Algorithms and Implementations in Computational Algebraic Geometry
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Using Macaulay2 Effectively in Practice
Using Macaulay2 effectively in practice Mike Stillman ([email protected]) Department of Mathematics Cornell 22 July 2019 / IMA Sage/M2 Macaulay2: at a glance Project started in 1993, Dan Grayson and Mike Stillman. Open source. Key computations: Gr¨obnerbases, free resolutions, Hilbert functions and applications of these. Rings, Modules and Chain Complexes are first class objects. Language which is comfortable for mathematicians, yet powerful, expressive, and fun to program in. Now a community project Journal of Software for Algebra and Geometry (started in 2009. Now we handle: Macaulay2, Singular, Gap, Cocoa) (original editors: Greg Smith, Amelia Taylor). Strong community: including about 2 workshops per year. User contributed packages (about 200 so far). Each has doc and tests, is tested every night, and is distributed with M2. Lots of activity Over 2000 math papers refer to Macaulay2. History: 1976-1978 (My undergrad years at Urbana) E. Graham Evans: asked me to write a program to compute syzygies, from Hilbert's algorithm from 1890. Really didn't work on computers of the day (probably might still be an issue!). Instead: Did computation degree by degree, no finishing condition. Used Buchsbaum-Eisenbud \What makes a complex exact" (by hand!) to see if the resulting complex was exact. Winfried Bruns was there too. Very exciting time. History: 1978-1983 (My grad years, with Dave Bayer, at Harvard) History: 1978-1983 (My grad years, with Dave Bayer, at Harvard) I tried to do \real mathematics" but Dave Bayer (basically) rediscovered Groebner bases, and saw that they gave an algorithm for computing all syzygies. I got excited, dropped what I was doing, and we programmed (in Pascal), in less than one week, the first version of what would be Macaulay. -
Research in Algebraic Geometry with Macaulay2
Research in Algebraic Geometry with Macaulay2 1 Macaulay2 a software system for research in algebraic geometry resultants the community, packages, the journal, and workshops 2 Tutorial the ideal of a monomial curve elimination: projecting the twisted cubic quotients and saturation 3 the package PHCpack.m2 solving polynomial systems numerically MCS 507 Lecture 39 Mathematical, Statistical and Scientific Software Jan Verschelde, 25 November 2019 Scientific Software (MCS 507) Macaulay2 L-39 25November2019 1/32 Research in Algebraic Geometry with Macaulay2 1 Macaulay2 a software system for research in algebraic geometry resultants the community, packages, the journal, and workshops 2 Tutorial the ideal of a monomial curve elimination: projecting the twisted cubic quotients and saturation 3 the package PHCpack.m2 solving polynomial systems numerically Scientific Software (MCS 507) Macaulay2 L-39 25November2019 2/32 Macaulay2 D. R. Grayson and M. E. Stillman: Macaulay2, a software system for research in algebraic geometry, available at https://faculty.math.illinois.edu/Macaulay2. Funded by the National Science Foundation since 1992. Its source is at https://github.com/Macaulay2/M2; licence: GNU GPL version 2 or 3; try it online at http://habanero.math.cornell.edu:3690. Several workshops are held each year. Packages extend the functionality. The Journal of Software for Algebra and Geometry (at https://msp.org/jsag) started its first volume in 2009. The SageMath interface to Macaulay2 requires its binary M2 to be installed on your computer. Scientific -
An Alternative Algorithm for Computing the Betti Table of a Monomial Ideal 3
AN ALTERNATIVE ALGORITHM FOR COMPUTING THE BETTI TABLE OF A MONOMIAL IDEAL MARIA-LAURA TORRENTE AND MATTEO VARBARO Abstract. In this paper we develop a new technique to compute the Betti table of a monomial ideal. We present a prototype implementation of the resulting algorithm and we perform numerical experiments suggesting a very promising efficiency. On the way of describing the method, we also prove new constraints on the shape of the possible Betti tables of a monomial ideal. 