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Using the Interval-Symbol Method with Zero Rewriting to factor polynomials over algebraic number fields

ACM COMMUNICATIONS IN computer algebra 2023年第2期57卷 21-30页

作者： Okuda, Kazuki Shirayanagi, Kiyoshi Toho Univ 2-2-1 Miyama Funabashi Chiba 2748510 Japan

The ISZ method (Interval-Symbol method with Zero rewriting) based on stabilization theory was proposed to reduce the amount of exact computations as much as possible but obtain the exact results by aid of floating-point computations. In this paper, we applied the ISZ method to Trager's algorithm which factors univariate polynomials over algebraic number fields. By Maple experiments, we show the efficiency of the ISZ method over the purely exact approach which uses exact computations throughout the execution of the algorithm. Furthermore, we propose a new method called the ISZ* method, which is similar to the ISZ method but beforehand excludes insufficient precisions of floating-point approximation by checking the correctness of the obtained supports. We confirmed that the ISZ* method is more *** than the ISZ method when the initially set precision is not sufficiently high.

关键词： algebra

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Fully Private Grouped Matrix Multiplication with Colluding Workers

Fully Private Grouped Matrix Multiplication with Colluding W...

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2023 IEEE International Symposium on Information theory, ISIT 2023

作者： Tauz, Lev Dolecek, Lara University of California Department of Electrical and Computer Engineering Los Angeles United States

ISBN: (纸本)9781665475549

In this paper, we present a novel variation of the coded matrix multiplication problem which we refer to as fully private grouped matrix multiplication (FPGMM). In FPGMM, a master wants to compute a group of matrix products between two matrix libraries that can be accessed by all workers while ensuring that any number of prescribed colluding workers learn nothing about which matrix products the master desires, nor the number of matrix products. We present an achievable scheme using a variant of Cross-Subspace Alignment (CSA) codes that offers flexibility in communication and computation cost. Additionally, we demonstrate how our scheme can outperform naive applications of schemes used in a related privacy focused coded matrix multiplication problem. © 2023 IEEE.

关键词： Matrix algebra

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Galois groups of polynomials and neurosymbolic networks

arXiv

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arXiv 2025年

作者： Shaska, Elira Shaska, Tony Department of Computer Science Oakland University Rochester United States Department of Mathematics and Statistics Oakland University Rochester United States

This paper introduces a novel approach to understanding Galois theory, one of the foundational areas of algebra, through the lens of machine learning. By analyzing polynomial equations with machine learning techniques, we aim to streamline the process of determining solvability by radicals and explore broader applications within Galois theory. This summary encapsulates the background, methodology, potential applications, and challenges of using data science in Galois theory. More specifically, we design a neurosymbolic network to classify Galois groups and show how this is more efficient than usual neural networks. We discover some very interesting distribution of polynomials for groups not isomorphic to the symmetric groups and alternating groups. Copyright © 2025, The Authors. All rights reserved.

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n-Dimensional Polynomial Chaotic System With applications

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I-REGULAR PAPERS 2022年第2期69卷 784-797页

作者： Hua, Zhongyun Zhang, Yinxing Bao, Han Huang, Hejiao Zhou, Yicong Harbin Inst Technol Sch Comp Sci & Technol Shenzhen 518055 Peoples R China Changzhou Univ Sch Microelect & Control Engn Changzhou 213164 Peoples R China Univ Macau Dept Comp & Informat Sci Macau 999078 Peoples R China

Designing high-dimensional chaotic maps with expected dynamic properties is an attractive but challenging task. The dynamic properties of a chaotic system can be reflected by the Lyapunov exponents (LEs). Using the inherent relationship between the parameters of a chaotic map and its LEs, this paper proposes an n-dimensional polynomial chaotic system (nD-PCS) that can generate nD chaotic maps with any desired LEs. The nD-PCS is constructed from n parametric polynomials with arbitrary orders, and its parameter matrix is configured using the preliminaries in linear algebra. Theoretical analysis proves that the nD-PCS can produce high-dimensional chaotic maps with any desired LEs. To show the effects of the nD-PCS, two high-dimensional chaotic maps with hyperchaotic behaviors were generated. A microcontroller-based hardware platform was developed to implement the two chaotic maps, and the test results demonstrated the randomness properties of their chaotic signals. Performance evaluations indicate that the high-dimensional chaotic maps generated from nD-PCS have the desired LEs and more complicated dynamic behaviors compared with other high-dimensional chaotic maps. In addition, to demonstrate the applications of nD-PCS, we developed a chaos-based secure communication scheme. Simulation results show that nD-PCS has a stronger ability to resist channel noise than other high-dimensional chaotic maps.

关键词： Chaotic communication Complexity theory Hardware Degradation Cats Eigenvalues and eigenfunctions Dynamical systems Chaotic system hardware implementation random number generator secure communication nonlinear system

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24th International Workshop on computer algebra in Scientific Computing, CASC 2022

24th International Workshop on Computer Algebra in Scientifi...

