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Journal articles on the topic 'Base du Gröbner'

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Published: 25 May 2024

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1

Bokut, L. A., Yuqun Chen, and Zerui Zhang. "Gröbner–Shirshov bases method for Gelfand–Dorfman–Novikov algebras." Journal of Algebra and Its Applications 16, no. 01 (January 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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2

Steiner, Matthias Johann. "Solving Degree Bounds for Iterated Polynomial Systems." IACR Transactions on Symmetric Cryptology 2024, no. 1 (March 1, 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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3

Jha, Ranjan, Damien Chablat, and Luc Baron. "Influence of design parameters on the singularities and workspace of a 3-RPS parallel robot." Transactions of the Canadian Society for Mechanical Engineering 42, no. 1 (March 1, 2018): 30–37. http://dx.doi.org/10.1139/tcsme-2017-0011.

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This paper presents variations in the workspace, singularities, and joint space with respect to design parameter k, which is the ratio of the dimensions of the mobile platform to the dimensions of the base of a 3-RPS parallel manipulator. The influence of the design parameters on parasitic motion, which is important when selecting a manipulator for a desired task, is also studied. The cylindrical algebraic decomposition method and Gröbner-based computations are used to model the workspace and joint space with parallel singularities in 2R1T (two rotational and one translational) and 3T (three translational) projection spaces, where the orientation of the mobile platform is represented using quaternions. These computations are useful in selecting the optimum value for the design parameter k such that the parasitic motions can be limited to specific values. Three designs of the 3-RPS parallel robot, based on different values of k, are analyzed.

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4

Yoshida, Hiroshi. "A model for analyzing phenomena in multicellular organisms with multivariable polynomials: Polynomial life." International Journal of Biomathematics 11, no. 01 (January 2018): 1850007. http://dx.doi.org/10.1142/s1793524518500079.

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Most of life maintains itself through turnover, namely cell proliferation, movement and elimination. Hydra’s cells, for example, disappear continuously from the ends of tentacles, but these cells are replenished by cell proliferation within the body. Inspired by such a biological fact, and together with various operations of polynomials, I here propose polynomial-life model toward analysis of some phenomena in multicellular organisms. Polynomial life consists of multicells that are expressed as multivariable polynomials. A cell is expressed as a term of polynomial, in which point [Formula: see text] is described as a term [Formula: see text] and the condition is described as its coefficient. Starting with a single term and following reductions by set of polynomials, I simulate the development from a cell to a multicell. In order to confirm uniqueness of the eventual multicell-pattern, Gröbner base can be used, which has been conventionally used to ensure uniqueness of normal form in the mathematical context. In this framework, I present various patterns through the polynomial-life model and discuss patterns maintained through turnover. Cell elimination seems to play an important role in turnover, which may shed some light on cancer or regenerative medicine.

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5

Gräbe, Hans-Gert, and Franz Pauer. "A remark on Hodge algebras and Gröbner bases." Czechoslovak Mathematical Journal 42, no. 2 (1992): 331–38. http://dx.doi.org/10.21136/cmj.1992.128327.

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6

Bellini, Emanuele, Massimiliano Sala, and Ilaria Simonetti. "Nonlinearity of Boolean Functions: An Algorithmic Approach Based on Multivariate Polynomials." Symmetry 14, no. 2 (January 22, 2022): 213. http://dx.doi.org/10.3390/sym14020213.

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We review and compare three algebraic methods to compute the nonlinearity of Boolean functions. Two of them are based on Gröbner basis techniques: the first one is defined over the binary field, while the second one over the rationals. The third method improves the second one by avoiding the Gröbner basis computation. We also estimate the complexity of the algorithms, and, in particular, we show that the third method reaches an asymptotic worst-case complexity of O(n2n) operations over the integers, that is, sums and doublings. This way, with a different approach, the same asymptotic complexity of established algorithms, such as those based on the fast Walsh transform, is reached.

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7

Eder, Christian. "Improving incremental signature-based Gröbner basis algorithms." ACM Communications in Computer Algebra 47, no. 1/2 (July 15, 2013): 1–13. http://dx.doi.org/10.1145/2503697.2503699.

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8

Eder, Christian, and Jean-Charles Faugère. "A survey on signature-based Gröbner basis computations." ACM Communications in Computer Algebra 49, no. 2 (August 14, 2015): 61. http://dx.doi.org/10.1145/2815111.2815156.

