Journal of Computational Algebra最新文献

英文中文

Computational and theoretical aspects of rational parametrization of generalized tubular surfaces 广义管状曲面合理参数化的计算与理论研究

Journal of Computational Algebra

Pub Date : 2025-03-01 DOI: 10.1016/j.jaca.2025.100030

J. William Hoffman , Haohao Wang

This paper consists of two components - a computational part and a theoretical part. The former targets the computer-aided geometric design of tubular surfaces. The latter focuses on the algebraic geometry of a family of conic curves. At the application level, we provide a straightforward and easy to implement computational algorithm to rationally parametrize generalized real tubular surfaces via moving lines. We discover that syzygies, i.e., moving lines, can be calculated directly from a given implicit equation of a projective conic. Specifically, we describe two linear polynomial vectors in 3-space whose entries are formulated in terms of the coefficients of the given implicit equation of the conic. We then prove that these two vectors are, in fact, a μ-basis, the generators for the syzygy module of the given conic, and furnish the rational parametrization of the given conic. At the theoretical level, we first briefly review the classical projection method for a rational parametrization of a generic non-degenerate conic. This is compared to the syzygy method, i.e., moving lines. We conclude the paper with an illustrative figure that depicts and compares the classical projection method and our moving line method.

本文由计算部分和理论部分两部分组成。前者针对的是管状表面的计算机辅助几何设计。后者侧重于一组二次曲线的代数几何。在应用层面，我们提供了一种简单易行的计算算法，通过移动线合理地参数化广义实管曲面。我们发现可以从给定的投影二次曲线的隐式方程直接计算出合子，即移动线。具体地说，我们描述了三维空间中的两个线性多项式向量，它们的项是用给定的二次曲线隐式方程的系数表示的。然后证明了这两个向量实际上是一个μ基，是给定二次曲线的合模的生成子，并给出了给定二次曲线的有理参数化。在理论层面上，我们首先简要回顾了一般非退化二次曲线的有理参数化的经典投影方法。这与syzygy方法（即移动线条）相比较。最后，我们用一个插图来描述和比较经典投影法和我们的移动线法。

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引用次数: 0

Differential uniformity properties of some classes of permutation polynomials 几类置换多项式的微分均匀性

Journal of Computational Algebra

Pub Date : 2025-03-01 DOI: 10.1016/j.jaca.2025.100035

Kirpa Garg , Sartaj Ul Hasan , Pantelimon Stănică

The notion of c-differential uniformity has received a lot of attention since its proposal [5], and recently a characterization of perfect c-nonlinear functions in terms of difference sets in some quasigroups was obtained in [1]. Moreover, in a very recent manuscript by Pal and Stănică [19], an intriguing connection was discovered showing that in fact, the boomerang uniformity for an odd APN function (odd characteristic) equals its c-differential uniformity when

c = - 1

, if the function is a permutation, otherwise it is the maximum of the

(- 1)

-DDT entries disregarding the first row/column. The construction of functions, especially permutations, with low c-differential uniformity is an interesting and difficult mathematical problem in this area, and recent work has focused heavily in this direction. We provide a few classes of permutation polynomials with low c-differential uniformity. The used technique involves handling various Weil sums, as well as analyzing some equations in finite fields, and we believe these can be of independent interest, from a mathematical perspective.

自提出[5]以来，c-微分均匀性的概念受到了广泛的关注，最近在[1]中得到了一些准群中完备c-非线性函数在差分集上的刻画。此外，在Pal和striturnicei[19]最近的一篇手稿中，发现了一个有趣的联系，表明事实上，当c= - 1时，奇数APN函数（奇数特征）的回旋均匀性等于它的c-微分均匀性，如果函数是一个排列，否则它是(- 1)-DDT项的最大值，不管第一行/列。低c微分均匀性函数的构造，特别是排列的构造，是该领域中一个有趣而困难的数学问题，最近的工作主要集中在这个方向上。我们提供了几类低c微分均匀性的置换多项式。所使用的技术包括处理各种Weil和，以及分析有限域中的一些方程，我们相信从数学的角度来看，这些可能是独立的兴趣。

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引用次数: 0

Zeros of S-characters 0个s字符

Journal of Computational Algebra

Pub Date : 2025-03-01 DOI: 10.1016/j.jaca.2025.100031

Thomas Breuer , Michael Joswig , Gunter Malle

The concept of S-characters of finite groups was introduced by Zhmud' as a generalisation of transitive permutation characters. Any non-trivial S-character takes a zero value on some group element. By a deep result depending on the classification of finite simple groups a non-trivial transitive permutation character even vanishes on some element of prime power order. J-P. Serre asked whether this generalises to S-characters. We provide a translation of this question into the language of polyhedral geometry and thereby construct many counterexamples, using the capabilities of the computer algebra system OSCAR.