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24th International Workshop on computer algebra in Scientific Computing, CASC 2022

ISBN: (纸本)9783031147876

The proceedings contain 21 papers. The special focus in this conference is on computer algebra in Scientific Computing. The topics include: A General Method of Finding New Symplectic Schemes for Hamiltonian Mechanics;a Mechanical Method for Isolating Locally Optimal Points of Certain Radical Functions;subresultant Chains Using Bézout Matrices;application of Symbolic-Numerical Modeling Tools for Analysis of Gyroscopic Stabilization of Gyrostat Equilibria;computer Science for Continuous Data: Survey, Vision, theory, and Practice of a computer Analysis System;computational Aspects of Equivariant Hilbert Series of Canonical Rings for algebraic Curves;symbolic-Numeric Algorithm for Calculations in Geometric Collective Model of Atomic Nuclei;analyses and Implementations of Chordality-Preserving Top-Down Algorithms for Triangular Decomposition;accelerated Subdivision for Clustering Roots of polynomials Given by Evaluation Oracles;preface;survey on Generalizations of the Intermediate Value Theorem and applications;on Equilibrium Positions in the Problem of the Motion of a System of Two Bodies in a Uniform Gravity Field;an Interpolation Algorithm for Computing Dixon Resultants;distance Evaluation to the Set of Matrices with Multiple Eigenvalues;on Boundary Conditions Parametrized by Analytic Functions;computing the Integer Hull of Convex Polyhedral Sets;a Comparison of Algorithms for Proving Positivity of Linearly Recurrent Sequences;stability Analysis of Periodic Motion of the Swinging Atwood Machine;new Heuristic to Choose a Cylindrical algebraic Decomposition Variable Ordering Motivated by Complexity Analysis.

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EXTENDED ROTA-BAXTER algebraS, DIAGONALLY COLORED DELANNOY PATHS AND HOPF algebraS

arXiv

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arXiv 2024年

作者： Zheng, Shanghua Guo, Li Qiu, Huizhen School of Mathematics and Statistics Jiangxi Normal University Jiangxi Nanchang330022 China Department of Mathematics and Computer Science Rutgers University NewarkNJ07102 United States

MSC Codes 16W99, 17B37, 16S10, 16T10, 16T30, 57R56The Rota-Baxter operator and the modified Rota-Baxter operator on various algebras are both important in mathematics and mathematical physics. The former is originated from the integration-by-parts formula and probability with applications to the renormalization of quantum field theory and the classical Yang-Baxter equation. The latter originated from Hilbert transformations with applications to ergodic theory and the modified Yang-Baxter equation. Their merged form, called the extended Rota-Baxter operator, has also found interesting applications recently. This paper presents a systematic study of the extended Rota-Baxter operator. We show that extended Rota-Baxter operators have properties similar to Rota-Baxter operatos and provide a linear structure that unifies Rota-Baxter operators and modified Rota-Baxter operators. Examples of extended Rota-Baxter operators are also given, especially from polynomials and Laurent series due to their importance in (q-)integration and the renormalization in quantum field theory. We then construct free commutative extended Rota-Baxter algebras by a generalization of the quasi-shuffle product. The multiplication of the initial object in the category of commutative extended Rota-Baxter algebras allows a combinatorial interpretation in terms of a color-enrichment of Delannoy paths. Applying its universal property, we equip a free commutative extended Rota-Baxter algebra with a coproduct which has a cocycle condition, yielding a bialgebraic structure. We then show that this bialgebra on a free extended Rota-Baxter algebra possesses an increasing filtration and a connectedness property, culminating at a Hopf algebraic structure on a free commutative extended Rota-Baxter algebra. Copyright © 2024, The Authors. All rights reserved.

关键词： algebra

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Counting invariant subspaces and decompositions of additive polynomials

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JOURNAL OF SYMBOLIC COMPUTATION 2021年 105卷 214-233页

作者： Gathen, Joachim von zur Giesbrecht, Mark Ziegler, Konstantin Univ Bonn B IT D-53115 Bonn Germany Univ Waterloo Cheriton Sch Comp Sci Waterloo ON N2L 3G1 Canada Univ Appl Sci Landshut D-84036 Landshut Germany

The functional (de)composition of polynomials is a topic in pure and computer algebra with many applications. The structure of decompositions of (suitably normalized) polynomials f = g circle h in F[x] over a field F is well understood in many cases, but less well when the degree of f is divisible by the positive characteristic p of F. This work investigates the decompositions of r-additive polynomials, where every exponent and also the field size is a power of r, which itself is a power of p. The decompositions of an r-additive polynomial f are intimately linked to the Frobenius-invariant subspaces of its root space V in the algebraic closure of F. We present an efficient algorithm to compute the rational Jordan form of the Frobenius automorphism on V. A formula of Fripertinger (2011) then counts the number of Frobenius-invariant subspaces of a given dimension and we derive the number of decompositions with prescribed degrees. (C) 2020 Elsevier Ltd. All rights reserved.