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9

Francis, Maria, and Thibaut Verron. "A Signature-Based Algorithm for Computing Gröbner Bases over Principal Ideal Domains." Mathematics in Computer Science 14, no. 2 (December 17, 2019): 515–30. http://dx.doi.org/10.1007/s11786-019-00432-5.

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AbstractSignature-based algorithms have become a standard approach for Gröbner basis computations for polynomial systems over fields, but how to extend these techniques to coefficients in general rings is not yet as well understood. In this paper, we present a proof-of-concept signature-based algorithm for computing Gröbner bases over commutative integral domains. It is adapted from a general version of Möller’s algorithm (J Symb Comput 6(2–3), 345–359, 1988) which considers reductions by multiple polynomials at each step. This algorithm performs reductions with non-decreasing signatures, and in particular, signature drops do not occur. When the coefficients are from a principal ideal domain (e.g. the ring of integers or the ring of univariate polynomials over a field), we prove correctness and termination of the algorithm, and we show how to use signature properties to implement classic signature-based criteria to eliminate some redundant reductions. In particular, if the input is a regular sequence, the algorithm operates without any reduction to 0. We have written a toy implementation of the algorithm in Magma. Early experimental results suggest that the algorithm might even be correct and terminate in a more general setting, for polynomials over a unique factorization domain (e.g. the ring of multivariate polynomials over a field or a PID).

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10

Eder, Christian. "An analysis of inhomogeneous signature-based Gröbner basis computations." Journal of Symbolic Computation 59 (December 2013): 21–35. http://dx.doi.org/10.1016/j.jsc.2013.08.001.

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11

Zheng, Licui, Jinwang Liu, Weijun Liu, and Dongmei Li. "A new signature-based algorithms for computing Gröbner bases." Journal of Systems Science and Complexity 28, no. 1 (January 13, 2015): 210–21. http://dx.doi.org/10.1007/s11424-015-2260-z.

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12

Roanes-Lozano, E., L. M. Laita, E. Roanes-Macı́as, V. Maojo, S. Corredor, A. de la Vega, and A. Zamora. "A Gröbner bases-based shell for rule-based expert systems development." Expert Systems with Applications 18, no. 3 (April 2000): 221–30. http://dx.doi.org/10.1016/s0957-4174(99)00064-0.

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13

Piury, Josefina, Luis M. Laita, Eugenio Roanes-Lozano, Antonio Hernando, Francisco-Javier Piury-Alonso, José M. Gómez-Argüelles, and Laura Laita. "A Gröbner bases-based rule based expert system for fibromyalgia diagnosis." Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas 106, no. 2 (March 25, 2012): 443–56. http://dx.doi.org/10.1007/s13398-012-0064-8.

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14

Galkin, V. V. "Simple signature based iterative algorithm for calculation of Gröbner bases." Moscow University Mathematics Bulletin 68, no. 5 (September 2013): 231–36. http://dx.doi.org/10.3103/s0027132213050033.

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15

Schauenburg, Peter. "A Gröbner-based treatment of elimination theory for affine varieties." Journal of Symbolic Computation 42, no. 9 (September 2007): 859–70. http://dx.doi.org/10.1016/j.jsc.2007.06.003.

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16

Eder, Christian, and Jean-Charles Faugère. "A survey on signature-based algorithms for computing Gröbner bases." Journal of Symbolic Computation 80 (May 2017): 719–84. http://dx.doi.org/10.1016/j.jsc.2016.07.031.

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17

Shirayanagi, Kiyoshi, and Hiroshi Sekigawa. "A new Gröbner basis conversion method based on stabilization techniques." Theoretical Computer Science 409, no. 2 (December 2008): 311–17. http://dx.doi.org/10.1016/j.tcs.2008.09.007.

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18

Maletzky, Alexander. "A generic and executable formalization of signature-based Gröbner basis algorithms." Journal of Symbolic Computation 106 (September 2021): 23–47. http://dx.doi.org/10.1016/j.jsc.2020.12.001.

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19

He, Honghui, and Jinzhao Wu. "A New Approach to Nonlinear Invariants for Hybrid Systems Based on the Citing Instances Method." Information 11, no. 5 (May 2, 2020): 246. http://dx.doi.org/10.3390/info11050246.

