Journal of Symbolic Computation最新文献

英文中文

Wronski pairs of honeycomb curves 蜂窝曲线的朗斯基对

IF 1.1 4区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Journal of Symbolic Computation

Pub Date : 2025-11-17 DOI: 10.1016/j.jsc.2025.102528

Laura Casabella , Michael Joswig , Rafael Mohr

We study certain generic systems of real polynomial equations associated with triangulations of convex polytopes and investigate their number of real solutions. Our main focus is set on pairs of plane algebraic curves which form a so-called Wronski system. The computational tasks arising in the analysis of such Wronski pairs lead us to the frontiers of current computer algebra algorithms and their implementations, both via Gröbner bases and numerical algebraic geometry.

研究了一类与凸多面体三角剖分相关的实多项式方程组，并研究了它们的实解个数。我们的主要焦点集中在平面代数曲线对上，它们形成了一个所谓的Wronski系统。在分析这些Wronski对中产生的计算任务将我们带到了当前计算机代数算法及其实现的前沿，包括Gröbner基和数值代数几何。

引用次数: 0

Equi-affine minimal-degree moving frames for polynomial curves 多项式曲线的等仿射最小度运动框架

IF 1.1 4区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Journal of Symbolic Computation

Pub Date : 2025-11-17 DOI: 10.1016/j.jsc.2025.102530

Hoon Hong, Irina A. Kogan

We develop a theory and an algorithm for constructing minimal-degree polynomial moving frames for polynomial curves in an affine space. The algorithm is equivariant under volume-preserving affine transformations of the ambient space and parameter shifts. We show that any matrix-completion algorithm can be turned into an equivariant moving frame algorithm via an equivariantization procedure that we develop. We prove that if a matrix-completion algorithm is of minimal degree, so is the resulting equivariant moving frame algorithm. We propose a novel minimal-degree matrix-completion algorithm, complementing the existing body of literature on this topic.

提出了一种构造仿射空间中多项式曲线的最小次多项式运动框架的理论和算法。该算法在环境空间的保体积仿射变换和参数移位下是等变的。我们证明了任何矩阵补全算法都可以通过我们开发的等变过程转化为等变移动帧算法。我们证明了如果矩阵补全算法是最小度的，那么由此得到的等变移动帧算法也是最小度的。我们提出了一种新的最小度矩阵补全算法，补充了关于该主题的现有文献。

引用次数: 0

Postulation of lines in P3 revisited 重新考虑P3中的行假设

IF 1.1 4区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Journal of Symbolic Computation

Pub Date : 2025-11-17 DOI: 10.1016/j.jsc.2025.102525

Marcin Dumnicki , Mikołaj Le Van , Grzegorz Malara , Tomasz Szemberg , Justyna Szpond , Halszka Tutaj-Gasińska

The purpose of the present note is to provide a new proof of the well-known result due to Hartshorne and Hirschowitz to the effect that general lines in projective spaces have good postulation. Our approach uses specialization to a hyperplane and thus opens a door to study postulation of general codimension 2 linear subspaces in projective spaces.

本文的目的是为Hartshorne和Hirschowitz的著名结论提供一个新的证明，证明投影空间中的一般线具有良好的公设。我们的方法对超平面进行专门化，从而为研究射影空间中一般余维数为2的线性子空间的假设打开了大门。

引用次数: 0

Generalised Buchberger and Schreyer algorithms for strongly discrete coherent rings 强离散相干环的广义Buchberger和Schreyer算法

IF 1.1 4区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Journal of Symbolic Computation

Pub Date : 2025-11-14 DOI: 10.1016/j.jsc.2025.102526

Henri Lombardi , Stefan Neuwirth , Ihsen Yengui

Let M be a finitely generated submodule of a free module over a multivariate polynomial ring with coefficients in a discrete coherent ring. We prove that its module

MLT (M)

of leading terms is countably generated and give an algorithm for computing explicitly a generating set. This result is also useful when

MLT (M)

is not finitely generated. Suppose that the base ring is strongly discrete coherent. We provide a Buchberger-like algorithm and prove that it converges if and only if the module of leading terms is finitely generated. We also state a constructive version of Hilbert's syzygy theorem by following Schreyer's method.

