Artykuły w czasopismach: „Gröbner bases application”

1

Xia, Shengxiang, i Gaoxiang Xia. "AN APPLICATION OF GRÖBNER BASES". Mathematics Enthusiast 6, nr 3 (1.07.2009): 381–94. http://dx.doi.org/10.54870/1551-3440.1159.

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HASHEMI, AMIR, i PARISA ALVANDI. "APPLYING BUCHBERGER'S CRITERIA FOR COMPUTING GRÖBNER BASES OVER FINITE-CHAIN RINGS". Journal of Algebra and Its Applications 12, nr 07 (16.05.2013): 1350034. http://dx.doi.org/10.1142/s0219498813500345.

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Norton and Sălăgean [Strong Gröbner bases and cyclic codes over a finite-chain ring, in Proc. Workshop on Coding and Cryptography, Paris, Electronic Notes in Discrete Mathematics, Vol. 6 (Elsevier Science, 2001), pp. 391–401] have presented an algorithm for computing Gröbner bases over finite-chain rings. Byrne and Fitzpatrick [Gröbner bases over Galois rings with an application to decoding alternant codes, J. Symbolic Comput.31 (2001) 565–584] have simultaneously proposed a similar algorithm for computing Gröbner bases over Galois rings (a special kind of finite-chain rings). However, they have not incorporated Buchberger's criteria into their algorithms to avoid unnecessary reductions. In this paper, we propose the adapted version of these criteria for polynomials over finite-chain rings and we show how to apply them on Norton–Sălăgean algorithm. The described algorithm has been implemented in Maple and experimented with a number of examples for the Galois rings.

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3

Kolesnikov, P. S. "Gröbner–Shirshov Bases for Replicated Algebras". Algebra Colloquium 24, nr 04 (15.11.2017): 563–76. http://dx.doi.org/10.1142/s1005386717000372.

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We establish a universal approach to solutions of the word problem in the varieties of di- and tri-algebras. This approach, for example, allows us to apply Gröbner–Shirshov bases method for Lie algebras to solve the ideal membership problem in free Leibniz algebras (Lie di-algebras). As another application, we prove an analogue of the Poincaré–Birkhoff–Witt Theorem for universal enveloping associative tri-algebra of a Lie tri-algebra (CTD!-algebra).

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4

BORISOV, A. V., A. V. BOSOV i A. V. IVANOV. "APPLICATION OF COMPUTER SIMULATION TO THE ANONYMIZATION OF PERSONAL DATA: STATE-OF-THE-ART AND KEY POINTS". Программирование, nr 4 (1.07.2023): 58–74. http://dx.doi.org/10.31857/s0132347423040040.

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A new version of GInv (Gröbner Involutive) for computing involutive Gröbner bases is presented as a library in C++11. GInv uses object-oriented memory reallocation for dynamic data structures, such as lists, red-black trees, binary trees, and GMP libraries for arbitrary-precision integer calculations. The interface of the package is designed as a Python3 module.

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5

Ohsugi, Hidefumi, i Takayuki Hibi. "Prestable ideals and Sagbi bases". MATHEMATICA SCANDINAVICA 96, nr 1 (1.03.2005): 22. http://dx.doi.org/10.7146/math.scand.a-14942.

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In order to find a reasonable class of squarefree monomial ideals $I$ for which the toric ideal of the Rees algebra of $I$ has a quadratic Gröbner basis, the concept of prestable ideals will be introduced. Prestable ideals arising from finite pure posets together with their application to Sagbi bases will be discussed.

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6

Qiu, Jianjun, i Yuqun Chen. "Free Lie differential Rota–Baxter algebras and Gröbner–Shirshov bases". International Journal of Algebra and Computation 27, nr 08 (grudzień 2017): 1041–60. http://dx.doi.org/10.1142/s0218196717500485.

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7

Conca, A., E. De Negri i E. Gorla. "Universal Gröbner Bases and Cartwright–Sturmfels Ideals". International Mathematics Research Notices 2020, nr 7 (25.04.2018): 1979–91. http://dx.doi.org/10.1093/imrn/rny075.

