scholarly journals Numerical algebraic geometry

2011 ◽  
Vol 3 (1) ◽  
pp. 5-10 ◽  
Author(s):  
Anton Leykin
Keyword(s):  
Algebraic Geometry ◽  
Numerical Algebraic Geometry

Applying Numerical Algebraic Geometry to Kinematics

2013 ◽  
pp. 125-159 ◽  
Author(s):  
Charles W. Wampler ◽  
Andrew J. Sommese
Keyword(s):  
Algebraic Geometry ◽  
Numerical Algebraic Geometry

scholarly journals Theta Surfaces

2020 ◽  
Author(s):  
Daniele Agostini ◽  
Türkü Özlüm Çelik ◽  
Julia Struwe ◽  
Bernd Sturmfels
Keyword(s):  
Algebraic Geometry ◽  
Theta Functions ◽  
Minkowski Sum ◽  
Zero Set ◽  
Analytic Surface ◽  
Numerical Algebraic Geometry ◽  
Abelian Integrals ◽  
Space Curves ◽  
Quartic Curves ◽  
Special Plane

Abstract A theta surface in affine 3-space is the zero set of a Riemann theta function in genus 3. This includes surfaces arising from special plane quartics that are singular or reducible. Lie and Poincaré showed that any analytic surface that is the Minkowski sum of two space curves in two different ways is a theta surface. The four space curves that generate such a double translation structure are parametrized by abelian integrals, so they are usually not algebraic. This paper offers a new view on this classical topic through the lens of computation. We present practical tools for passing between quartic curves and their theta surfaces, and we develop the numerical algebraic geometry of degenerations of theta functions.

Numerical algebraic geometry and optimization

2018 ◽  
Vol 65 (10) ◽  
pp. 1
Author(s):  
Jonathan D. Hauenstein
Keyword(s):  
Algebraic Geometry ◽  
Numerical Algebraic Geometry

scholarly journals Identifiability and numerical algebraic geometry

PLoS ONE ◽  
2019 ◽  
Vol 14 (12) ◽  
pp. e0226299
Author(s):  
Daniel J. Bates ◽  
Jonathan D. Hauenstein ◽  
Nicolette Meshkat
Keyword(s):  
Algebraic Geometry ◽  
Numerical Algebraic Geometry

Numerical irreducible decomposition over a number field

2018 ◽  
Vol 17 (10) ◽  
pp. 1850195
Author(s):  
Timothy M. McCoy ◽  
Chris Peterson ◽  
Andrew J. Sommese
Keyword(s):  
Algebraic Geometry ◽  
Number Field ◽  
Polynomial System ◽  
Field Extension ◽  
General Point ◽  
Numerical Algebraic Geometry ◽  
Irreducible Decomposition ◽  
Algebraic Set ◽  
Field Of Definition ◽  
Witness Set

Let [Formula: see text] be a set of elements in the polynomial ring [Formula: see text], let [Formula: see text] denote the ideal generated by the elements of [Formula: see text], and let [Formula: see text] denote the radical of [Formula: see text]. There is a unique decomposition [Formula: see text] with each [Formula: see text] a prime ideal corresponding to a minimal associated prime of [Formula: see text] over [Formula: see text]. Let [Formula: see text] denote the reduced algebraic set corresponding to the common zeroes of the elements of [Formula: see text]. Techniques from numerical algebraic geometry can be used to determine the numerical irreducible decomposition of [Formula: see text] over [Formula: see text]. This corresponds to producing a witness set for [Formula: see text] for each [Formula: see text] together with the degree and dimension of [Formula: see text] (a point in a witness set for [Formula: see text] can be considered as a numerical approximation for a general point on [Formula: see text]). The purpose of this paper is to show how to extend these results taking into account the field of definition for the polynomial system. In particular, let [Formula: see text] be a number field (i.e. a finite field extension of [Formula: see text]) and let [Formula: see text] be a set of elements in [Formula: see text]. We show how to extend techniques from numerical algebraic geometry to determine the numerical irreducible decomposition of [Formula: see text] over [Formula: see text].

scholarly journals What is numerical algebraic geometry?

2017 ◽  
Vol 79 ◽  
pp. 499-507 ◽  
Author(s):  
Jonathan D. Hauenstein ◽  
Andrew J. Sommese
Keyword(s):  
Algebraic Geometry ◽  
Numerical Algebraic Geometry

scholarly journals Numerical algebraic geometry: a new perspective on gauge and string theories

2012 ◽  
Vol 2012 (7) ◽  
Author(s):  
Dhagash Mehta ◽  
Yang-Hui He ◽  
Jonathan D. Hauenstein
Keyword(s):  
Algebraic Geometry ◽  
Numerical Algebraic Geometry ◽  
New Perspective ◽  
String Theories
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