设M是离散相干环上具有系数的多元多项式环上自由模的有限生成子模。证明了其前导项的模MLT(M)是可数生成的，并给出了显式计算生成集的算法。当MLT(M)不是有限地生成时，这个结果也很有用。假设基环是强离散相干的。我们提供了一个类buchberger算法，并证明当且仅当前导项模块有限生成时它收敛。根据施赖尔的方法，给出了希尔伯特syzygy定理的构造式。

引用次数: 0

Efficient detection of redundancies in systems of linear inequalities 线性不等式系统中冗余的有效检测

IF 1.1 4区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Journal of Symbolic Computation

Pub Date : 2025-11-14 DOI: 10.1016/j.jsc.2025.102527

Rui-Juan Jing , Marc Moreno Maza , Chirantan Mukherjee , Yan-Feng Xie , Chun-Ming Yuan

Fourier-Motzkin elimination is a fundamental operation in polyhedral geometry. It can be performed by several equivalent procedures, and can be regarded as an adaptation of Gaussian elimination to systems of linear inequalities. These procedures tend to generate large numbers of redundant inequalities. Efficiently detecting these redundancies is essential for obtaining software implementation of practical interest. In this paper, we propose a novel detection technique. We demonstrate its benefits over alternative approaches. A detailed experimentation is reported.

傅里叶-莫兹金消去是多面体几何中的一个基本运算。它可以通过几个等效过程来实现，并且可以看作是对线性不等式系统的高斯消去的一种适应。这些程序往往会产生大量的冗余不等式。有效地检测这些冗余对于获得实用的软件实现至关重要。本文提出了一种新的检测技术。我们将展示其优于其他方法的优点。并进行了详细的实验。

引用次数: 0

q-analogues of π-formulas due to Ramanujan and Guillera Ramanujan和Guillera的π-公式的q-类似物

IF 1.1 4区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Journal of Symbolic Computation

Pub Date : 2025-11-14 DOI: 10.1016/j.jsc.2025.102531

John M. Campbell

The first known q-analogues for any of the 17 formulas for

\frac{1}{π}

due to Ramanujan were introduced in 2018 by Guo and Liu (2018), via the q-Wilf–Zeilberger method. Through a “normalization” method, which we refer to as EKHAD-normalization, based on the q-polynomial coefficients involved in first-order difference equations obtained from the q-version of Zeilberger's algorithm, we introduce q-WZ pairs that extend WZ pairs introduced by Guillera, 2002, Guillera, 2006. We apply our EKHAD-normalization method to prove four new q-analogues for three of Ramanujan's formulas for

\frac{1}{π}

along with q-analogues of Guillera's first two series for

\frac{1}{π^{2}}

. Our normalization method does not seem to have been previously considered in any equivalent way in relation to q-series, and this is substantiated through our survey on previously known q-analogues of Ramanujan-type series for

\frac{1}{π}

and of Guillera's series for

\frac{1}{π^{2}}

. We conclude by showing how our method can be adapted to further extend Guillera's WZ pairs by introducing hypergeometric expansions for

\frac{1}{π^{2}}

郭和刘（2018）在2018年通过q-Wilf-Zeilberger方法引入了拉马努金的17个1π公式中的任何一个已知的q-类似物。通过一种“归一化”方法，我们称之为ekhad归一化，基于Zeilberger算法的q-version得到的一阶差分方程中涉及的q-多项式系数，我们引入了q-WZ对，扩展了Guillera， 2002, Guillera， 2006引入的WZ对。我们应用ekhad归一化方法证明了Ramanujan的三个1π公式的四个新的q-类似物以及Guillera的前两个1π2级数的q-类似物。我们的归一化方法似乎以前没有考虑过与q级数相关的任何等效方式，这一点通过我们对已知的1π的ramanujan型级数和1π2的Guillera级数的q-类似物的调查得到证实。最后，我们展示了我们的方法如何通过引入1π2的超几何展开来进一步扩展Guillera的WZ对。