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Abstract The main theoretical contribution of the paper is the description of two classes of multigraded ideals named after Cartwright and Sturmfels and the study of their surprising properties. Among other things we prove that these classes of ideals have very special multigraded generic initial ideals and are closed under several operations including arbitrary multigraded hyperplane sections. As a main application we describe the universal Gröbner basis of the ideal of maximal minors and the ideal of 2-minors of a multigraded matrix of linear forms generalizing earlier results of various authors including Bernstein, Sturmfels, Zelevinsky, and Boocher.

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8

Gao, Xing, i Tianjie Zhang. "Averaging algebras, rewriting systems and Gröbner–Shirshov bases". Journal of Algebra and Its Applications 17, nr 07 (13.06.2018): 1850130. http://dx.doi.org/10.1142/s021949881850130x.

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In this paper, we study the averaging operator by assigning a rewriting system to it. We obtain some basic results on the kind of rewriting system we used. In particular, we obtain a sufficient and necessary condition for the confluence. We supply the relationship between rewriting systems and Gröbner–Shirshov bases based on bracketed polynomials. As an application, we give a basis of the free unitary averaging algebra on a nonempty set.

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9

Matsuda, Kazunori, Hidefumi Ohsugi i Kazuki Shibata. "Toric Rings and Ideals of Stable Set Polytopes". Mathematics 7, nr 7 (10.07.2019): 613. http://dx.doi.org/10.3390/math7070613.

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In the present paper, we study the normality of the toric rings of stable set polytopes, generators of toric ideals of stable set polytopes, and their Gröbner bases via the notion of edge polytopes of finite nonsimple graphs and the results on their toric ideals. In particular, we give a criterion for the normality of the toric ring of the stable set polytope and a graph-theoretical characterization of the set of generators of the toric ideal of the stable set polytope for a graph of stability number two. As an application, we provide an infinite family of stable set polytopes whose toric ideal is generated by quadratic binomials and has no quadratic Gröbner bases.

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10

BOKUT, L. A., i A. A. KLEIN. "SERRE RELATIONS AND GRÖBNER-SHIRSHOV BASES FOR SIMPLE LIE ALGEBRAS II". International Journal of Algebra and Computation 06, nr 04 (sierpień 1996): 401–12. http://dx.doi.org/10.1142/s0218196796000234.

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Gröbner-Shirshov bases for the Lie algebras Bn, Cn, Dn, abstractly defined by generators and the Serre relations for the corresponding Cartan matrices over a field of characteristic ≠2, 3, are constructed. As an application, we obtain that each of the previous algebras is isomorphic to a classical simple Lie algebra so2n+1(k), sp2n(k), so2nk respectively.

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11

Sawada, Hiroyuki, i Xiu-Tian Yan. "Application of Gröbner bases and quantifier elimination for insightful engineering design". Mathematics and Computers in Simulation 67, nr 1-2 (wrzesień 2004): 135–48. http://dx.doi.org/10.1016/j.matcom.2004.05.014.

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12

Ioakimidis, N. I., i E. G. Anastasselou. "Application of Gröbner bases to problems of movement of a particle". Computers & Mathematics with Applications 27, nr 3 (luty 1994): 51–57. http://dx.doi.org/10.1016/0898-1221(94)90046-9.

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13

Li, Huishi. "The General PBW Property". Algebra Colloquium 14, nr 04 (grudzień 2007): 541–54. http://dx.doi.org/10.1142/s1005386707000508.

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For ungraded quotients of an arbitrary ℤ-graded ring, we define the general PBW property, that covers the classical PBW property and the N-type PBW property studied via the N-Koszulity by several authors (see [2–4]). In view of the noncommutative Gröbner basis theory, we conclude that every ungraded quotient of a path algebra (or a free algebra) has the general PBW property. We remark that an earlier result of Golod [5] concerning Gröbner bases can be used to give a homological characterization of the general PBW property in terms of Shafarevich complex. Examples of application are given.

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14

Byrne, Eimear, i Patrick Fitzpatrick. "Gröbner Bases over Galois Rings with an Application to Decoding Alternant Codes". Journal of Symbolic Computation 31, nr 5 (maj 2001): 565–84. http://dx.doi.org/10.1006/jsco.2001.0442.

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BOKUT, L. A., i A. A. KLEIN. "SERRE RELATIONS AND GRÖBNER-SHIRSHOV BASES FOR SIMPLE LIE ALGEBRAS I". International Journal of Algebra and Computation 06, nr 04 (sierpień 1996): 389–400. http://dx.doi.org/10.1142/s0218196796000222.