有限群s字的概念是由朱木作为传递置换字的推广引入的。任何非平凡的s字符在某个群元素上取零值。通过一个依赖于有限单群分类的深层结果，在一些素数幂次元上，一个非平凡传递置换特征甚至消失。j]。Serre问这是否适用于s字符。我们将这个问题翻译成多面体几何语言，从而利用计算机代数系统OSCAR的功能构建了许多反例。

引用次数: 0

Symmetric perfect 2-colorings of J(10,3) J(10,3) 的对称完全 2 色

Journal of Computational Algebra

Pub Date : 2024-11-19 DOI: 10.1016/j.jaca.2024.100028

Paul Tricot

引用次数: 0

Improved computation of polynomial roots over number fields when using complex embeddings 使用复嵌入时改进数域上多项式根的计算

Journal of Computational Algebra

Pub Date : 2024-10-21 DOI: 10.1016/j.jaca.2024.100026

Andrea Lesavourey , Thomas Plantard , Willy Susilo

We explore a fairly generic method to compute roots of polynomials over number fields through complex embeddings. Our main contribution is to show how to use a structure of a relative extension to decode in a subfield. Additionally we describe several heuristic options to improve practical efficiency. We provide experimental data from our implementation and compare our methods to the state of the art algorithm implemented in Pari/Gp.

我们探索了一种相当通用的方法，通过复嵌入计算数域上多项式的根。我们的主要贡献在于展示了如何利用相对扩展结构在子字段中解码。此外，我们还介绍了几种启发式方案，以提高实际效率。我们提供了实现过程中的实验数据，并将我们的方法与在 Pari/Gp 中实现的最先进算法进行了比较。

引用次数: 0

Signature-based algorithm under non-compatible term orders and its application to change of ordering 非相容词序下基于签名的算法及其在更改词序中的应用

Journal of Computational Algebra

Pub Date : 2024-10-21 DOI: 10.1016/j.jaca.2024.100027

Masayuki Noro

The notion of the compatibility between a term order in a polynomial ring R and a module term order in

R^{l}

is crucial to ensure the termination of a signature-based algorithm for general input ideals. However, it is shown experimentally that the compatibility does not necessarily imply efficient computation. Our experiments show that combining non-compatible term orders can improve performance for computing Gröbner bases with respect to some term orders. In such cases, we can use the Hilbert function to guarantee the termination. The Hilbert function can be computed by using a Gröbner basis with respect to some term order and thus the resulting algorithm is considered a change of ordering algorithm. In this paper, we give the details of the new change of ordering algorithm and we compare its performance with that of the usual Hilbert-driven Buchberger algorithm and the Gröbner walk algorithm.

多项式环 R 中的阶次与 Rl 中的模块阶次之间的兼容性概念对于确保基于签名的一般输入理想算法的终止至关重要。然而，实验表明，兼容性并不一定意味着高效计算。我们的实验表明，结合不兼容的项阶可以提高计算格罗伯纳基时某些项阶的性能。在这种情况下，我们可以使用希尔伯特函数来保证终止。希尔伯特函数可以通过使用相对于某些术语阶的格罗伯纳基计算出来，因此由此产生的算法被认为是一种改变阶算法。在本文中，我们给出了新的改序算法的细节，并将其性能与通常的希尔伯特驱动布赫伯格算法和格罗伯纳行走算法进行了比较。

引用次数: 0

Rational Askey–Wilson Bernstein bases and a multirational Askey–Wilson blossom 有理阿斯基-威尔逊伯恩斯坦基和多有理阿斯基-威尔逊花

Journal of Computational Algebra

Pub Date : 2024-10-21 DOI: 10.1016/j.jaca.2024.100025

Plamen Simeonov , Ron Goldman

We introduce and study the properties of new negative degree rational Bernstein bases associated with the Askey–Wilson operator and we use these bases to define new types of rational Bernstein-Bézier curves. We also introduce a new type of blossom, the multirational Askey–Wilson blossom. We prove that four axioms uniquely characterize this blossom and we provide an explicit formula for this multirational blossom involving a right inverse of the Askey–Wilson operator. A formula for the coefficients of a function expanded in a rational Askey–Wilson Bernstein basis in terms of certain values of the Askey–Wilson operator is derived. We also establish a dual functional property that expresses the coefficients of these new types of rational Bernstein–Bézier curves in terms of values of their multirational Askey–Wilson blossom.

我们介绍并研究了与阿斯基-威尔逊算子相关的新负度有理伯恩斯坦基的性质，并利用这些基定义了新型有理伯恩斯坦-贝塞尔曲线。我们还引入了一种新的绽放，即多有理阿斯基-威尔逊绽放。我们证明了四条公理唯一地描述了这种绽放，并提供了涉及阿斯基-威尔逊算子右逆的多向绽放的明确公式。根据阿斯基-威尔逊算子的某些值，我们得出了在有理阿斯基-威尔逊伯恩斯坦基础上展开的函数系数公式。我们还建立了一个对偶函数性质，用它们的多ational Askey-Wilson 开花的值来表示这些新型有理伯恩斯坦-贝塞尔曲线的系数。

引用次数: 0

Factoring perfect reconstruction filter banks into causal lifting matrices: A Diophantine approach 将完美重构滤波器组分解为因果提升矩阵：一种二阶方法

