a one-step deflation technique, from the description of the multiplicity structure in terms of differentials. The deflated system can be used in Newton-based iterative schemes with quadratic convergence. Starting from a polynomial system and a small-enough neighborhood, we obtain a criterion for the existence and uniqueness of a singular root of a given multiplicity structure, applying a well-chosen symbolic perturbation. Standard verification methods, based eg. on interval arithmetic and a fixed point theorem, are employed to certify that there exists a unique perturbed system with a singular root in the domain. Applications to topological degree computation and to the analysis of real branches of an implicit curve illustrate the method.

cs.SC↗
arXiv2013

A Symbolic Transformation Language and its Application to a Multiscale Method

The context of this work is the design of a software, called MEMSALab, dedicated to the automatic derivation of multiscale models of arrays of micro- and nanosystems. In this domain a model is a partial differential equation. Multiscale methods approximate it by another partial differential equation which can be numerically simulated in a reasonable time. The challenge consists in taking into account a wide range of geometries combining thin and periodic structures with the possibility of multiple nested scales. In this paper we present a transformation language that will make the development of MEMSALab more feasible. It is proposed as a Maple package for rule-based programming, rewriting strategies and their combination with standard Maple code. We illustrate the practical interest of this language by using it to encode two examples of multiscale derivations, namely the two-scale limit of the derivative operator and the two-scale model of the stationary heat equation.

cs.SC↗
arXiv2011

A Generalized Criterion for Signature Related Gröbner Basis Algorithms

A generalized criterion for signature related algorithms to compute Gröbner basis is proposed in this paper. Signature related algorithms are a popular kind of algorithms for computing Gröbner basis, including the famous F5 algorithm, the extended F5 algorithm and the GVW algorithm. The main purpose of current paper is to study in theory what kind of criteria is correct in signature related algorithms and provide a generalized method to develop new criteria. For this purpose, a generalized criterion is proposed. The generalized criterion only relies on a general partial order defined on a set of polynomials. When specializing the partial order to appropriate specific orders, the generalized criterion can specialize to almost all existing criteria of signature related algorithms. For {\em admissible} partial orders, a complete proof of the correctness of the algorithm based on this generalized criterion is also presented. This proof has no extra requirements on the computing order of critical pairs, and is also valid for non-homogeneous polynomial systems. More importantly, the partial orders implied by existing criteria are admissible. Besides, one can also check whether a new criterion is correct in signature related algorithms or even develop new criteria by using other admissible partial orders in the generalized criterion.

cs.SC↗
arXiv2011

Signature-based algorithms to compute Groebner bases

This paper describes a Buchberger-style algorithm to compute a Groebner basis of a polynomial ideal, allowing for a selection strategy based on "signatures". We explain how three recent algorithms can be viewed as different strategies for the new algorithm, and how other selection strategies can be formulated. We describe a fourth as an example. We analyze the strategies both theoretically and empirically, leading to some surprising results.

math.AC↗
arXiv2011

Multiplicity Preserving Triangular Set Decomposition of Two Polynomials

In this paper, a multiplicity preserving triangular set decomposition algorithm is proposed for a system of two polynomials. The algorithm decomposes the variety defined by the polynomial system into unmixed components represented by triangular sets, which may have negative multiplicities. In the bivariate case, we give a complete algorithm to decompose the system into multiplicity preserving triangular sets with positive multiplicities. We also analyze the complexity of the algorithm in the bivariate case. We implement our algorithm and show the effectiveness of the method with extensive experiments.

cs.SC↗
arXiv2011

Diversification improves interpolation

We consider the problem of interpolating an unknown multivariate polynomial with coefficients taken from a finite field or as numerical approximations of complex numbers. Building on the recent work of Garg and Schost, we improve on the best-known algorithm for interpolation over large finite fields by presenting a Las Vegas randomized algorithm that uses fewer black box evaluations. Using related techniques, we also address numerical interpolation of sparse polynomials with complex coefficients, and provide the first provably stable algorithm (in the sense of relative error) for this problem, at the cost of modestly more evaluations. A key new technique is a randomization which makes all coefficients of the unknown polynomial distinguishable, producing what we call a diverse polynomial. Another departure from most previous approaches is that our algorithms do not rely on root finding as a subroutine. We show how these improvements affect the practical performance with trial implementations.

cs.SC↗
arXiv2011

Univariate real root isolation in an extension field

We present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial in $B_α \in L[y]$, where $L=\QQ(α)$ is a simple algebraic extension of the rational numbers. We consider two approaches for tackling the problem. In the first approach using resultant computations we perform a reduction to a polynomial with integer coefficients. We compute separation bounds for the roots, and using them we deduce that we can isolate the real roots of $B_α$ in $\sOB(N^{10})$, where $N$ is an upper bound on all the quantities (degree and bitsize) of the input polynomials. In the second approach we isolate the real roots working directly on the polynomial of the input. We compute improved separation bounds for real roots and we prove that they are optimal, under mild assumptions. For isolating the roots we consider a modified Sturm's algorithm, and a modified version of \func{descartes}' algorithm introduced by Sagraloff. For the former we prove a complexity bound of $\sOB(N^8)$ and for the latter a bound of $\sOB(N^{7})$. We implemented the algorithms in \func{C} as part of the core library of \mathematica and we illustrate their efficiency over various data sets. Finally, we present complexity results for the general case of the first approach, where the coefficients belong to multiple extensions.