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A Gröbner-Shirshov basis for the Lie algebra An, abstractly defined by generators hi, xi, yi, i=1,..., n and the Serre relations for the Cartan matrix An, over a field k of characteristic ≠2 is constructed. It consists of the Serre relations for An together with the following relations: [Formula: see text] with j≥1, i≥2, i+j≤n and the same relations for y1,…, yn, where by [z1z2…zm] we mean [z1[z2… zm]]. As an application we get a direct proof that An, as defined, is isomorphic to sℓn+1(k).

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16

Georg, Klein, i Zargeh Chia. "Operadic approach to HNN-extensions of Leibniz algebras". Quasigroups and Related Systems 30, nr 1(47) (maj 2022): 101–14. http://dx.doi.org/10.56415/qrs.v30.08.

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We construct HNN-extensions of Lie di-algebras in the variety of di-algebras and provide a presentation for the replicated HNN-extension of a Lie di-algebras. Then, by applying the method of Gröbner-Shirshov bases for replicated algebras, we obtain a linear basis. As an application of HNN-extensions, we prove that Lie di-algebras are embedded in their HNNextension.

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17

Mora, Teo, i Massimiliano Sala. "On the Gröbner bases of some symmetric systems and their application to coding theory". Journal of Symbolic Computation 35, nr 2 (luty 2003): 177–94. http://dx.doi.org/10.1016/s0747-7171(02)00131-1.

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18

Scheicher, Martin. "Gröbner bases and their application to the Cauchy problem on finitely generated affine monoids". Journal of Symbolic Computation 80 (maj 2017): 416–50. http://dx.doi.org/10.1016/j.jsc.2016.07.002.

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19

Chou, Shang-Ching, William F. Schelter i Jin-Gen Yang. "An algorithm for constructing gröbner bases from characteristic sets and its application to geometry". Algorithmica 5, nr 1-4 (czerwiec 1990): 147–54. http://dx.doi.org/10.1007/bf01840382.

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20

PAUER, FRANZ, i SANDRO ZAMPIERI. "Gröbner Bases with Respect to Generalized Term Orders and their Application to the Modelling Problem". Journal of Symbolic Computation 21, nr 2 (luty 1996): 155–68. http://dx.doi.org/10.1006/jsco.1996.0007.

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21

Pauer, Franz, i Andreas Unterkircher. "Gröbner Bases for Ideals in Laurent Polynomial Rings and their Application to Systems of Difference Equations". Applicable Algebra in Engineering, Communication and Computing 9, nr 4 (1.02.1999): 271–91. http://dx.doi.org/10.1007/s002000050108.

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22

Yildirim, Necmettin, Nurullah Ankaralioglu, Derya Yildrim i Fatih Akcay. "Application of Gröbner Bases theory to derive rate equations for enzyme catalysed reactions with two or more substrates or products". Applied Mathematics and Computation 137, nr 1 (maj 2003): 67–76. http://dx.doi.org/10.1016/s0096-3003(02)00084-x.

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23

Bokut, L. A., Yuqun Chen i Abdukadir Obul. "Some new results on Gröbner–Shirshov bases for Lie algebras and around". International Journal of Algebra and Computation 28, nr 08 (grudzień 2018): 1403–23. http://dx.doi.org/10.1142/s0218196718400027.

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We review Gröbner–Shirshov bases for Lie algebras and survey some new results on Gröbner–Shirshov bases for [Formula: see text]-Lie algebras, Gelfand–Dorfman–Novikov algebras, Leibniz algebras, etc. Some applications are given, in particular, some characterizations of extensions of groups, associative algebras and Lie algebras are given.

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24

Huang, Juwei, i Yuqun Chen. "Gröbner–Shirshov Bases Theory for Trialgebras". Mathematics 9, nr 11 (26.05.2021): 1207. http://dx.doi.org/10.3390/math9111207.

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We establish a method of Gröbner–Shirshov bases for trialgebras and show that there is a unique reduced Gröbner–Shirshov basis for every ideal of a free trialgebra. As applications, we give a method for the construction of normal forms of elements of an arbitrary trisemigroup, in particular, A.V. Zhuchok’s (2019) normal forms of the free commutative trisemigroups are rediscovered and some normal forms of the free abelian trisemigroups are first constructed. Moreover, the Gelfand–Kirillov dimension of finitely generated free commutative trialgebra and free abelian trialgebra are calculated, respectively.