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引用次数: 0

Fast in-place accumulation 就地快速堆积

IF 1.1 4区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Journal of Symbolic Computation

Pub Date : 2025-10-15 DOI: 10.1016/j.jsc.2025.102523

Jean-Guillaume Dumas, Bruno Grenet

<div><div>This paper deals with simultaneously fast and in-place algorithms for formulae where the result has to be linearly accumulated: some output variables are also input variables, linked by a linear dependency. Fundamental examples include the in-place accumulated multiplication of polynomials or matrices, <math><mi>C</mi><mo>+</mo><mo>=</mo><mi>A</mi><mi>B</mi></math> (that is with only <math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math> extra space). The difficulty is to combine in-place computations with fast algorithms: those usually come at the expense of (potentially large) extra temporary space, but with accumulation the output variables are not even available to store intermediate values. We first propose a novel automatic design of fast and in-place accumulating algorithms for any bilinear formulae (and thus for polynomial and matrix multiplication) and then extend it to any linear accumulation of a collection of functions. For this, we relax the in-place model to any algorithm allowed to modify its inputs, provided that those are restored to their initial state afterwards. This allows us to ultimately derive unprecedented in-place accumulating algorithms for fast polynomial multiplications and for Strassen-like matrix multiplications.</div><div>We then consider the simultaneously fast and in-place computation of the Euclidean polynomial modular remainder <math><mi>R</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>≡</mo><mi>A</mi><mo>(</mo><mi>X</mi><mo>)</mo><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mi>B</mi><mo>(</mo><mi>X</mi><mo>)</mo></math>. Fast algorithms for this usually also come at the expense of a linear amount of extra temporary space. In particular, they require one to first compute and store the whole quotient <math><mi>Q</mi><mo>(</mo><mi>X</mi><mo>)</mo></math> such that <math><mi>A</mi><mo>=</mo><mi>B</mi><mi>Q</mi><mo>+</mo><mi>R</mi></math>. We here propose an in-place algorithm to compute the remainder only. If A and B have respective degree <math><mi>m</mi><mo>+</mo><mi>n</mi></math> and n, and <math><mi>M</mi><mo>(</mo><mi>k</mi><mo>)</mo></math> denotes the complexity of a (not-in-place) algorithm to multiply two degree-k polynomials, our algorithm uses at most <math><mi>O</mi><mrow><mo>(</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow></mfrac><mi>M</mi><mo>(</mo><mi>n</mi><mo>)</mo><mi>log</mi><mo>⁡</mo><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo></mrow></math> arithmetic operations. In this particular case this is a factor <math><mi>log</mi><mo>⁡</mo><mo>(</mo><mi>n</mi><mo>)</mo></math> more than the not-in-place algorithm. But if <math><mi>M</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mi>Θ</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>ϵ</mi></mrow></msup><mo>)</mo></math> for some <ma