1. Introduction Since many years syzygies, and more generally free resolutions, are central in purely theo- retical aspects of algebraic geometry; more recently, after the connection between algebra and statistics have been initiated by Diaconis and Sturmfels in [DS98], free resolutions have also become an important tool in statistics (for instance, see [D11, SW09]). As a consequence, it is fundamental to have efficient algorithms to compute them. The usual approach uses Gr¨obner bases and exploits a result of Schreyer (for more details see [Sc80, Sc91] or [Ei95, Chapter 15, Section 5]). The packages for free resolutions of the most used computer algebra systems, like [Macaulay2, Singular, CoCoA], are based on these techniques. In this paper, we introduce a new algorithmic method to compute the minimal graded free resolution of any finitely generated graded module over a polynomial ring such that some (possibly non- minimal) graded free resolution is known a priori. We describe this method and we present the resulting algorithm in the case of monomial ideals in a polynomial ring, in which situ- ation we always have a starting nonminimal graded free resolution. -
Computations in Algebraic Geometry with Macaulay 2
Computations in algebraic geometry with Macaulay 2 Editors: D. Eisenbud, D. Grayson, M. Stillman, and B. Sturmfels Preface Systems of polynomial equations arise throughout mathematics, science, and engineering. Algebraic geometry provides powerful theoretical techniques for studying the qualitative and quantitative features of their solution sets. Re- cently developed algorithms have made theoretical aspects of the subject accessible to a broad range of mathematicians and scientists. The algorith- mic approach to the subject has two principal aims: developing new tools for research within mathematics, and providing new tools for modeling and solv- ing problems that arise in the sciences and engineering. A healthy synergy emerges, as new theorems yield new algorithms and emerging applications lead to new theoretical questions. This book presents algorithmic tools for algebraic geometry and experi- mental applications of them. It also introduces a software system in which the tools have been implemented and with which the experiments can be carried out. Macaulay 2 is a computer algebra system devoted to supporting research in algebraic geometry, commutative algebra, and their applications. The reader of this book will encounter Macaulay 2 in the context of concrete applications and practical computations in algebraic geometry. The expositions of the algorithmic tools presented here are designed to serve as a useful guide for those wishing to bring such tools to bear on their own problems. A wide range of mathematical scientists should find these expositions valuable. This includes both the users of other programs similar to Macaulay 2 (for example, Singular and CoCoA) and those who are not interested in explicit machine computations at all. -
Prizes and Awards Session
PRIZES AND AWARDS SESSION Wednesday, July 12, 2021 9:00 AM EDT 2021 SIAM Annual Meeting July 19 – 23, 2021 Held in Virtual Format 1 Table of Contents AWM-SIAM Sonia Kovalevsky Lecture ................................................................................................... 3 George B. Dantzig Prize ............................................................................................................................. 5 George Pólya Prize for Mathematical Exposition .................................................................................... 7 George Pólya Prize in Applied Combinatorics ......................................................................................... 8 I.E. Block Community Lecture .................................................................................................................. 9 John von Neumann Prize ......................................................................................................................... 11 Lagrange Prize in Continuous Optimization .......................................................................................... 13 Ralph E. Kleinman Prize .......................................................................................................................... 15 SIAM Prize for Distinguished Service to the Profession ....................................................................... 17 SIAM Student Paper Prizes .................................................................................................................... -
What Can Computer Algebraic Geometry Do Today?
