Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany oneor two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

Modern computer algebra

From the reviews of the first edition: "It is certainly no exaggeration to say that A Singular Introduction to Commutative Algebra aims to lead a further stage in the computational revolution in commutative algebra . Among the great strengths and most distinctive features is a new, completely unified treatment of the global and local theories. making it one of the most flexible and most efficient systems of its type....another strength of Greuel and Pfister's book is its breadth of coverage of theoretical topics in the portions of commutative algebra closest to algebraic geometry, with algorithmic treatments of almost every topic....Greuel and Pfister have written a distinctive and highly useful book that should be in the library of every commutative algebraist and algebraic geometer, expert and novice alike." J.B. Little, MAA, March 2004 The second edition is substantially enlarged by a chapter on Groebner bases in non-commtative rings, a chapter on characteristic and triangular sets with applications to primary decomposition and polynomial solving and an appendix on polynomial factorization including factorization over algebraic field extensions and absolute factorization, in the uni- and multivariate case.

A Singular Introduction to Commutative Algebra

Polynomials in one variable Grobner bases of zero-dimensional ideals Bernstein's theorem and fewnomials Resultants Primary decomposition Polynomial systems in economics Sums of squares Polynomial systems in statistics Tropical algebraic geometry Linear partial differential equations with constant coefficients Bibliography Index.

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Solving systems of polynomial equations

Modular elliptic curves and fermat's last theorem

Preface: Algebra and Geometry * Free Resolutions and Hilbert Functions * First Examples of Free Resolutions * Points in P^2 * Castelnuovo-Mumford Regularity * The Regularity of Projective Curves * Linear Series and 1-Generic Matrices * Linear Complexes and the Linear Syzygy Theorem * Curves of High Degree * Clifford Index and Canonical Embedding * Appendix 1: Introduction to Local Cohomology * Appendix 2: A Jog Through Commutative Algebra * References * Index

The Geometry of Syzygies: A Second Course in Commutative Algebra and Algebraic Geometry

The Hilbert series and degree bounds play significant roles in computational invariant theory In the modular case, neither of these tools is avrulable in general In this article three results are obtruned, which provide partial remedies for these shortcomings First, it is shown that the so-called extended Hilbert series, which can always be calculated by a MoHen type formula, yields strong constraints on the degrees of primary invariants Then it is shown that for a trivial source module the (ordinary) Hilbert series coincides with that of a lift to characteristic 0 and can hence be calculated by MoHen’s formula The last result is a generalization of Goobel’s degree bound to the case of monomial representations

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Primary Decomposition: Algorithms and Comparisons

We introduce an algebro-geometrically motived integration-by-parts (IBP) re- duction method for multi-loop and multi-scale Feynman integrals, using a framework for massively parallel computations in computer algebra. This framework combines the com- puter algebra system Singular with the workflow management system GPI-Space, which are being developed at the TU Kaiserslautern and the Fraunhofer Institute for Industrial Mathematics (ITWM), respectively. In our approach, the IBP relations are first trimmed by modern tools from computational algebraic geometry and then solved by sparse linear algebra and our new interpolation method. Modelled in terms of Petri nets, these steps are efficiently automatized and automatically parallelized by GPI-Space. We demonstrate the potential of our method at the nontrivial example of reducing two-loop five-point non- planar double-pentagon integrals. We also use GPI-Space to convert the basis of IBP reductions, and discuss the possible simplification of master-integral coefficients in a uni- formly transcendental basis.

https://link.springer.com/content/pdf/10.1007/JHEP02(2020)079.pdf

Integration-by-parts reductions of Feynman integrals using Singular and GPI-Space

Non-general type surfaces in P 4 : some remarks on bounds and constructions

An elliptic conic bundle in $\mathbb{P}^4$ arising from a stable rank-3 vector bundle

We introduce an algebro-geometrically motived integration-by-parts (IBP) reduction method for multi-loop and multi-scale Feynman integrals, using a framework for massively parallel computations in computer algebra. This framework combines the computer algebra system Singular with the workflow management system GPI-Space, which is being developed at the Fraunhofer Institute for Industrial Mathematics (ITWM). In our approach, the IBP relations are first trimmed by modern algebraic geometry tools and then solved by sparse linear algebra and our new interpolation methods. These steps are efficiently automatized and automatically parallelized by modeling the algorithm in GPI-Space using the language of Petri-nets. We demonstrate the potential of our method at the nontrivial example of reducing two-loop five-point nonplanar double-pentagon integrals. We also use GPI-Space to convert the basis of IBP reductions, and discuss the possible simplification of IBP coefficients in a uniformly transcendental basis.

/pdf/integration-by-parts-reductions-of-feynman-integrals-using-32arbw605k.pdf

Wolfram Decker

Papers

Primary Decomposition: Algorithms and Comparisons

Integration-by-parts reductions of Feynman integrals using Singular and GPI-Space

Non-general type surfaces in P 4 : some remarks on bounds and constructions

An elliptic conic bundle in $\mathbb{P}^4$ arising from a stable rank-3 vector bundle

Integration-by-parts reductions of Feynman integrals using Singular and GPI-Space.