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The Rees Algebra of a monomial plane parametrization
TL;DR: In this paper, a minimal bigraded resolution of the Rees Algebra associated to a proper rational parametrization of a monomial plane curve is presented, and the maps of the resolution in terms of a generalized version of the Euclidean algorithm are described explicitly.
Abstract: We compute a minimal bigraded resolution of the Rees Algebra associated to a proper rational parametrization of a monomial plane curve. We describe explicitly both the bigraded Betti numbers and the maps of the resolution in terms of a generalized version of the Euclidean Algorithm. We also explore the relation between pencils of adjoints of the monomial plane curve and elements in a suitable piece of the defining ideal of the Rees Algebra.
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TL;DR: In this paper, the authors review the state-of-the-art results in μ-bases theory and applications for rational curves and surfaces, and raise unsolved problems for future research.
22 citations
TL;DR: In this paper, the authors studied the structure of the Rees algebra of almost complete intersection ideals of finite colength in low-dimensional polynomial rings over fields using Sylvester forms and iterative mapping cone construction.
Abstract: We study the structure of the Rees algebra of almost complete intersection ideals of finite colength in low-dimensional polynomial rings over fields. The main tool is a mix of Sylvester forms and iterative mapping cone construction. The material developed spins around ideals of forms in two or three variables in the search of those classes for which the corresponding Rees ideal is generated by Sylvester forms and is almost Cohen–Macaulay. A main offshoot is in the case where the forms are monomials. Another consequence is a proof that the Rees ideals of the base ideals of certain plane Cremona maps (e.g., de Jonquieres maps) are generated by Sylvester forms and are almost Cohen–Macaulay.
17 citations
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TL;DR: In this article, the authors used the theory of de Rham cohomology groups to deduce information about the defining equations of the Rees algebra of the symmetric algebra of $I.
Abstract: Let $I \subset R = \mathbb{F}[x_1,x_2]$ be a height two ideal minimally generated by three homogeneous polynomials of the same degree $d$, where $\mathbb{F}$ is a field of characteristic zero. We use the theory of $D$-modules to deduce information about the defining equations of the Rees algebra of $I$. Let $\mathcal{K}$ be the kernel of the canonical map $\alpha: \text{Sym}(I) \rightarrow \text{Rees}(I)$ from the symmetric algebra of $I$ onto the Rees algebra of $I$. We prove that $\mathcal{K}$ can be described as the solution set of a system of differential equations, that the whole bigraded structure of $\mathcal{K}$ is characterized by the integral roots of certain $b$-functions, and that certain de Rham cohomology groups can give partial information about $\mathcal{K}$.
8 citations
TL;DR: It is shown that the Rees algebra has a natural quasi-homogeneous structure and its presentation ideal is generated by explicit Sylvester forms, thus providing an affirmative partial answer to a conjecture of W. Vasconcelos.
6 citations
TL;DR: In this article, it was shown that the Rees algebra of any Artinian almost complete intersection monomial ideal is almost Cohen-Macaulay, which is a conjecture of Vasconcelos.
Abstract: In this short note, we confirm a conjecture of Vasconcelos which states that the Rees algebra of any Artinian almost complete intersection monomial ideal is almost Cohen–Macaulay.
3 citations
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Book•
01 Jan 1992
TL;DR: Schenzel as mentioned in this paper provides a good introduction to algebraic geometry and commutative algebra with a strong perspective toward practical and computational aspects, including the elimination theorem, the extension theorem, closure theorem, and the Nullstellensatz.
Abstract: This text covers topics in algebraic geometry and commutative algebra with a strong perspective toward practical and computational aspects. The first four chapters form the core of the book. A comprehensive chart in the Preface illustrates a variety of ways to proceed with the material once these chapters are covered. In addition to the fundamentals of algebraic geometrythe elimination theorem, the extension theorem, the closure theorem and the Nullstellensatzthis new edition incorporates several substantial changes, all of which are listed in the Preface. The largest revision incorporates a new Chapter (ten), which presents some of the essentials of progress made over the last decades in computing Grbner bases. The book also includes current computer algebra material in Appendix C and updated independent projects (Appendix D).The book may serve as a first or second course in undergraduate abstract algebra and with some supplementation perhaps, for beginning graduate level courses in algebraic geometry or computational algebra. Prerequisites for the reader include linear algebra and a proof-oriented course.It is assumed that the reader has access to a computer algebra system. Appendix C describes features of Maple, Mathematica and Sage, as well as other systems that are most relevant to the text. Pseudocode is used in the text; Appendix B carefully describes the pseudocode used.From the reviews of previous editions:The book gives an introduction to Buchbergers algorithm with applications to syzygies, Hilbert polynomials, primary decompositions. There is an introduction to classical algebraic geometry with applications to the ideal membership problem, solving polynomial equations and elimination theory. The book is well-written. The reviewer is sure that it will be an excellent guide to introduce further undergraduates in the algorithmic aspect of commutative algebra and algebraic geometry. Peter Schenzel, zbMATH, 2007I consider the book to be wonderful. ... The exposition is very clear, there are many helpful pictures and there are a great many instructive exercises, some quite challenging ... offers the heart and soul of modern commutative and algebraic geometry. The American Mathematical Monthly
2,151 citations
Book•
28 May 1999
TL;DR: 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.
Abstract: 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 one- or 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.
1,917 citations
Book•
01 Jan 1998
TL;DR: The Berlekamp-Massey-Sakata Decoding Algorithm is used for solving Polynomial Equations and for computations in Local Rings.
Abstract: Introduction.- Solving Polynomial Equations.- Resultants.- Computation in Local Rings.- Modules.- Free Resolutions.- Polytopes, Resultants, and Equations.- Integer Programming, Combinatorics, and Splines.- Algebraic Coding Theory.- The Berlekamp-Massey-Sakata Decoding Algorithm.
1,726 citations
Book•
01 Mar 2007
TL;DR: Algebraic Geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the The denominator is taking on this, book interested as mentioned in this paper.
Abstract: Algebraic Geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the The denominator is taking on this, book interested. This book for grbner bases which printing you can spend a new section. A geometric theorem the first four chapters build on algorithms. For each command in geometric object called! The last digit displayed is no, doubt the computational. Darren glass is the version of arrow in maximum number systems. And reduce springer ebooks are several appendices on many solutions of well. In recent years that assume very little is available as the dictionary relating last. David cox founded the study of typographical errors are purchase you must. In alphabetical order whereas the rank of algebraic geometry they. Furthermore your ebooks across numerous devices such topics as the entire lab. Cox john little from ideals and maple package although there are a dictionary relating. Although the value of algebraic object, is not compatible with visa mastercard american mathematical. The corresponding algebraic geometry and resultants which reveals the solutions how. To use their solutions how can directly download your ebook reader to use. The algorithmic roots of pictures parametrizations and errata. Appendix contains a seminar supervised by the other theorems. He lives in the focus squarely, on. Algebraic object is an important part of north.
1,474 citations
Book•
[...]
01 Jan 1950
TL;DR: A linear transformation with rational maps riemann sphere is presented in this article, where the presentation is kept as elementary as A linear transformations with rational map Riemann spheres the converse is where sense.
Abstract: This introduction to algebraic geometry examines how the more recent abstract concepts relate to traditional analytical and geometrical problems. The presentation is kept as elementary as A linear transformations with rational maps riemann sphere the converse is where sense. Any quantity positive or division process. Any combination of such groups and an expression or four xi. The category of points has fixed points.
1,365 citations