What can computer Algebraic Geometry do today? Gregory G. Smith Wolfram Decker Mike Stillman 14 July 2015 Essential Questions ̭ What can be computed? ̭ What software is currently available? ̭ What would you like to compute? ̭ How should software advance your research? Basic Mathematical Types ̭ Polynomial Rings, Ideals, Modules, ̭ Varieties (affine, projective, toric, abstract), ̭ Sheaves, Divisors, Intersection Rings, ̭ Maps, Chain Complexes, Homology, ̭ Polyhedra, Graphs, Matroids, ̯ Established Geometric Tools ̭ Elimination, Blowups, Normalization, ̭ Rational maps, Working with divisors, ̭ Components, Parametrizing curves, ̭ Sheaf Cohomology, ঠ-modules, ̯ Emerging Geometric Tools ̭ Classification of singularities, ̭ Numerical algebraic geometry, ̭ ैक़௴Ь, Derived equivalences, ̭ Deformation theory,Positivity, ̯ Some Geometric Successes ̭ GEOGRAPHY OF SURFACES: exhibiting surfaces with given invariants ̭ BOIJ-SÖDERBERG: examples lead to new conjectures and theorems ̭ MODULI SPACES: computer aided proofs of unirationality Some Existing Software ̭ GAP,Macaulay2, SINGULAR, ̭ CoCoA, Magma, Sage, PARI, RISA/ASIR, ̭ Gfan, Polymake, Normaliz, 4ti2, ̭ Bertini, PHCpack, Schubert, Bergman, an idiosyncratic and incomplete list Effective Software ̭ USEABLE: documented examples ̭ MAINTAINABLE: includes tests, part of a larger distribution ̭ PUBLISHABLE: Journal of Software for Algebra and Geometry; www.j-sag.org ̭ CITATIONS: reference software Recent Developments in Singular Wolfram Decker Janko B¨ohm, Hans Sch¨onemann, Mathias Schulze Mohamed Barakat TU Kaiserslautern July 14, 2015 Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 1 / 24 commutative and non-commutative algebra, singularity theory, and with packages for convex and tropical geometry. It is free and open-source under the GNU General Public Licence. -
Arxiv:2010.06953V2 [Math.RA] 23 Apr 2021
IDENTITIES AND BASES IN THE HYPOPLACTIC MONOID ALAN J. CAIN, ANTÓNIO MALHEIRO, AND DUARTE RIBEIRO Abstract. This paper presents new results on the identities satisfied by the hypoplactic monoid. We show how to embed the hypoplactic monoid of any rank strictly greater than 2 (including infinite rank) into a direct product of copies of the hypoplactic monoid of rank 2. This confirms that all hypoplactic monoids of rank greater than or equal to 2 satisfy exactly the same identities. We then give a complete characterization of those identities, and prove that the variety generated by the hypoplactic monoid has finite axiomatic rank, by giving a finite basis for it. 1. Introduction A (non-trivial) identity is a formal equality u ≈ v, where u and v are words over some alphabet of variables, which is not of the form u ≈ u. If a monoid is known to satisfy an identity, an important question is whether the set of identities it satisfies is finitely based, that is, if all these identities are consequences of those in some finite subset (see [37, 42]). The plactic monoid plac, also known as the monoid of Young tableaux, is an important algebraic structure, first studied by Schensted [38] and Knuth [23], and later studied in depth by Lascoux and Schützenberger [27]. It is connected to many different areas of Mathematics, such as algebraic combinatorics, symmetric functions [29], crystal bases [4] and representation theory [14]. In particular, the question of identities in the plactic monoid has received a lot of attention recently [26, 20]. Finitely-generated polynomial-growth groups are virtually nilpotent and so sat- isfy identities [16]. -
Vita BERND STURMFELS Department of Mathematics, University of California, Berkeley, CA 94720 Phone