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In generating invariants for hybrid systems, a main source of intractability is that transition relations are first-order assertions over current-state variables and next-state variables, which doubles the number of system variables and introduces many more free variables. The more variables, the less tractability and, hence, solving the algebraic constraints on complete inductive conditions by a comprehensive Gröbner basis is very expensive. To address this issue, this paper presents a new, complete method, called the Citing Instances Method (CIM), which can eliminate the free variables and directly solve for the complete inductive conditions. An instance means the verification of a proposition after instantiating free variables to numbers. A lattice array is a key notion in this paper, which is essentially a finite set of instances. Verifying that a proposition holds over a Lattice Array suffices to prove that the proposition holds in general; this interesting feature inspires us to present CIM. On one hand, instead of computing a comprehensive Gröbner basis, CIM uses a Lattice Array to generate the constraints in parallel. On the other hand, we can make a clever use of the parallelism of CIM to start with some constraint equations which can be solved easily, in order to determine some parameters in an early state. These solved parameters benefit the solution of the rest of the constraint equations; this process is similar to the domino effect. Therefore, the constraint-solving tractability of the proposed method is strong. We show that some existing approaches are only special cases of our method. Moreover, it turns out CIM is more efficient than existing approaches under parallel circumstances. Some examples are presented to illustrate the practicality of our method.

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20

Sakata, Kosuke. "An efficient reduction strategy for signature-based algorithms to compute Gröbner basis." ACM Communications in Computer Algebra 53, no. 3 (December 17, 2019): 81–92. http://dx.doi.org/10.1145/3377006.3377007.

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21

Sun, Yao, Dongdai Lin, and Dingkang Wang. "On implementing signature-based Gröbner basis algorithms using linear algebraic routines from M4RI." ACM Communications in Computer Algebra 49, no. 2 (August 14, 2015): 63–64. http://dx.doi.org/10.1145/2815111.2815161.

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22

Ioakimidis, N. I., and E. G. Anastasselou. "Computer-based manipulation of systems of equations in elasticity problems with Gröbner bases." Computer Methods in Applied Mechanics and Engineering 110, no. 1-2 (December 1993): 103–11. http://dx.doi.org/10.1016/0045-7825(93)90022-p.

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23

Fukasaku, Ryoya, Shutaro Inoue, and Yosuke Sato. "On QE Algorithms over an Algebraically Closed Field Based on Comprehensive Gröbner Systems." Mathematics in Computer Science 9, no. 3 (September 23, 2015): 267–81. http://dx.doi.org/10.1007/s11786-015-0237-x.

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24

Shany, Yaron, and Amit Berman. "A Gröbner-Bases Approach to Syndrome-Based Fast Chase Decoding of Reed–Solomon Codes." IEEE Transactions on Information Theory 68, no. 4 (April 2022): 2300–2318. http://dx.doi.org/10.1109/tit.2022.3140678.

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25

Matsumoto, Ryutaroh, Diego Ruano, and Olav Geil. "List decoding algorithm based on voting in Gröbner bases for general one-point AG codes." Journal of Symbolic Computation 79 (March 2017): 384–410. http://dx.doi.org/10.1016/j.jsc.2016.02.015.

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26

Eder, Christian, Pierre Lairez, Rafael Mohr, and Mohab Safey El Din. "Towards signature-based gröbner basis algorithms for computing the nondegenerate locus of a polynomial system." ACM Communications in Computer Algebra 56, no. 2 (June 2022): 41–45. http://dx.doi.org/10.1145/3572867.3572872.

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Problem statement. Let K be a field and K be an algebraic closure of K. Consider the polynomial ring R = K[ x 1 ,..., x n ] over K and a finite sequence of polynomials f 1 ,..., f c in R with c ≤ n. Let V ⊂ K n be the algebraic set defined by the simultaneous vanishing of the f i 's. Recall that V can be decomposed into finitely many irreducible components, whose codimension cannot be greater than c. The set V c which is the union of all these irreducible components of codimension exactly c is named further the nondegenerate locus of f 1 ,..., f c .

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27

Liu, Ruixian, Philippe Serré, Jean-François Rameau, and André Clément. "Generic Approach for the Generation of Symbolic Dimensional Variations Based on Gröbner Basis for Over-constrained Mechanical Assemblies." Procedia CIRP 27 (2015): 223–29. http://dx.doi.org/10.1016/j.procir.2015.04.070.

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28

Jangisarakul, P., and C. Charoenlarpnopparut. "Algebraic decoder of multidimensional convolutional code: constructive algorithms for determining syndrome decoder and decoder matrix based on Gröbner basis." Multidimensional Systems and Signal Processing 22, no. 1-3 (October 22, 2010): 67–81. http://dx.doi.org/10.1007/s11045-010-0139-7.