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25

Kelarev, Andrei, John Yearwood i Paul Watters. "INTERNET SECURITY APPLICATIONS OF GRÖBNER-SHIRSHOV BASES". Asian-European Journal of Mathematics 03, nr 03 (wrzesień 2010): 435–42. http://dx.doi.org/10.1142/s1793557110000283.

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This article is motivated by internet security applications of multiple classifiers designed for the detection of malware. Following a standard approach in data mining, Dazeley et al. (Asian-European J. Math. 2 (2009)(1) 41–56) used Gröbner-Shirshov bases to define a family of multiple classifiers and develop an algorithm optimizing their properties.The present article complements and strengthens these results. We consider a broader construction of classifiers and develop a new and more general algorithm for the optimization of their essential properties.

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26

Tran, Quoc-Nam, i Franz Winkler. "Special Issue on Applications of Gröbner Bases". Journal of Symbolic Computation 30, nr 4 (październik 2000): 339–40. http://dx.doi.org/10.1006/jsco.1999.0410.

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27

Liu, Jinwang, Dongmei Li i Weijun Liu. "Some criteria for Gröbner bases and their applications". Journal of Symbolic Computation 92 (maj 2019): 15–21. http://dx.doi.org/10.1016/j.jsc.2017.11.016.

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28

Silva, Sérgio Ricardo Xavier da, Leizer Schnitman i Vitalino Cesca Filho. "Analysis of computational efficiency for the solution of inverse kinematics problem of anthropomorphic robots using Gröbner bases theory". International Journal of Advanced Robotic Systems 18, nr 1 (1.01.2021): 172988142198954. http://dx.doi.org/10.1177/1729881421989542.

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This article presents an analysis of computational efficiency to solve the inverse kinematics problem of anthropomorphic robots. Two approaches are investigated: the first approach uses Paul’s method applied to the matrix obtained by the Denavit–Hartenberg algorithm and the second approach uses Gröbner bases theory. With each approach, the problem of inverse kinematics for an anthropomorphic robot will be solved. When comparing each method, this article will demonstrate that the method using Gröbner bases theory is more computationally efficient.

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29

BOKUT, L. A., YUQUN CHEN i JIAPENG HUANG. "GRÖBNER–SHIRSHOV BASES FOR L-ALGEBRAS". International Journal of Algebra and Computation 23, nr 03 (16.04.2013): 547–71. http://dx.doi.org/10.1142/s0218196713500094.

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In this paper, we first establish Composition-Diamond lemma for Ω-algebras. We give a Gröbner–Shirshov basis of the free L-algebra as a quotient algebra of a free Ω-algebra, and then the normal form of the free L-algebra is obtained. Second we establish Composition-Diamond lemma for L-algebras. As applications, we give Gröbner–Shirshov bases of the free dialgebra and the free product of two L-algebras, and then we show four embedding theorems of L-algebras: (1) Every countably generated L-algebra can be embedded into a two-generated L-algebra. (2) Every L-algebra can be embedded into a simple L-algebra. (3) Every countably generated L-algebra over a countable field can be embedded into a simple two-generated L-algebra. (4) Three arbitrary L-algebras A, B, C over a field k can be embedded into a simple L-algebra generated by B and C if |k| ≤ dim (B * C) and |A| ≤ |B * C|, where B * C is the free product of B and C.

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30

Qiu, Jianjun, i Yuqun Chen. "Gröbner–Shirshov bases for Lie Ω-algebras and free Rota–Baxter Lie algebras". Journal of Algebra and Its Applications 16, nr 10 (20.09.2017): 1750190. http://dx.doi.org/10.1142/s0219498817501900.

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We generalize the Lyndon–Shirshov words to the Lyndon–Shirshov [Formula: see text]-words on a set [Formula: see text] and prove that the set of all the nonassociative Lyndon–Shirshov [Formula: see text]-words forms a linear basis of the free Lie [Formula: see text]-algebra on the set [Formula: see text]. From this, we establish Gröbner–Shirshov bases theory for Lie [Formula: see text]-algebras. As applications, we give Gröbner–Shirshov bases of a free [Formula: see text]-Rota–Baxter Lie algebra, of a free modified [Formula: see text]-Rota–Baxter Lie algebra, and of a free Nijenhuis Lie algebra and, then linear bases of these three algebras are obtained.

