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Noncommutative Computer Algebra with applications: Graded commutative algebra and related structures in Singular with applicati,Used
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This book that represents the author's Ph.D. thesis is devoted to constructive module theory of polynomial graded commutative algebras over a field. It treats the theory of Grbner bases, standard bases (SB) and syzygies as well as algorithms and their implementations over graded commutative algebras, which naturally unify exterior and commutative polynomial algebras. They are graded noncommutative, associative unital algebras over fields and may contain zerodivisors. In this book we try to make the most use out of apriori knowledge about their characteristic (supercommutative) structure in developing direct symbolic methods, algorithms and implementations, which are intrinsic to these algebras and practically efficient. We also tackle their central localizations by generalizing a variation of Mora algorithm. In this setting we prove a generalized Buchberger's criterion, which shows that syzygies of leading terms play the utmost important role in SB and syzygy computations. We develop a variation of the La ScalaStillman free resolution algorithm. Benchmarks show that our new algorithms and implementation are efficient. We give some applications of the developed framework.
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