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29

Khan, Muhammad Fahad, Khalid Saleem, Tariq Shah, Mohammad Mazyad Hazzazi, Ismail Bahkali, and Piyush Kumar Shukla. "Block Cipher’s Substitution Box Generation Based on Natural Randomness in Underwater Acoustics and Knight’s Tour Chain." Computational Intelligence and Neuroscience 2022 (May 20, 2022): 1–17. http://dx.doi.org/10.1155/2022/8338508.

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The protection of confidential information is a global issue, and block encryption algorithms are the most reliable option for securing data. The famous information theorist, Claude Shannon, has given two desirable characteristics that should exist in a strong cipher which are substitution and permutation in their fundamental research on “Communication Theory of Secrecy Systems.” block ciphers strictly follow the substitution and permutation principle in an iterative manner to generate a ciphertext. The actual strength of the block ciphers against several attacks is entirely based on its substitution characteristic, which is gained by using the substitution box (S-box). In the current literature, algebraic structure-based and chaos-based techniques are highly used for the construction of S-boxes because both these techniques have favourable features for S-box construction but also various attacks of these techniques have been identified including SAT solver, linear and differential attacks, Gröbner-based attacks, XSL attacks, interpolation attacks, XL-based attacks, finite precision effect, chaotic systems degradation, predictability, weak randomness, chaotic discontinuity, and limited control parameters. The main objective of this research is to design a novel technique for the dynamic generation of S-boxes that are safe against the cryptanalysis techniques of algebraic structure-based and chaos-based approaches. True randomness has been universally recognized as the ideal method for cipher primitives design because true random numbers are unpredictable, irreversible, and unreproducible. The biggest challenge we faced during this research was how can we generate the true random numbers and how can true random numbers utilized for strengthening the S-box construction technique. The basic concept of the proposed technique is the extraction of true random bits from underwater acoustic waves and to design a novel technique for the dynamic generation of S-boxes using the chain of knight’s tour. Rather than algebraic structure- and chaos-based techniques, our proposed technique depends on inevitable high-quality randomness which exists in underwater acoustics waves. The proposed method satisfies all standard evaluation tests of S-boxes construction and true random numbers generation. Two million bits have been analyzed using the NIST randomness test suite, and the results show that underwater sound waves are an impeccable entropy source for true randomness. Additionally, our dynamically generated S-boxes have better or equal strength, over the latest published S-boxes (2020 to 2021). According to our knowledge first time, this type of research has been conducted, in which natural randomness of underwater acoustic waves has been used for the construction of block cipher’s substitution box.

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30

WALKLING, ADRIAN P., and G. RUSSELL COOPE. "Climatic reconstructions from the Eemian/Early Weichselian transition in Central Europe based on the coleopteran record from Gröbern, Germany." Boreas 25, no. 3 (January 16, 2008): 145–59. http://dx.doi.org/10.1111/j.1502-3885.1996.tb00843.x.

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31

Mohammadi, Fatemeh, and Farbod Shokrieh. "Divisors on graphs, Connected flags, and Syzygies." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AS,..., Proceedings (January 1, 2013). http://dx.doi.org/10.46298/dmtcs.2351.

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International audience We study the binomial and monomial ideals arising from linear equivalence of divisors on graphs from the point of view of Gröbner theory. We give an explicit description of a minimal Gröbner basis for each higher syzygy module. In each case the given minimal Gröbner basis is also a minimal generating set. The Betti numbers of $I_G$ and its initial ideal (with respect to a natural term order) coincide and they correspond to the number of ``connected flags'' in $G$. Moreover, the Betti numbers are independent of the characteristic of the base field.

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32

Zhao, Xiangui, and Yang Zhang. "A signature-based algorithm for computing Gröbner-Shirshov bases in skew solvable polynomial rings." Open Mathematics 13, no. 1 (May 6, 2015). http://dx.doi.org/10.1515/math-2015-0028.

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AbstractSignature-based algorithms are efficient algorithms for computing Gröbner-Shirshov bases in commutative polynomial rings, and some noncommutative rings. In this paper, we first define skew solvable polynomial rings, which are generalizations of solvable polynomial algebras and (skew) PBW extensions. Then we present a signature-based algorithm for computing Gröbner-Shirshov bases in skew solvable polynomial rings over fields.