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31

Zerz, E. "Some Applications of Gröbner Bases in Multidimensional Systems Theory". ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik 81, S3 (2001): 635–36. http://dx.doi.org/10.1002/zamm.20010811593.

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32

Hillar, Christopher J., i Seth Sullivant. "Finite Gröbner bases in infinite dimensional polynomial rings and applications". Advances in Mathematics 229, nr 1 (styczeń 2012): 1–25. http://dx.doi.org/10.1016/j.aim.2011.08.009.

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33

Liu, Mulan, i Lei Hu. "Properties of Gröbner Bases and Applications to Doubly Periodic Arrays". Journal of Symbolic Computation 26, nr 3 (wrzesień 1998): 301–14. http://dx.doi.org/10.1006/jsco.1998.0213.

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34

Kubler, Felix, i Karl Schmedders. "Tackling Multiplicity of Equilibria with Gröbner Bases". Operations Research 58, nr 4-part-2 (sierpień 2010): 1037–50. http://dx.doi.org/10.1287/opre.1100.0819.

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35

Bokut, L. A., Yuqun Chen i Zerui Zhang. "Gröbner–Shirshov bases method for Gelfand–Dorfman–Novikov algebras". Journal of Algebra and Its Applications 16, nr 01 (styczeń 2017): 1750001. http://dx.doi.org/10.1142/s0219498817500013.

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We establish Gröbner–Shirshov base theory for Gelfand–Dorfman–Novikov algebras over a field of characteristic [Formula: see text]. As applications, a PBW type theorem in Shirshov form is given and we provide an algorithm for solving the word problem of Gelfand–Dorfman–Novikov algebras with finite homogeneous relations. We also construct a subalgebra of one generated free Gelfand–Dorfman–Novikov algebra which is not free.

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36

Qiu, Jianjun. "Gröbner–Shirshov bases for commutative algebras with multiple operators and free commutative Rota–Baxter algebras". Asian-European Journal of Mathematics 07, nr 02 (czerwiec 2014): 1450033. http://dx.doi.org/10.1142/s1793557114500338.

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In this paper, the Composition-Diamond lemma for commutative algebras with multiple operators is established. As applications, the Gröbner–Shirshov bases and linear bases of free commutative Rota–Baxter algebra, free commutative λ-differential algebra and free commutative λ-differential Rota–Baxter algebra are given, respectively. Consequently, these three free algebras are constructed directly by commutative Ω-words.

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37

Zhiping Lin, L. Xu i N. K. Bose. "A Tutorial on GrÖbner Bases With Applications in Signals and Systems". IEEE Transactions on Circuits and Systems I: Regular Papers 55, nr 1 (luty 2008): 445–61. http://dx.doi.org/10.1109/tcsi.2007.914007.

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38

Gritzmann, Peter, i Bernd Sturmfels. "Minkowski Addition of Polytopes: Computational Complexity and Applications to Gröbner Bases". SIAM Journal on Discrete Mathematics 6, nr 2 (maj 1993): 246–69. http://dx.doi.org/10.1137/0406019.

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Lin, Zhiping, Li Xu i Qinghe Wu. "Applications of Gröbner bases to signal and image processing: a survey". Linear Algebra and its Applications 391 (listopad 2004): 169–202. http://dx.doi.org/10.1016/j.laa.2004.01.008.

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Heyworth, Anne. "One-Sided Noncommutative Gröbner Bases with Applications to Computing Green's Relations". Journal of Algebra 242, nr 2 (sierpień 2001): 401–16. http://dx.doi.org/10.1006/jabr.2001.8801.

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Ioakimidis, N. I., i E. G. Anastasselou. "Gröbner bases in truss problems with maple". Computers & Structures 52, nr 5 (wrzesień 1994): 1093–96. http://dx.doi.org/10.1016/0045-7949(94)90093-0.

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QIU, JIANJUN, i YUQUN CHEN. "COMPOSITION-DIAMOND LEMMA FOR λ-DIFFERENTIAL ASSOCIATIVE ALGEBRAS WITH MULTIPLE OPERATORS". Journal of Algebra and Its Applications 09, nr 02 (kwiecień 2010): 223–39. http://dx.doi.org/10.1142/s0219498810003859.