Vita BERND STURMFELS Department of Mathematics, University of California, Berkeley, CA 94720 Phone: (510) 642 4687, Fax: (510) 642 8204, [email protected] M.A. [Diplom] TH Darmstadt, Germany, Mathematics and Computer Science, 1985 Ph.D. [Dr. rer. nat.] TH Darmstadt, Germany, Mathematics, 1987 Ph.D. University of Washington, Seattle, Mathematics, 1987 Professional Experience: 1987–1988 Postdoctoral Fellow, I.M.A., University of Minnesota, Minneapolis 1988–1989 Assistant Professor, Research Institute for Symbolic Computation, (RISC-Linz), Linz, Austria 1989–1991 Assistant Professor, Department of Mathematics, Cornell University 1992–1996 Associate Professor, Department of Mathematics, Cornell University 1994–2001 Professor, Department of Mathematics, University of California, Berkeley 2001– Professor, Department of Mathematics and Computer Science, UC Berkeley Academic Honors: 1986 - 1987 Alfred P. Sloan Doctoral Dissertation Fellowship 1991 - 1993 Alfred P. Sloan Research Fellow 1992 - 1997 National Young Investigator (NSF) 1992 - 1997 David and Lucile Packard Fellowship 1999 Lester R. Ford Prize for Expository Writing (MAA) 2000-2001 Miller Research Professorship, UC Berkeley Spring 2003 John von Neumann Professor, Technical University M¨unchen 2003-2004 Hewlett-Packard Research Professor at MSRI Berkeley July 2004 Clay Mathematics Institute Senior Scholar Research Interests: Computational Algebra, Combinatorics, Algebraic Geometry Selected Professional Activities: Visiting Positions: D´epartment de Math´ematiques, Universit´ede Nice, France, Spring 1989 Mathematical Sciences Research Institute, Berkeley, Fall 1992 Courant Institute, New York University, 1994–95 RIMS, Kyoto University, Japan, 1997–98 Current Editorial Board Membership: Journal of the American Mathematical Society, Duke Mathematical Journal, Collecteana Mathematica, Beitr¨agezur Geometrie und Algebra, Order, Discrete and Computational Geometry, Applicable Algebra (AAECC) Journal of Combinatorial Theory (Ser. -
GNU Texmacs User Manual Joris Van Der Hoeven
GNU TeXmacs User Manual Joris van der Hoeven To cite this version: Joris van der Hoeven. GNU TeXmacs User Manual. 2013. hal-00785535 HAL Id: hal-00785535 https://hal.archives-ouvertes.fr/hal-00785535 Preprint submitted on 6 Feb 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. GNU TEXMACS user manual Joris van der Hoeven & others Table of contents 1. Getting started ...................................... 11 1.1. Conventionsforthismanual . .......... 11 Menuentries ..................................... 11 Keyboardmodifiers ................................. 11 Keyboardshortcuts ................................ 11 Specialkeys ..................................... 11 1.2. Configuring TEXMACS ..................................... 12 1.3. Creating, saving and loading documents . ............ 12 1.4. Printingdocuments .............................. ........ 13 2. Writing simple documents ............................. 15 2.1. Generalities for typing text . ........... 15 2.2. Typingstructuredtext ........................... ......... 15 2.3. Content-basedtags -
The Reesalgebra Package in Macaulay2
Journal of Software for Algebra and Geometry The ReesAlgebra package in Macaulay2 DAVID EISENBUD vol 8 2018 JSAG 8 (2018), 49–60 The Journal of Software for dx.doi.org/10.2140/jsag.2018.8.49 Algebra and Geometry The ReesAlgebra package in Macaulay2 DAVID EISENBUD ABSTRACT: This note introduces Rees algebras and some of their uses, with illustrations from version 2.2 of the Macaulay2 package ReesAlgebra.m2. INTRODUCTION. A central construction in modern commutative algebra starts from an ideal I in a commutative ring R, and produces the Rees algebra R.I / VD R ⊕ I ⊕ I 2 ⊕ I 3 ⊕ · · · D∼ RTI tU ⊂ RTtU; where RTtU denotes the polynomial algebra in one variable t over R. For basics on Rees algebras, see[Vasconcelos 1994] and[Swanson and Huneke 2006], and for some other research, see[Eisenbud and Ulrich 2018; Kustin and Ulrich 1992; Ulrich 1994], and[Valabrega and Valla 1978]. From the point of view of algebraic geometry, the Rees algebra R.I / is a homo- geneous coordinate ring for the graph of a rational map whose total space is the blowup of Spec R along the scheme defined by I. (In fact, the “Rees algebra” is sometimes called the “blowup algebra”.) Rees algebras were first studied in the algebraic context by David Rees, in the now-famous paper[Rees 1958]. Actually, Rees mainly studied the ring RTI t; t−1U, now also called the extended Rees algebra of I. Mike Stillman and I wrote a Rees algebra script for Macaulay classic. It was aug- mented, and made into the[Macaulay2] package ReesAlgebra.m2 around 2002, to study a generalization of Rees algebras to modules described in[Eisenbud et al. -