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33

Arikawa, Keisuke. "Kinematic Analysis of Mechanisms Based on Parametric Polynomial System: Basic Concept of a Method Using Gröbner Cover and Its Application to Planar Mechanisms." Journal of Mechanisms and Robotics 11, no. 2 (February 22, 2019). http://dx.doi.org/10.1115/1.4042475.

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Many kinematic problems in mechanisms can be represented by polynomial systems. By algebraically analyzing the polynomial systems, we can obtain the kinematic properties of the mechanisms. Among these algebraic methods, approaches based on Gröbner bases are effective. Usually, the analyses are performed for specific mechanisms; however, we often encounter phenomena for which, even within the same class of mechanisms, the kinematic properties differ significantly. In this research, we consider the cases where the parameters are included in the polynomial systems. The parameters are used to express link lengths, displacements of active joints, hand positions, and so on. By analyzing a parametric polynomial system (PPS), we intend to comprehensively analyze the kinematic properties of mechanisms represented by these parameters. In the proposed method, we first express the kinematic constraints in the form of PPS. Subsequently, by calculating the Gröbner cover of the PPS, we obtain the segmentation of the parameter space and valid Gröbner bases for each segment. Finally, we interpret the meaning of the segments and their corresponding Gröbner bases. We analyzed planar four- and five-bar linkages and five-bar truss structures using the proposed method. We confirmed that it was possible to enumerate the assembly and working modes and to identify the geometrical conditions that enable overconstrained motions.

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34

Roanes-Lozano, Eugenio, and Eugenio Roanes-Macias. "Maple-based introductory visual guide to Gröbner bases." Maple Transactions 2, no. 1 (September 5, 2022). http://dx.doi.org/10.5206/mt.v2i1.14425.

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In 1975 the Consejo Superior de Investigaciones Científicas (the main Spanish institution for scientific research) published the monograph [14] by the second author (by the way, father and Ph.D. advisor of the first author). Its title could be translated as "Geometric Interpretation of Ideal Theory" (nowadays Ideal Theory is not normally used, in favour of Commutative Algebra). It somehow illustrated the geometric ideas underlying the basics of the classic books of the period (like [2, 11, 16]) and was a success: although written in Spanish, the edition was sold out.Of course there are much more modern books on ideals and varieties than [2, 11, 16], such as the famous [7] or [8], that illustrate the theory with images. Moreover, there are introductory works to Gröbner bases such as [3, 9, 12, 13, 15], as well as books on the topic like [1], and articles about applications, like the early [4]. Even a summary in English of the original Ph.D. Thesis by Bruno Buchberger is available [5]. Nevertheless, we believe that there is a place for a visual guide to Gröbner bases, as there was a place for [14]. For instance, statistical packages are probably the pieces of mathematical software best known by non-mathematicians, and they are frequently used as black boxes by users with a slight knowledge of the theory behind. Meanwhile, Gröbner bases, the most common exact method behind non-linear polynomial systems (algebraic systems) solving, although incorporated to all computer algebra systems, are only known by a relatively small ratio of the members of the scientific community, most of them mathematicians. This article presents in an intuitive and visual way an illustrative selection of ideals and their Gröbner bases, together with the plots of the (real part) of their corresponding algebraic varieties, computed and plotted with Maple [6, 10]. A minimum amount of theoretical details is given. We believe that exact algebraic systems solving could also be used as a black box by non-mathematicians just understanding the basic ideas underlying commutative algebra and computer algebra.

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35

G Karpuz, Eylem, Firat Ates, A. Sinan Çevik, and I. Naci Cangul. "The graph based on Gröbner-Shirshov bases of groups." Fixed Point Theory and Applications 2013, no. 1 (March 26, 2013). http://dx.doi.org/10.1186/1687-1812-2013-71.

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36

Lyskov, Denis. "A Generalization of Operads Based on Subgraph Contractions." International Mathematics Research Notices, May 20, 2024. http://dx.doi.org/10.1093/imrn/rnae096.

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Abstract We introduce a generalization of the notion of operad that we call a contractad, whose set of operations is indexed by connected graphs and whose composition rules are numbered by contractions of connected subgraphs. We show that many classical operads, such as the operad of commutative algebras, Lie algebras, associative algebras, pre-Lie algebras, the little disks operad, and the operad of moduli spaces of stable curves $\operatorname{\overline{{\mathcal{M}}}}_{0,n+1}$, admit generalizations to contractads. We explain that standard tools like Koszul duality and the machinery of Gröbner bases can be easily generalized to contractads. We verify the Koszul property of the commutative, Lie, associative, and Gerstenhaber contractads.