本文同时处理公式的快速和就地算法，其中结果必须是线性累积的：一些输出变量也是输入变量，通过线性依赖关系连接。基本的例子包括多项式或矩阵的就地累积乘法，C+=AB（即只有O(1)个额外空间）。困难之处在于将就地计算与快速算法结合起来：这些计算通常以（可能很大的）额外临时空间为代价，但是随着累积，输出变量甚至无法用于存储中间值。我们首先提出了一种新的快速和就地积累算法的自动设计，适用于任何双线性公式（从而适用于多项式和矩阵乘法），然后将其扩展到函数集合的任何线性积累。为此，我们将就地模型放宽为允许修改其输入的任何算法，前提是这些输入随后恢复到初始状态。这使我们最终能够为快速多项式乘法和Strassen-like矩阵乘法推导出前所未有的就地累积算法。然后我们考虑欧几里得多项式模余数R(X)≡A(X)modB(X)的同时快速就地计算。快速算法通常也以额外的线性临时空间为代价。特别是，它们要求首先计算并存储整个商Q(X)，使A=BQ+R。我们在此提出一种只计算余数的就地算法。如果A和B的阶分别为m+n和n， m (k)表示两个k阶多项式相乘的（非原位）算法的复杂度，则我们的算法最多使用O(mnM（n)log (n)）个算术运算。在这种特殊情况下，它比非原地算法大log (n)个因子。但是，如果M(n)=Θ（n1+ λ）对于某些ϵ>；0，那么我们的算法确实符合O（mnM(n)）的非原位复杂性界限。我们还提出了计算变体-仍然在原地并且具有相同类型的复杂性界限-过位余数A(X)≡A(X)modB(X)，累积余数R(X)+=A(X)modB(X)和累积模乘法R(X)+=A(X)C(X)modB(X)，即在有限域的多项式扩展中的乘法。为了实现这一点，我们开发了Toeplitz矩阵运算，广义卷积，短乘积和幂级数除法和余数的技术，其输出也是输入的一部分。

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引用次数: 0

Towards automated generation of fast and accurate algorithms for recursive matrix multiplication 实现递归矩阵乘法快速准确算法的自动生成

IF 1.1 4区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Journal of Symbolic Computation

Pub Date : 2025-10-15 DOI: 10.1016/j.jsc.2025.102524

Jean-Guillaume Dumas , Clément Pernet , Alexandre Sedoglavic

We propose a strategy for the generation of fast and accurate versions of non-commutative recursive matrix multiplication algorithms. To generate these algorithms, we consider matrix and tensor norm bounds governing the stability and accuracy of numerical matrix multiplication. We start by a unification on known max-norm bounds on matrix multiplication stability and then extend them to further norms and more generally to recursive bilinear algorithms and the alternative basis matrix multiplication algorithms. Then our strategy has three phases. First, we reduce those bounds by minimizing a growth factor along the orbits of the associated matrix multiplication tensor decomposition. Second, we develop heuristics that minimize the number of operations required to realize a bilinear formula, while further improving its accuracy. Third, we perform an alternative basis sparsification that improves on the time complexity constant and mostly preserves the overall accuracy. Let

〈 m \times k \times n : r 〉

denote a non-commutative algorithm multiplying an

m \times k

matrix with an

k \times n

matrix, using r coefficient products. Our strategy allows us to propose a

〈 2 \times 2 \times 2 : 7 〉

algorithm that reaches simultaneously a better accuracy in practice, compared to previously known such fast

〈 2 \times 2 \times 2 : 7 〉

Strassen-like algorithms, and a time complexity bound with the best currently known leading term (obtained via alternative basis sparsification). We also present detailed results of our technique on Smirnov's

〈 3 \times 3 \times 6 : 40 〉

family of algorithms.

我们提出了一种生成快速而准确的非交换递归矩阵乘法算法的策略。为了生成这些算法，我们考虑了控制数值矩阵乘法的稳定性和准确性的矩阵和张量范数边界。我们从矩阵乘法稳定性的已知最大范数界的统一开始，然后将其推广到进一步的范数，更一般地推广到递归双线性算法和备选基矩阵乘法算法。我们的战略分为三个阶段。首先，我们通过最小化相关矩阵乘法张量分解轨道上的生长因子来减小这些边界。其次，我们开发了启发式算法，以最大限度地减少实现双线性公式所需的操作次数，同时进一步提高其准确性。第三，我们执行了一种替代的基稀疏化，它提高了时间复杂度常数，并在很大程度上保持了总体精度。令< m×k×n:r >表示使用r系数乘积将m×k矩阵与k×n矩阵相乘的非交换算法。我们的策略允许我们提出一种< 2×2×2:7 >算法，与之前已知的快速< 2×2×2:7 > Strassen-like算法相比，该算法在实践中同时达到更好的精度，并且具有当前已知的最佳领先项的时间复杂度（通过替代基稀疏化获得）。我们还介绍了我们的技术在Smirnov的< 3×3×6:40 >系列算法上的详细结果。