SMT Solving in a Nutshell
SAT and SMT Solving in a Nutshell Erika Abrah´ am´ RWTH Aachen University, Germany LuFG Theory of Hybrid Systems February 27, 2020 Erika Abrah´ am´ - SAT and SMT solving 1 / 16 What is this talk about? Satisfiability problem The satisfiability problem is the problem of deciding whether a logical formula is satisfiable. We focus on the automated solution of the satisfiability problem for first-order logic over arithmetic theories, especially using SAT and SMT solving. Erika Abrah´ am´ - SAT and SMT solving 2 / 16 CAS SAT SMT (propositional logic) (SAT modulo theories) Enumeration Computer algebra DP (resolution) systems [Davis, Putnam’60] DPLL (propagation) [Davis,Putnam,Logemann,Loveland’62] Decision procedures NP-completeness [Cook’71] for combined theories CAD Conflict-directed [Shostak’79] [Nelson, Oppen’79] backjumping Partial CAD Virtual CDCL [GRASP’97] [zChaff’04] DPLL(T) substitution Watched literals Equalities and uninterpreted Clause learning/forgetting functions Variable ordering heuristics Bit-vectors Restarts Array theory Arithmetic Decision procedures for first-order logic over arithmetic theories in mathematical logic 1940 Computer architecture development 1960 1970 1980 2000 2010 Erika Abrah´ am´ - SAT and SMT solving 3 / 16 SAT SMT (propositional logic) (SAT modulo theories) Enumeration DP (resolution) [Davis, Putnam’60] DPLL (propagation) [Davis,Putnam,Logemann,Loveland’62] Decision procedures NP-completeness [Cook’71] for combined theories Conflict-directed [Shostak’79] [Nelson, Oppen’79] backjumping CDCL [GRASP’97] [zChaff’04] -
Numerical Algebraic Geometry
JSAG 3 (2011), 5 – 10 The Journal of Software for Algebra and Geometry Numerical Algebraic Geometry ANTON LEYKIN ABSTRACT. Numerical algebraic geometry uses numerical data to describe algebraic varieties. It is based on numerical polynomial homotopy continuation, which is an alternative to the classical symbolic approaches of computational algebraic geometry. We present a package, whose primary purpose is to interlink the existing symbolic methods of Macaulay2 and the powerful engine of numerical approximate computations. The core procedures of the package exhibit performance competitive with the other homotopy continuation software. INTRODUCTION. Numerical algebraic geometry [SVW,SW] is a relatively young subarea of compu- tational algebraic geometry that originated as a blend of the well-understood apparatus of classical algebraic geometry over the field of complex numbers and numerical polynomial homotopy continua- tion methods. Recently steps have been made to extend the reach of numerical algorithms making it possible not only for complex algebraic varieties, but also for schemes, to be represented numerically. What we present here is a description of “stage one” of a comprehensive project that will make the machinery of numerical algebraic geometry available to the wide community of users of Macaulay2 [M2], a dominantly symbolic computer algebra system. Our open-source package [L1, M2] was first released in Macaulay2 distribution version 1.3.1. Stage one has been limited to implementation of algorithms that solve the most basic problem: Given polynomials f1;:::; fn 2 C[x1;:::;xn] that generate a 0-dimensional ideal I = ( f1;:::; fn), find numerical approximations of all points of the underlying n variety V(I) = fx 2 C j f(x) = 0g.