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Showing 1 to 16 of 16 for “"Gröbner Bases"”.
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Gröbner Bases and Syzygy Modules
The theory of Gröbner bases has become a useful tool in computational commutative algebra. In this paper, we outline some basic results, as they are found in [1], including the concepts of terms ordering, multivariable polynomial division, Gröbner bases, Buchberger's algorithm, and syzygy modules. …
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Non-commutative Gröbner Bases and Applications
Commutative Gröbner bases have a lot of applications in theory and practice, because they have many nice properties, they are computable, and there exist many efficient improvements of their computations. Non-commutative Gröbner bases also have many useful properties. However, applications of …
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Gröbner bases in rational homotopy theory
… converging to the homotopy of a pushout. We use Gröbner-Shirshov bases, an analogue of Gröbner bases for free Lie algebras, to compute directly the E2 term for pushouts of wedges of spheres. In particular, for a cofiber sequence A --> X --> C where A and X are wedges of spheres, we use this …
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A survey of primary decomposition using Gröbner bases
Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1995.
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Algorithms and Orders for Finding Noncummutative Gröbner Bases
… and good admissible orders for computing Gröbner bases in noncommutative algebras is considered. Gröbner bases are an important tool that make many problems in polynomial algebra computationally tractable. However, the computation of Gröbner bases is expensive, and in noncommutative …
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Weyl Gröbner Basis Cryptosystems
… certain class of algebraic cryptosystems called Gröbner Basis Cryptosystems. In 1994, Koblitz introduced the Polly Cracker cryptosystem that is based on the theory of Gröbner basis in commutative polynomials rings. The security of this cryptosystem relies on the fact that the computation of …
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A Classification of some Quadratic Algebras
… to be Koszul. Techniques include the use of Gröbner bases and development of Koszul resolutions. When the quadratic algebra is Koszul, we provide the associated linear resolution of the field. When not Koszul, we describe the maps of the resolution up to the instance of nonlinearity.
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Exploiting chordal structure in systems of polynomial equations
… that relies in elimination theory and Gröbner bases. Chordal elimination can be seen as a generalization of sparse linear algebra. Unlike the linear case, the elimination process may not be exact. Nonetheless, we show that our methods are well-behaved for a large family of problems. We …
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Gröbner basis techniques for certain problems in coding and systems theory
… multivariable interpolation and list decoding. Gröbner basis theory has come into its own in the context of software and algorithm development. By generalizing the concept of polynomial degree, term orders are provided for multivariable polynomial rings and free modules over polynomial rings. …
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Arrangement of minors in the positive Grassmannian
… appeared in the context of alcoved polytopes and Gröbner bases. Maximal arrangements of this form correspond to simplices of the alcoved triangulation of the hypersimplex; and the number of such arrangements equals the Eulerian number. On the other hand, we prove in many cases that arrangements of …
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On the Complexity of Computing Galois Groups of Differential Equations
… complexity, in the worst case, of computing a Gröbner basis is doubly exponential in the number of variables. Feng chose to represent the radical of the ideal generated by the defining equations of a proto-Galois group by its Gröbner basis. Hence, a double-exponential degree bound for computing …
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Algebraic theory for discrete models in systems biology
… which can be efficiently solved by computing Gröbner bases. Section 5 introduces a bit-wise implementation of a Gröbner basis algorithm for Boolean polynomials. This implementation has been incorporated into the core engine of Macaulay 2. Chapter 6 discusses bistability for Boolean models …
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The Combinatorics of Involutive Bases: Theory, algorithms and applications
… and combinatorics is the use of involutive bases, along with other types of Gröbner bases that exhibit additional combinatorial properties. In particular, the nice combinatorial properties that come along with these bases enable applications in other areas, such as system reliability, …
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Algebraic and Logic Solving Methods for Cryptanalysis
… and techniques that are related to border bases and Gröbner bases. In particular, we describe in detail the Border Basis Algorithm and discuss its specialized version for Boolean polynomials called the Boolean Border Basis Algorithm. In the second part of the thesis, we deal with connecting …
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Eine Anwendung der Invariantentheorie auf das Korrespondenzproblem lokaler Bildmerkmale
… von Unteralgebren werden, analog zu Gröbner-Basen als Erzeugendensysteme von Idealen, SAGBI-Basen genannt ("Subalgebra Analogs to Gröbner Bases for Ideals"). SAGBI-Basen werden hier insbesondere aus algorithmischer Sicht behandelt, d.h. die Berechnung von SAGBI-Basen steht im …
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Polynomial Models for Systems Biology: Data Discretization and Term Order Effect on Dynamics
… data points and uniqueness of Gröbner bases is studied in this dissertation. More specifically, an algorithm is developed for finding all minimal sets of points that, added to the given set, have a corresponding ideal of points with a unique Gröbner basis. This question is of …