This book that represents the authors Ph.D. thesis is devoted to constructive module theory of polynomial graded commutative algebras over a field. It treats the theory of Gröbner 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 non-commutative, associative unital algebras over fields and may contain zero-divisors. In this book we try to make the most use out of a-priori knowledge about their characteristic (super-commutative) 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 Buchbergers 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 Scala-Stillman 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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Non–commutative Computer Algebra for polynomial algebras: Gröbner bases, applications and implementation. Viktor Levandovskyy. Vom Fachbereich ... 4 Aug 2011 ... Non-commutative Computer Algebra with applications, 978-3-8381-2752-1, This book that represents the author's Ph.D. thesis is devoted to ... Myoung An, Jim Byrnes, William Moran, B. Saffari, Harold S. Shapiro, Richard Tolimieri. Pages 155-196. PDF · Clifford Algebras as Unified Language for Image ...