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37

Bhayani, Snehal, Janne Heikkilä, and Zuzana Kukelova. "Sparse Resultant-Based Minimal Solvers in Computer Vision and Their Connection with the Action Matrix." Journal of Mathematical Imaging and Vision, March 23, 2024. http://dx.doi.org/10.1007/s10851-024-01182-1.

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AbstractMany computer vision applications require robust and efficient estimation of camera geometry from a minimal number of input data measurements. Minimal problems are usually formulated as complex systems of sparse polynomial equations. The systems usually are overdetermined and consist of polynomials with algebraically constrained coefficients. Most state-of-the-art efficient polynomial solvers are based on the action matrix method that has been automated and highly optimized in recent years. On the other hand, the alternative theory of sparse resultants based on the Newton polytopes has not been used so often for generating efficient solvers, primarily because the polytopes do not respect the constraints amongst the coefficients. In an attempt to tackle this challenge, here we propose a simple iterative scheme to test various subsets of the Newton polytopes and search for the most efficient solver. Moreover, we propose to use an extra polynomial with a special form to further improve the solver efficiency via Schur complement computation. We show that for some camera geometry problems our resultant-based method leads to smaller and more stable solvers than the state-of-the-art Gröbner basis-based solvers, while being significantly smaller than the state-of-the-art resultant-based methods. The proposed method can be fully automated and incorporated into existing tools for the automatic generation of efficient polynomial solvers. It provides a competitive alternative to popular Gröbner basis-based methods for minimal problems in computer vision. Additionally, we study the conditions under which the minimal solvers generated by the state-of-the-art action matrix-based methods and the proposed extra polynomial resultant-based method, are equivalent. Specifically, we consider a step-by-step comparison between the approaches based on the action matrix and the sparse resultant, followed by a set of substitutions, which would lead to equivalent minimal solvers.

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38

Zhu, Ganmin, Shimin Wei, Duanling Li, Yingli Wang, and Qizheng Liao. "CGA-based geometric modeling method for forward displacement analysis of 6-4 Stewart platforms." Journal of Mechanisms and Robotics, September 27, 2023, 1–19. http://dx.doi.org/10.1115/1.4063501.

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Abstract This paper presents a novel geometric modeling method for direct displacement analysis of 6-4 Stewart platforms based on the conformal geometric algebra (CGA). Firstly, a geometric constraint relationship of four lines and a plane intersecting at a point is first published. Secondly, a new coordinate-invariant geometric constraint equation of 6-4 Stewart platforms is deduced by CGA operation. Thirdly, five polynomial equations are established by CGA theory. Fourthly, Based on the above six equations, a 5 × 5 Sylvester's matrix is formulated by using Sylvester's Dialytic elimination method and Gröbner bases method under the graded reverse lexicographical order. Finally, the coordinates of four points on the moving platform are revealed. Besides, a numerical example is used to prove the validity of the proposed method. The novelties of this study is that a whole geometric modeling method by geometric constraint relationship of four lines and a plane intersecting at a point is put forward under the CGA framework, which has good intuition and offers a novel idea for solving the other complex mechanisms. At the same time, the Sylvester's matrix constructed by this method is the smallest one in the known literature for forward displacement analysis of 6-4 Stewart platforms.

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Kong, Xianwen. "Classification of a 3-RER Parallel Manipulator Based on the Type and Number of Operation Modes." Journal of Mechanisms and Robotics, August 31, 2020, 1–17. http://dx.doi.org/10.1115/1.4048262.

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Abstract The type/number of operation modes of a parallel manipulator (PM) may vary with the link parameters of the PM. This paper presents a systematic classification of a 3-RER PM based on the type/number of operation modes. The 3-RER PM was proposed as a 4-DOF (degree-of-freedom) 3T1R PM in the literature. Using the proposed method, the classification of a PM based on the type/number of operation modes can be carried out in four steps, including formulation of constraint equations of the PM, preliminary classification of the PM using Gröbner Cover, operation mode analysis of all the types of PMs using primary decomposition of ideals, and identification redundant types of PMs. Classification of the 3-RER PM shows that it has 19 types. Besides the two 4-DOF 3T1R operation modes, different types of 3-RER PMs may have up to two more 3-DOF or other types of 4- DOF operation modes. This work is the first systematic study on the impact of link parameters on the operation modes of the 3-RER PM and provide a solid foundation for further research on the design and control of 3-RER PMs and other multi-mode (or reconfigurable) PMs.

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