cs.SC↗
arXiv2017

Independence of hyperlogarithms over function fields via algebraic combinatorics

We obtain a necessary and sufficient condition for the linear independence of solutions of differential equations for hyperlogarithms. The key fact is that the multiplier (i.e. the factor $M$ in the differential equation $dS=MS$) has only singularities of first order (Fuchsian-type equations) and this implies that they freely span a space which contains no primitive. We give direct applications where we extend the property of linear independence to the largest known ring of coefficients.

math.CO↗
arXiv2011

A Note on the Group-theoretic Approach to Fast Matrix Multiplication

In 2003 COHN and UMANS introduced a group-theoretic approach to fast matrix multiplication. This involves finding large subsets S, T and U of a group G satisfying the Triple Product Property (TPP) as a means to bound the exponent $ω$ of the matrix multiplication. We show that S, T and U may be be assumed to contain the identity and be otherwise disjoint. We also give a much shorter proof of the upper bound |S|+|T|+|U| <= |G|+2.

math.GR↗
arXiv2011

Computing Semi-algebraic Invariants for Polynomial Dynamical Systems

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. The main idea is encoding relevant dynamical properties as conditions on the high order Lie derivatives of polynomials occurring in the SAI. A direct consequence of this criterion is a relatively complete method of SAI generation based on template assumption and semi-algebraic constraint solving. Relative completeness means if there is an SAI in the form of a predefined template, then our method can indeed find one using this template.

cs.SC↗
arXiv2011

Generalized companion matrix for approximate GCD

We study a variant of the univariate approximate GCD problem, where the coefficients of one polynomial f(x)are known exactly, whereas the coefficients of the second polynomial g(x)may be perturbed. Our approach relies on the properties of the matrix which describes the operator of multiplication by gin the quotient ring C[x]=(f). In particular, the structure of the null space of the multiplication matrix contains all the essential information about GCD(f; g). Moreover, the multiplication matrix exhibits a displacement structure that allows us to design a fast algorithm for approximate GCD computation with quadratic complexity w.r.t. polynomial degrees.

cs.SC↗
arXiv2011

Root Isolation of Zero-dimensional Polynomial Systems with Linear Univariate Representation

In this paper, a linear univariate representation for the roots of a zero-dimensional polynomial equation system is presented, where the roots of the equation system are represented as linear combinations of roots of several univariate polynomial equations. The main advantage of this representation is that the precision of the roots can be easily controlled. In fact, based on the linear univariate representation, we can give the exact precisions needed for roots of the univariate equations in order to obtain the roots of the equation system to a given precision. As a consequence, a root isolation algorithm for a zero-dimensional polynomial equation system can be easily derived from its linear univariate representation.

cs.SC↗
arXiv2011

SqFreeEVAL: An (almost) optimal real-root isolation algorithm

Let f be a univariate polynomial with real coefficients, f in R[X]. Subdivision algorithms based on algebraic techniques (e.g., Sturm or Descartes methods) are widely used for isolating the real roots of f in a given interval. In this paper, we consider a simple subdivision algorithm whose primitives are purely numerical (e.g., function evaluation). The complexity of this algorithm is adaptive because the algorithm makes decisions based on local data. The complexity analysis of adaptive algorithms (and this algorithm in particular) is a new challenge for computer science. In this paper, we compute the size of the subdivision tree for the SqFreeEVAL algorithm. The SqFreeEVAL algorithm is an evaluation-based numerical algorithm which is well-known in several communities. The algorithm itself is simple, but prior attempts to compute its complexity have proven to be quite technical and have yielded sub-optimal results. Our main result is a simple O(d(L+ln d)) bound on the size of the subdivision tree for the SqFreeEVAL algorithm on the benchmark problem of isolating all real roots of an integer polynomial f of degree d and whose coefficients can be written with at most L bits. Our proof uses two amortization-based techniques: First, we use the algebraic amortization technique of the standard Mahler-Davenport root bounds to interpret the integral in terms of d and L. Second, we use a continuous amortization technique based on an integral to bound the size of the subdivision tree. This paper is the first to use the novel analysis technique of continuous amortization to derive state of the art complexity bounds.

cs.DS↗
arXiv2011

Special Values of Generalized Log-sine Integrals

We study generalized log-sine integrals at special values. At $π$ and multiples thereof explicit evaluations are obtained in terms of Nielsen polylogarithms at $\pm1$. For general arguments we present algorithmic evaluations involving Nielsen polylogarithms at related arguments. In particular, we consider log-sine integrals at $π/3$ which evaluate in terms of polylogarithms at the sixth root of unity. An implementation of our results for the computer algebra systems Mathematica and SAGE is provided.

math.CA↗