Computational Algebraic Geometry Today

2001 ◽  
pp. 65-119 ◽  
Author(s):  
W. Decker ◽  
F.-O. Schreyer
Keyword(s):  
Algebraic Geometry ◽  
Computational Algebraic Geometry

scholarly journals A Simple Introduction to Gröbner Basis Methods in String Phenomenology

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
James Gray
Keyword(s):  
Algebraic Geometry ◽  
Gröbner Basis ◽  
Grobner Basis ◽  
Simple Description ◽  
Computational Algebraic Geometry ◽  
String Phenomenology ◽  
The Subject ◽  
Electroweak Sector ◽  
String Models ◽  
Vacuum Space

I give an elementary introduction to the key algorithm used in recent applications of computational algebraic geometry to the subject of string phenomenology. I begin with a simple description of the algorithm itself and then give 3 examples of its use in physics. I describe how it can be used to obtain constraints on flux parameters, how it can simplify the equations describing vacua in 4D string models, and lastly how it can be used to compute the vacuum space of the electroweak sector of the MSSM.

Computational algebraic geometry and global analysis of regional manipulators

2014 ◽  
Vol 232 ◽  
pp. 820-835 ◽  
Author(s):  
Teijo Arponen ◽  
Andreas Müller ◽  
Samuli Piipponen ◽  
Jukka Tuomela
Keyword(s):  
Algebraic Geometry ◽  
Global Analysis ◽  
Computational Algebraic Geometry

scholarly journals Computational Algebraic Geometry and Quantum Mechanics: An Initiative toward Post-Contemporary Quantum Chemistry

2019 ◽  
Author(s):  
Ichio Kikuchi ◽  
Akihito Kikuchi
Keyword(s):  
Electronic Structure ◽  
Quantum Mechanics ◽  
Algebraic Geometry ◽  
Quantum Chemistry ◽  
Symbolic Computation ◽  
Algebraic Approach ◽  
Polynomial Systems ◽  
Orbital Theory ◽  
Computational Algebraic Geometry ◽  
Future Direction

A new framework in quantum chemistry has been proposed recently (``An approach to first principles electronic structure calculation by symbolic-numeric computation'' by A. Kikuchi). It is based on the modern technique of computational algebraic geometry, viz. the symbolic computation of polynomial systems. Although this framework belongs to molecular orbital theory, it fully adopts the symbolic method. The analytic integrals in the secular equations are approximated by the polynomials. The indeterminate variables of polynomials represent the wave-functions and other parameters for the optimization, such as atomic positions and contraction coefficients of atomic orbitals. Then the symbolic computation digests and decomposes the polynomials into a tame form of the set of equations, to which numerical computations are easily applied. The key technique is Gr\"obner basis theory, by which one can investigate the electronic structure by unraveling the entangled relations of the involved variables. In this article, at first, we demonstrate the featured result of this new theory. Next, we expound the mathematical basics concerning computational algebraic geometry, which are necessitated in our study. We will see how highly abstract ideas of polynomial algebra would be applied to the solution of the definite problems in quantum mechanics. We solve simple problems in ``quantum chemistry in algebraic variety'' by means of algebraic approach. Finally, we review several topics related to polynomial computation, whereby we shall have an outlook for the future direction of the research.

scholarly journals Computing Euler Obstruction Functions Using Maximum Likelihood Degrees

2020 ◽  
Vol 2020 (20) ◽  
pp. 6699-6712
Author(s):  
Jose Israel Rodriguez ◽  
Botong Wang
Keyword(s):  
Algebraic Geometry ◽  
Maximum Likelihood ◽  
Numerical Algorithm ◽  
Algebraic Statistics ◽  
Computational Algebraic Geometry ◽  
Euler Obstruction

Abstract We give a numerical algorithm computing Euler obstruction functions using maximum likelihood degrees. The maximum likelihood degree is a well-studied property of a variety in algebraic statistics and computational algebraic geometry. In this article we use this degree to give a new way to compute Euler obstruction functions. We define the maximum likelihood obstruction function and show how it coincides with the Euler obstruction function. With this insight, we are able to bring new tools of computational algebraic geometry to study Euler obstruction functions.

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