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引用次数: 0

New dimension polynomials of inversive difference field extensions and inversive difference modules 逆差分域扩展和逆差分模的新维多项式

IF 1.1 4区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Journal of Symbolic Computation

Pub Date : 2025-10-14 DOI: 10.1016/j.jsc.2025.102521

Alexander Levin

In this paper we introduce new types of characteristic sets in rings of inversive difference polynomials over an inversive difference field K and in inversive difference K-modules. These characteristic sets are defined via reductions associated with partitions of the basic set of automorphisms σ and with a generalization of the concept of the effective order of an ordinary difference polynomial. Using these characteristic sets, we provide an effective construction of multivariate difference dimension polynomials of a new type. They describe the transcendence degrees of intermediate fields of inversive difference field extensions and dimensions of submodules of inversive difference modules that are obtained by adjoining transforms of the generators whose orders with respect to the components of the partition of σ are bounded by two sequences of natural numbers. We show that such dimension polynomials carry substantially more invariants (that is, characteristics that do not depend on the set of generators) than standard (univariate) difference dimension polynomials. We also show how the obtained results can be applied to the equivalence problem for systems of algebraic difference equations and to the isomorphism problem for inversive difference modules.

本文引入了逆差分域K和逆差分K模上逆差分多项式环上的新类型特征集。这些特征集是通过自同构的基本集σ的划分和常差分多项式有效阶的概念的推广来定义的。利用这些特征集，我们提供了一种新的多元差分维多项式的有效构造方法。它们描述了逆差分域扩展的中间域的超越度和逆差分模的子模的维数，这些子模是由对σ的分块的阶由两个自然数序列有界的发生器的相邻变换得到的。我们表明，与标准（单变量）差分维多项式相比，这种维多项式具有更多的不变量（即，不依赖于生成器集的特征）。我们还展示了如何将所得结果应用于代数差分方程系统的等价问题和逆差分模的同构问题。

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引用次数: 0

Sasbi-standard bases of modules sasbi标准的模块基础

IF 1.1 4区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Journal of Symbolic Computation

Pub Date : 2025-10-14 DOI: 10.1016/j.jsc.2025.102522

Nazia Jabeen, Junaid Alam Khan

The theory of Standard bases of modules over the localization of a polynomial ring (with respect to a local ordering >) is well-developed. For a polynomial subalgebra A, the concept of “Subalgebra bases” of

A_{>}

(localization of A with respect to >) is well-defined, known as Sasbi bases (subalgebra analogue of Standard bases for ideals). In this paper, we have developed and proposed a theory for Standard bases of modules over

A_{>}

(admitting a finite Sasbi basis), termed as Sasbi-Standard bases. The main goal is to develop a constructive algorithm for the computation of these bases, which we implement in the computer algebra system SINGULAR. We also present an application of these bases to compute singularity invariants: the monoid of values of an algebroid curve and the monomodule of values of the Kähler's differentials.

多项式环的局部化上模的标准基理论（关于局部序）得到了很好的发展。对于多项式子代数a， A>的“子代数基”（a相对于>的局部化）的概念是定义良好的，称为Sasbi基（理想标准基的子代数模拟）。在本文中，我们发展并提出了A>上模块的标准基理论（承认有限的Sasbi基），称为Sasbi-标准基。本文的主要目标是开发一种计算这些基的构造算法，并在计算机代数系统SINGULAR中实现。我们也给出了这些基在计算奇异不变量的一个应用：代数曲线的值的单模和Kähler的微分值的单模。

引用次数: 0

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