关键词： Univariate polynomial decomposition Additive polynomials Finite fields Rational Jordan form

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BOUNDS ON THE REALIZATIONS OF ZERO-NONZERO PATTERNS AND SIGN CONDITIONS OF polynomials RESTRICTED TO VARIETIES AND applications

arXiv

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arXiv 2024年

作者： Basu, Saugata Parida, Laxmi Department of Mathematics Purdue University West LafayetteIN47906 United States IBM T.J. Watson Research Center Yorktown HeightsNY10598 United States

We prove upper bounds which are independent of the dimension of the ambient space, on the number of realizable zero-nonzero patterns as well as sign conditions (when the field of coefficients is ordered) of a finite set of polynomials P restricted to some algebraic subset V . Our bounds (which are tight) depend on the number and the degrees of the polynomials in P, as well as the degree (of the embedding) of V and the dimension of V , but are independent of the dimension of the space in which V is embedded. This last feature of our bounds is useful in situations where the ambient dimension could be much larger than dim V . We give several applications of our results. We generalize existing results on bounding the speeds of algebraically defined classes of graphs, as well as lower bounds in terms of the number of connected components for testing membership in semi-algebraic sets using algebraic computation trees. Motivated by quantum complexity theory we introduce a notion of relative rank (additive as well as multiplicative) in finite dimensional vector spaces and algebras relative to a fixed algebraic subset in the vector space or algebra – which generalizes the classical definition of ranks of tensors. We prove a very general lower bound on the maximum relative rank of finite subsets relative to algebraic subsets of bounded degree and dimension which is independent of the dimension of the vector space or algebra. We show how our lower bound implies a quantum analog of a classical lower bound result of Shannon for Boolean circuits – that almost all Boolean functions require (classical) circuits of size at least Ω(2n/n).MSC Codes Primary 68Q12, Secondary 14P10, 81P68 © 2024, CC BY.

关键词： Boolean functions

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Linked Partition Ideals and a Schur-Type Identity of Andrews 19th

Linked Partition Ideals and a Schur-Type Identity of Andrews

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19th Workshops on Combinatorial and Additive number theory, CANT 2021

作者： Chern, Shane Department of Mathematics and Statistics Dalhousie University HalifaxNSB3H 4R2 Canada

ISBN: (纸本)9783031107955

In his contribution to the Proceedings of the 1998 AMS–IMS–SIAM Joint Summer Research Conference on q-Series, Combinatorics, and computer algebra, Andrews considered a variant of Schur’s partition theorem, concerning partitions in which odd parts appear at most once, even parts appear at most twice, and the difference between two parts can never be 1 and can be 2 only if both are odd. Fitting into the framework of linked partition ideals, we obtain a non-standard trivariate generating function for such partitions that counts both the number of parts and the number of different parts that appear twice;the latter statistic plays an important role in Andrews’ identity. In particular, we are led to an application of using q-Borel operators in solving certain q-difference equations. Finally, we show that the regular trivariate generating function for such partitions has an interesting connection with the continuous q-Hermite polynomials. © 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.

关键词： polynomials

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Sums of binomial coefficients evaluated at a ∈(Q)over-bar, and applications

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JOURNAL OF algebra 2021年 587卷 171-214页

作者： Kalinov, Daniil Mandelshtam, Andrei

The additive monoid R+(x) is defined as the set of all nonnegative integer linear combinations of binomial polynomials ((x)(n)) for n is an element of Z(+). This paper is concerned with the inquiry into the structure of R+(alpha) for complex numbers alpha. Particularly interesting is the case of algebraic awhich are not non-negative integers. This question is motivated by the study of functors between Deligne categories Rep(S-t)(and also Rep(GL(t))) for t is an element of C\Z(+). We prove that this object is a ring if and only if ais an algebraic number that is not a nonnegative integer. Furthermore, we show that all algebraic integers generated by alpha, i.e. all elements of O-Q(a), are also contained in this ring. We also give two explicit representations of R+(alpha) for both algebraic integers and general algebraic numbers alpha. One is in terms of inequalities for the valuations with respect to certain prime ideals and the other is in terms of explicitly constructed generators. We show how these results work in the context of the study of symmetric monoidal functors between Deligne categories in positive characteristic. Moreover, this leads to a particularly simple description of R+(alpha) for both quadratic algebraic numbers and roots of unity. (C) 2021 Elsevier Inc. All rights reserved.

关键词： algebraic number theory number fields Integer-valued polynomials Deligne categories Symmetric groups

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