The Rees Algebra of a monomial plane parametrization

doi:10.1016/J.JSC.2014.09.026

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The Rees Algebra of a monomial plane parametrization

Journal Article•DOI•

The Rees Algebra of a monomial plane parametrization

Teresa Cortadellas Benítez¹, Carlos D'Andrea¹•Institutions (1)

University of Barcelona¹

01 Sep 2015-Journal of Symbolic Computation (Academic Press, Inc.)-Vol. 70, pp 71-105

TL;DR: A minimal bigraded resolution of the Rees Algebra associated to a proper rational parametrization of a monomial plane curve is computed in terms of a generalized version of the Euclidean Algorithm.

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About: This article is published in Journal of Symbolic Computation.The article was published on 2015-09-01 and is currently open access. It has received 11 citations till now. The article focuses on the topics: Rees algebra & Monomial.

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Journal Article•DOI•

Survey on the theory and applications of μ-bases for rational curves and surfaces

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Xiaohong Jia¹, Xiaoran Shi², Falai Chen³•Institutions (3)

Chinese Academy of Sciences¹, Harbin Institute of Technology², University of Science and Technology of China³

01 Feb 2018-Journal of Computational and Applied Mathematics

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.

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22 citations

Cites background from "The Rees Algebra of a monomial plan..."

...See work on Rees algebra of rational plane curves in [44][43][39] [8] [4], of rational space curves in [59][52] [62], of rational surfaces in [63][3][51] [16][11][12][18]....
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Journal Article•DOI•

The ubiquity of Sylvester forms in almost complete intersections

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Aron Simis, Stefan O. Tohaneanu¹•Institutions (1)

University of Idaho¹

01 Jan 2015-Collectanea Mathematica

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.

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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.

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17 citations

Journal Article•DOI•

On a conjecture of Vasconcelos via Sylvester forms

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Ricardo Burity¹, Aron Simis², Stefan O. Tohaneanu³•Institutions (3)

Universidade Federal Rural de Pernambuco¹, Federal University of Pernambuco², University of Idaho³

01 Nov 2016-Journal of Symbolic Computation

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.

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6 citations

Cites methods or result from "The Rees Algebra of a monomial plan..."

...They have been largely used in many sources, such as (Busé, 2009; Cortadellas and D’Andrea, 2014, 2015; Cox, 2008; Cox et al., 2008; Hassanzadeh and Simis, 2014; Jouanolou, 1997; Hong et al., 2008, 2012, 2013; Simis and Tohǎneanu, 2015)....
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...D’Andrea (Cortadellas and D’Andrea, 2015), and independently, of work by the present second and third authors (Simis and Tohǎneanu, 2015)....
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...Recently, different proofs were established in the binary case of monomials of the same degree as a consequence of work by T.B. Cortadellas and C. D’Andrea (Cortadellas and D’Andrea, 2015), and independently, of work by the present second and third authors (Simis and Tohǎneanu, 2015)....
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Journal Article•DOI•

Bounding the degrees of a minimal μ-basis for a rational surface parametrization

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Yairon Cid-Ruiz¹•Institutions (1)

University of Barcelona¹

01 Nov 2019-Journal of Symbolic Computation

TL;DR: In this article, the existence of a μ-basis with polynomials bounded in degree by O(d 33 ) was shown for an arbitrary rational surface parametrization P ( s, t ) = ( a 1 ( s, t ), a 2 (s, t ), a 3 ( s), t ), a 4 ( s and t ) ∈ F [ s, T ] 4 over an infinite field F.

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5 citations

Journal Article•DOI•

On the conjecture of vasconcelos for artinian almost complete intersection monomial ideals

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Kuei-Nuan Lin¹, Yi-Huang Shen²•Institutions (2)

Pennsylvania State University¹, University of Science and Technology of China²

01 Sep 2021-Nagoya Mathematical Journal

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.

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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.

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3 citations

References

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Book•

Modern Computer Algebra

[...]

Joachim von zur Gathen, Jürgen Gerhard

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.

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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.

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1,917 citations

Book•

Using Algebraic Geometry

[...]

David A. Cox, John Little, Donal O'Shea

01 Jan 1998

TL;DR: The Berlekamp-Massey-Sakata Decoding Algorithm is used for solving Polynomial Equations and for computations in Local Rings.

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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.

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1,726 citations

Book•

Algebraic curves

[...]

Robert John Walker

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.

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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.

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1,365 citations

"The Rees Algebra of a monomial plan..." refers methods in this paper

...multiplicities, which concludes the proof. Remark 8.2. Lema 8.1 implies that the singularities of Cu,d are not ordinary, as if this were the case, then by applying the genus formula (see for instance [Wal50]) we would get u(u−1)+(d −u)(d −u−1) = (d −1)(d −2), which is impossible unless u+1 = d, contradicting the fact u < d 2. The following result will be useful to compute dimensions of pencils of adjo...
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Book•

An Introduction to Gröbner Bases

[...]

William W. Adams¹, Philippe Loustaunau²•Institutions (2)

University of Maryland, College Park¹, George Mason University²

01 Jan 2012

TL;DR: In this paper, the basic theory of Grobner bases is presented and a well-ordering and induction algorithm for well-ordered Grobners over rings is presented, along with a list of symbols.

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Abstract: Basic theory of Grobner bases Applications of Grobner bases Modules and Grobner bases Grobner bases over rings Appendix A. Computations and algorithms Appendix B. Well-ordering and induction References List of symbols Index.

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842 citations

"The Rees Algebra of a monomial plan..." refers background in this paper

...s on Sm and syzygies In this section we will recall deﬁnitions and properties of Gr¨obner bases of submodules of Sm for m ∈ N. All the known material is classical, we refer the reader to Chapter 3 in [AL94] for proofs and further references. Denote with {e1,...,em} the canonical basis of Sm. Recall that a monomial in Sm is a vector of the type TαXβ e i, 1 ≤ i ≤ m, with TαXβ being a monomial in S. A term...
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Book•

Plane algebraic curves

[...]

Gerd Fischer¹•Institutions (1)

University of Düsseldorf¹

01 Jan 2001

TL;DR: The projective closure of algebraic curves and their equations are discussed in this article, along with a discussion of the implicit function theorem and the Harnack inequality of singularities.

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Abstract: Introduction Affine algebraic curves and their equations The projective closure Tangents and singularities Polars and Hessian curves The dual curve and the Plucker formulas The ring of convergent power series Parametrizing the branches of a curve by Puiseux series Tangents and intersection multiplicities of germs of curves The Riemann surface of an algebraic curve The resultant Covering maps The implicit function theorem The Newton polygon A numerical invariant of singularities of curves Harnack's inequality Bibliography Subject index List of symbols.

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710 citations