Journal of Computational Algebra

Pub Date : 2024-10-09 DOI: 10.1016/j.jaca.2024.100024

Christopher M. Brislawn

The elementary theory of bivariate linear Diophantine equations over polynomial rings is used to construct causal lifting factorizations (elementary matrix decompositions) for causal two-channel FIR perfect reconstruction transfer matrices and wavelet transforms. The Diophantine approach generates causal factorizations satisfying certain polynomial degree-reducing inequalities, enabling a new factorization strategy called the Causal Complementation Algorithm. This provides a causal (i.e., polynomial, hence realizable) alternative to the noncausal lifting scheme developed by Daubechies and Sweldens using the Extended Euclidean Algorithm for Laurent polynomials. The new approach replaces the Euclidean Algorithm with Gaussian elimination employing a slight generalization of polynomial division that ensures existence and uniqueness of quotients whose remainders satisfy user-specified divisibility constraints. The Causal Complementation Algorithm is shown to be more general than the causal version of the Euclidean Algorithm approach by generating additional causal lifting factorizations beyond those obtainable using the polynomial Euclidean Algorithm.

多项式环上的双变量线性二叉方程的基本理论被用于构建因果双通道 FIR 完美重构传递矩阵和小波变换的因果提升因式分解（基本矩阵分解）。Diophantine 方法生成的因果因数化满足某些多项式程度递减不等式，从而实现了一种名为 "因果补全算法 "的新因数化策略。这为道贝奇斯（Daubechies）和斯韦尔登斯（Sweldens）利用扩展欧几里得算法为劳伦多项式开发的非因果提升方案提供了因果（即多项式，因此可实现）替代方案。新方法用高斯消元法取代欧几里得算法，采用多项式除法的轻微广义化，确保余数满足用户指定的可分性约束的商的存在性和唯一性。因果互补算法比欧几里得算法的因果版本更具通用性，因为它可以生成多项式欧几里得算法所无法获得的因果提升因式。

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引用次数: 0

New condensation methods with applications to the computation of Brauer character tables 应用于计算布劳尔字符表的新凝聚方法

Journal of Computational Algebra

Pub Date : 2024-10-02 DOI: 10.1016/j.jaca.2024.100023

Klaus Lux , A.J.E. Ryba

Condensation is a technique that can often predict a Brauer character table of a finite group with a very high degree of confidence, but without a proof of correctness. In this paper we describe a strategy that can give such a proof. We introduce and apply two novel condensation methods: virtual tensor condensation and the condensation of bilinear forms. We illustrate our strategy and new techniques with examples taken from our computation of the 5-modular Brauer character table of the sporadic simple Lyons group.

凝缩是一种技术，通常能以极高的置信度预测有限群的布劳尔特征表，但却无法证明其正确性。在本文中，我们描述了一种可以给出这种证明的策略。我们引入并应用了两种新颖的凝聚方法：虚拟张量凝聚和双线性形式凝聚。我们以计算零星简单里昂群的 5 模布劳尔特征表为例，说明我们的策略和新技术。

引用次数: 0

Monomial-agnostic computation of vanishing ideals 消失理想的一元对立计算

Journal of Computational Algebra

Pub Date : 2024-09-01 DOI: 10.1016/j.jaca.2024.100022

Hiroshi Kera , Yoshihiko Hasegawa

Approximate basis computation of vanishing ideals has recently been studied extensively in computational algebra and data-driven applications such as machine learning. However, symbolic computation and the dependency on term order remain essential gaps between the two fields. In this study, we present the first monomial-agnostic basis computation, which works fully numerically with proper normalization and without term order. This is realized by gradient normalization, a newly proposed data-dependent normalization that normalizes a polynomial with the magnitude of gradients at given points. Its data-dependent nature brings various advantages: i) efficient resolution of the spurious vanishing problem, the scale-variance issue of approximately vanishing polynomials, without accessing coefficients of terms, ii) scaling-consistent basis computation, ensuring that input scaling does not lead to an essential change in the output, and iii) robustness against input perturbations, where the upper bound of error is determined only by the magnitude of the perturbations. Existing studies did not achieve any of these. As further applications of gradient information, we propose a monomial-agnostic basis reduction method and a regularization method to manage positive-dimensional ideals.

近来，计算代数和数据驱动应用（如机器学习）对消失理想的近似基础计算进行了广泛研究。然而，符号计算和对项阶的依赖仍然是这两个领域之间的重要差距。在本研究中，我们首次提出了单项式无关基础计算，它可以通过适当的归一化实现完全数值计算，且无需项阶。这是通过梯度归一化实现的，梯度归一化是一种新提出的依赖数据的归一化，它根据给定点的梯度大小对多项式进行归一化。它与数据相关的特性带来了各种优势：i) 有效解决虚假消失问题，即近似消失多项式的尺度方差问题，而无需访问项的系数；ii) 与缩放一致的基础计算，确保输入缩放不会导致输出的本质变化；iii) 对输入扰动的鲁棒性，误差上限仅由扰动的大小决定。现有的研究并没有实现上述任何一点。作为梯度信息的进一步应用，我们提出了一种与单项式无关的基础缩减方法和一种管理正维理想的正则化方法。

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Journal of Computational Algebra

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