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In this paper, we establish the Composition-Diamond lemma for λ-differential associative algebras over a field K with multiple operators. As applications, we obtain Gröbner–Shirshov bases of free λ-differential Rota–Baxter algebras. In particular, linear bases of free λ-differential Rota–Baxter algebras are obtained and consequently, the free λ-differential Rota–Baxter algebras are constructed by words.

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43

Steiner, Matthias Johann. "Solving Degree Bounds for Iterated Polynomial Systems". IACR Transactions on Symmetric Cryptology 2024, nr 1 (1.03.2024): 357–411. http://dx.doi.org/10.46586/tosc.v2024.i1.357-411.

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For Arithmetization-Oriented ciphers and hash functions Gröbner basis attacks are generally considered as the most competitive attack vector. Unfortunately, the complexity of Gröbner basis algorithms is only understood for special cases, and it is needless to say that these cases do not apply to most cryptographic polynomial systems. Therefore, cryptographers have to resort to experiments, extrapolations and hypotheses to assess the security of their designs. One established measure to quantify the complexity of linear algebra-based Gröbner basis algorithms is the so-called solving degree. Caminata & Gorla revealed that under a certain genericity condition on a polynomial system the solving degree is always upper bounded by the Castelnuovo-Mumford regularity and henceforth by the Macaulay bound, which only takes the degrees and number of variables of the input polynomials into account. In this paper we extend their framework to iterated polynomial systems, the standard polynomial model for symmetric ciphers and hash functions. In particular, we prove solving degree bounds for various attacks on MiMC, Feistel-MiMC, Feistel-MiMC-Hash, Hades and GMiMC. Our bounds fall in line with the hypothesized complexity of Gröbner basis attacks on these designs, and to the best of our knowledge this is the first time that a mathematical proof for these complexities is provided. Moreover, by studying polynomials with degree falls we can prove lower bounds on the Castelnuovo-Mumford regularity for attacks on MiMC, Feistel-MiMC and Feistel-MiMCHash provided that only a few solutions of the corresponding iterated polynomial system originate from the base field. Hence, regularity-based solving degree estimations can never surpass a certain threshold, a desirable property for cryptographic polynomial systems.

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44

Sabzrou, Hossein. "A determinantal formula for circuits of integer lattices". Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica 20, nr 1 (1.01.2021): 121–27. http://dx.doi.org/10.2478/aupcsm-2021-0008.

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Abstract Let L be a not necessarily saturated lattice in ℤ n with a defining matrix B. We explicitly compute the set of circuits of L in terms of maximal minors of B. This has a variety of applications from toric to tropical geometry, from Gröbner to Graver bases, and from linear to binomial ideals.

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45

Iima, Kei-ichiro, i Yuji Yoshino. "Gröbner Bases for the Polynomial Ring with Infinite Variables and Their Applications". Communications in Algebra 37, nr 10 (9.10.2009): 3424–37. http://dx.doi.org/10.1080/00927870802502878.

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46

Ioakimidis, N. I., i E. G. Anastasselou. "Computer-based manipulation of systems of equations in elasticity problems with Gröbner bases". Computer Methods in Applied Mechanics and Engineering 110, nr 1-2 (grudzień 1993): 103–11. http://dx.doi.org/10.1016/0045-7825(93)90022-p.

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Shany, Yaron, i Amit Berman. "A Gröbner-Bases Approach to Syndrome-Based Fast Chase Decoding of Reed–Solomon Codes". IEEE Transactions on Information Theory 68, nr 4 (kwiecień 2022): 2300–2318. http://dx.doi.org/10.1109/tit.2022.3140678.

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Brickenstein, Michael, Alexander Dreyer, Gert-Martin Greuel, Markus Wedler i Oliver Wienand. "New developments in the theory of Gröbner bases and applications to formal verification". Journal of Pure and Applied Algebra 213, nr 8 (sierpień 2009): 1612–35. http://dx.doi.org/10.1016/j.jpaa.2008.11.043.

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Liu, Mulan. "Applications of the theory of Gröbner bases to the study of linear recurring arrays". Chinese Science Bulletin 46, nr 14 (lipiec 2001): 1149–51. http://dx.doi.org/10.1007/bf02900589.

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Naumowicz, Adam. "Interfacing external CA systems for Gröbner bases computation in Mizarproof checking". International Journal of Computer Mathematics 87, nr 1 (styczeń 2010): 1–11. http://dx.doi.org/10.1080/00207160701864459.

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