Computational Commutative Algebra 1

Abstract

The use of general descriptive names, registered names, trademarks, etc. in this publication does not tective laws and regulations and therefore free for general use. is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broad-Law of September 9, 1965, in its current version, and permission for use must always be obtained from imply, even in the absence of a specific statement, that such names are exempt from the relevant pro-fur Fakult t

Algebraic and Differential Invariants

“…This is a founding stone in algebraic computing. Introduced inthe mid 60s they have been the subject of intensive research and now textbooks [1,3,12,20,37]. They are implemented in most computer algebra systems.…”

Regularization and Matrix Computation in Numerical Polynomial Algebra

“…Most advanced software packages are perhaps the polynomial system solvers based on the homotopy continuation method [57,82], including PHCPACK [35,89], BERTINI [4,5] and MIXEDVOL [31]. There are also specialized implementations of algorithms such as approximate GCD finder QRGCD [14] that is bundled in recent Maple releases, approximate factorization [48], multiplicity structure [6,88], SNAP package [41] bundled in Maple, univariate factorization and multiple root solver MULTROOT [97], SYNAPS package [66], and COCOA [9,55,56], etc. Those packages provide a broad range of versatile tools for applications and academic research.…”

Ranking committees, income streams or multisets

“…Then one of the main theorems about Farey fractions states that in this case s + ℓ = j and t + h = k + 1 (25) (see [19], Theorem 29 or [7] Theorem 9). In particular, t < k + 1 and h < k + 1.…”

“…It has to be noted that orders on multisets and its subsets which extend a given linear order on the underlying set on which multisets are constructed have been instrumental in proofs of program termination, in equational reasoning algorithms based on term rewriting systems, in computer algebra, the theory of invariants, and the theory of partitions. We refer the reader to the two surveys by Martin [27] and Dershowitz [8] and to the book by Kreuzer and Robbiano [25] on these topics.…”