Journal of Symbolic Computation最新文献

This paper proposes ideas to speed up the process of creative telescoping, particularly when the telescoper is reducible. One can interpret telescoping as computing an annihilator $L\in D$ for an element m in a D-module M. The main idea in this paper is to look for submodules of M. If N is a non-trivial submodule of M, constructing the minimal annihilator R of the image of m in $M/N$ gives a right-factor of L in D. Then $L=L^{\prime }R$ where the left-factor $L^{\prime }$ is the telescoper of $R\left(m\right)\in N$. To expedite computing $L^{\prime }$, compute the action of D on a natural basis of N, then obtain $L^{\prime }$ with a cyclic vector computation.

The next main idea is to construct submodules from automorphisms, if we can find some. An automorphism with distinct eigenvalues can be used to decompose N as a direct sum $N_1\oplus \cdots \oplus N_k$. Then $L^{\prime }$ is the LCLM (Least Common Left Multiple) of $L_1,\dots ,L_k$ where $L_i$ is the telescoper of the projection of $R\left(m\right)$ on $N_i$. An LCLM can greatly increase the degrees of coefficients, so $L^{\prime }$ and L can be much larger expressions than the factors $L_1,\dots ,L_k$ and R. Examples show that computing each factor $L_i$ and R separately can save a lot of CPU time compared to computing L in expanded form with standard creative telescoping.

We propose a way to split a given bivariate P-recursive sequence into a summable part and a non-summable part in such a way that the non-summable part is minimal in some sense. This decomposition gives rise to a new reduction-based creative telescoping algorithm based on the concept of integral bases.

We develop a new eigenvalue method for solving structured polynomial equations over any field. The equations are defined on a projective algebraic variety which admits a rational parameterization by a Khovanskii basis, e.g., a Grassmannian in its Plücker embedding. This generalizes established algorithms for toric varieties, and introduces the effective use of Khovanskii bases in computer algebra. We investigate regularity questions and discuss several applications.

Creative telescoping is an algorithmic method initiated by Zeilberger to compute definite sums by synthesizing summands that telescope, called certificates. We describe a creative telescoping algorithm that computes telescopers for definite sums of D-finite functions as well as the associated certificates in a compact form. The algorithm relies on a discrete analogue of the generalized Hermite reduction, or equivalently, a generalization of the Abramov-Petkovšek reduction. We provide a Maple implementation with good timings on a variety of examples.

We introduce hypergeometric-type sequences. They are linear combinations of interlaced hypergeometric sequences (of arbitrary interlacements). We prove that they form a subring of the ring of holonomic sequences. An interesting family of sequences in this class are those defined by trigonometric functions with linear arguments in the index and π, such as Chebyshev polynomials, ${({\sin }^2(n\pi /4)\cdot \cos (n\pi /6))}_n$, and compositions like ${(\sin (\cos (n\pi /3\left)\pi \right))}_n$.

We describe an algorithm that computes a hypergeometric-type normal form of a given holonomic nth term whenever it exists. Our implementation enables us to generate several identities for terms defined via trigonometric functions.

We present the Julia package ToricAtiyahBott.jl, providing an easy way to perform the Atiyah–Bott formula on the moduli space of genus 0 stable maps ${\bar{M}}_{0,m}(X,\beta )$ where X is any smooth projective toric variety, and β is any effective 1-cycle. The list of the supported cohomological cycles contains the most common ones, and it is extensible. We provide a detailed explanation of the algorithm together with many examples and applications. The toric variety X, as well as the cohomology class β, must be defined using the package Oscar.jl.

In this paper, we examine the structure of systems that are weighted homogeneous for several systems of weights, and how it impacts the computation of Gröbner bases. We present several linear algebra algorithms for computing Gröbner bases for systems with this structure, either directly or by reducing to existing structures. We also present suitable optimization techniques.

As an opening towards complexity studies, we discuss potential definitions of regularity and prove that they are generic if non-empty. Finally, we present experimental data from a prototype implementation of the algorithms in SageMath.

An Infinite and finite dimensional generalized Hilbert tensor with a is positive definite if and only if $a>0$. The infinite dimensional generalized Hilbert tensor related operators $F_{\infty }$ and $T_{\infty }$ are bounded, continuous and positively homogeneous. A generalized Cauchy tensor of which generating vectors are $c,d$ is positive definite if and only if every element of vector d is not zero and each element of vector c is positive and mutually distinct. The 4th order n-dimensional generalized Cauchy tensor is matrix positive semi-definite if and only if every element of generating vector c is positive. Finally, the other properties of generalized Cauchy tensor are presented.

In this paper, we study parametric semidefinite programs (SDPs) where the solution space of both the primal and dual problems change simultaneously. Given a bounded set, we aim to find the a priori unknown maximal permissible perturbation set within it where the semidefinite program problem has a unique optimum and is analytic with respect to the parameters. Our approach reformulates the parametric SDP as a system of partial differential equations (PDEs) where this maximal analytical permissible set (MAPS) is the set on which the system of PDEs is well-posed. A sweeping Euler scheme is developed to approximate this a priori unknown perturbation set. We prove local and global error bounds for this second-order sweeping Euler scheme and demonstrate the method in comparison to existing SDP solvers and its performance on several two-parameter and three-parameter SDPs for which the MAPS can be visualized.

A cofactor representation of an ideal element, that is, a representation in terms of the generators, can be considered as a certificate for ideal membership. Such a representation is typically not unique, and some can be a lot more complicated than others. In this work, we consider the problem of computing sparsest cofactor representations, i.e., representations with a minimal number of terms, of a given element in a polynomial ideal. While we focus on the more general case of noncommutative polynomials, all results also apply to the commutative setting.

We show that the problem of computing cofactor representations with a bounded number of terms is decidable and $\text{NP}$-complete. Moreover, we provide a practical algorithm for computing sparse (not necessarily optimal) representations by translating the problem into a linear optimization problem and by exploiting properties of signature-based Gröbner basis algorithms. We show that, for a certain class of ideals, representations computed by this method are actually optimal, and we present experimental data illustrating that it can lead to noticeably sparser cofactor representations.