What is numerical algebraic geometry?

Hauenstein, Jonathan D.; Sommese, Andrew J.

doi:10.1016/j.jsc.2016.07.015

Cited by 17 publications

(6 citation statements)

References 78 publications

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“…The recovery problem also makes sense for g = 3, both classically [4] and tropically [1, §7].This paper is organized as follows. In Section 2 we tackle the classical Schottky problem as a task in numerical algebraic geometry [6,14,24]. For g = 4, we utilize the software abelfunctions [25] to test whether the Schottky-Igusa modular form vanishes.…”

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confidence: 99%

Schottky algorithms: Classical meets tropical

2018

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We present a new perspective on the Schottky problem that links numerical computing with tropical geometry. The task is to decide whether a symmetric matrix defines a Jacobian, and, if so, to compute the curve and its canonical embedding. We offer solutions and their implementations in genus four, both classically and tropically. The locus of cographic matroids arises from tropicalizing the Schottky-Igusa modular form.arXiv:1707.08520v2 [math.AG] 30 Sep 2018 the latter, τ already passed that test, and we compute a curve whose Jacobian is given by τ . The recovery problem also makes sense for g = 3, both classically [4] and tropically [1, §7].This paper is organized as follows. In Section 2 we tackle the classical Schottky problem as a task in numerical algebraic geometry [6,14,24]. For g = 4, we utilize the software abelfunctions [25] to test whether the Schottky-Igusa modular form vanishes. In the affirmative case, we use a numerical version of Kempf's method [17] to compute a canonical embedding into P 3 . Our main results in Section 3 are Algorithms 3.3 and 3.5. Based on the work in [7,8,27,28], these furnish a computational solution to the tropical Schottky problem. Key ingredients are cographic matroids and the f-vectors of Voronoi polytopes.Section 4 links the classical and tropical Schottky scenarios. Theorem 4.2 expresses the edge lengths of a metric graph in terms of tropical theta constants, and Theorem 4.9 explains what happens to the Schottky-Igusa modular form in the tropical limit. We found it especially gratifying to discover how the cographic locus is encoded in the classical theory.The software we describe in this paper is made available at the supplementary website http://eecs.berkeley.edu/~chualynn/schottky (2) This contains several pieces of code for the tropical Schottky problem, as well as a more coherent Sage program for the classical Schottky problem that makes calls to abelfunctions.

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confidence: 99%

Schottky algorithms: Classical meets tropical

2018

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“…Numerical algebraic geometry [35,13] was pioneered by Sommese, Wampler, Verschelde and others (see [2,34] for references and background). They first considered the more accessible characterization of complex solution components of each possible dimension, by slicing the solution set with appropriate random planes, that intersected the solution components in complex points called witness points.…”

Section: Real Algebraic Geometrymentioning

confidence: 99%

Implicit Method for Degenerated Differential-Algebraic Equations and Applications

Yang¹,

Wu²,

Reid³

2021

Preprint

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Systems of differential-algebraic equations (daes) are routinely automatically produced by modeling enviroments such as Maplesim, System Modeler and Modelica. Structural methods are important for reducing the index and obtaining hidden constraints of such daes. This is especially the case for high index non-linear daes. Although such structural analysis is often successful for many dynamic systems, it may fail if the resulting Jacobian is still singular due to symbolic cancellation or numerical degeneration. Existing modified structural methods can handle some cases caused by symbolic cancellation, where assumes the determinant of a Jacobian matrix is identically zero. This paper removes such assumptions and provides numerical methods to analyze such degenerated cases using real algebraic geometry for polynomially nonlinear daes. Firstly, we provide a witness point method, which produces witness points on all components and can help to detect degeneration on all components of polynomially daes. Secondly, we propose an implicit index reduction (IIR) method which can restore a full rank Jacobian matrix for degenerated dae. Thirdly, based on IIR, we introduce an improved structural method, which can numerically solve degenerated daes on all components. Examples are given to illustrate our methods and show their advantages for degenerated daes.

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“…In this section we review the basics of homotopy continuation for polynomial systems, which plays a fundamental role in the subject of numerical algebraic geometry and is central to our approach for determining ε-local rigidity. For general references see any of [5,18,33]. The main idea is simple.…”

Section: Polynomial Homotopy Continuationmentioning

confidence: 99%

Epsilon local rigidity and numerical algebraic geometry

Frohmader,

Heaton

2020

Preprint

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A well-known combinatorial algorithm can decide generic rigidity in the plane by determining if the graph is of Pollaczek-Geiringer-Laman type. Methods from matroid theory have been used to prove other interesting results, again under the assumption of generic configurations. However, configurations arising in applications may not be generic. We present Theorem 7 and its corresponding Algorithm 1 which decide if a configuration is ε-locally rigid, a notion we define. This provides a partial answer to a problem discussed in the 2011 paper of Hauenstein, Sommese, and Wampler. The theorem and algorithm use results from a 2012 paper of Hauenstein. We also present Algorithm 2 which uses numerical algebraic geometry to find nearby valid configurations which are not obtained by rigid motions. When successful, this method demonstrates the failure of local rigidity by explicitly constructing a sequence of configurations which are a discrete-time sample of a continuous flex.

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What is numerical algebraic geometry?

Cited by 17 publications

References 78 publications

Schottky algorithms: Classical meets tropical

Schottky algorithms: Classical meets tropical

Implicit Method for Degenerated Differential-Algebraic Equations and Applications

Epsilon local rigidity and numerical algebraic geometry

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