Introduction to numerical algebraic geometry

@inproceedings{Sommese2005IntroductionTN,
  title={Introduction to numerical algebraic geometry},
  author={Andrew J. Sommese and Jan Verschelde and Charles W. Wampler},
  year={2005}
}
In a 1996 paper, Andrew Sommese and Charles Wampler began developing a new area, “Numerical Algebraic Geometry”, which would bear the same relation to “Algebraic Geometry” that “Numerical Linear Algebra” bears to “Linear Algebra”. 
View via Publisher
people.math.gatech.edu
68 Citations
Highly Influential Citations
1
Background Citations
22
Methods Citations
18

68 Citations

Numerical Algebraic Geometry for Macaulay2
  • A. Leykin
  • Mathematics, Computer Science
    ArXiv
  • 2009
TLDR
A package to interlink the existing symbolic methods of Macaulay2 and the powerful engine of numerical approximate computations to exhibit performance competitive with the other homotopy continuation software.
Computing Tropical Curves via Homotopy Continuation
TLDR
This work exploits a connection between amoebas and tropical curves to devise a method for computing tropical curves using numerical algebraic geometry and gives an implementation that is used to compute Newton polygons of A-polynomials of knots.
Nonlinear algebra in 3 D computer vision
I study computational algebraic geometry—an amusing, somewhat more broad name for this field is nonlinear algebra. This field is firmly anchored to its algorithmic techniques—traditionally, symbolic…
General witness sets for numerical algebraic geometry
TLDR
This work presents a general notion of witness set for subvarieties of a smooth complete complex algebraic variety using ideas from intersection theory, and introduces Schubert witness sets, which provide general witness sets for Grassmannians and flag manifolds.
Numerical algorithms for detecting embedded components
Eliminating dual spaces
Computing Puiseux series for algebraic surfaces
TLDR
The preliminary methods produce exact representations for solution sets of the cyclic n-roots problem, for n = m2, corresponding to a result of Backelin.
Signatures of algebraic curves via numerical algebraic geometry
Galois groups of Schubert problems via homotopy computation
TLDR
This work shows by direct computation that the Galois group of the Schubert problem of 3-planes in ℂ 8 meeting 15 fixed 5-planes non-trivially is the full symmetric group S 6006.
Regularization and Matrix Computation in Numerical Polynomial Algebra
TLDR
Regularization principles for reformulating the ill-posed algebraic problems, derive matrix computations arising in numerical polynomial algebra, as well as subspace strategies that substantially improve computational efficiency by reducing matrix sizes are developed.
...
...

References

SHOWING 1-10 OF 201 REFERENCES
Algebraic Geometry
Introduction to Algebraic Geometry.By Serge Lang. Pp. xi + 260. (Addison–Wesley: Reading, Massachusetts, 1972.)
Algebraic curves
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…
Algebraic curves
TLDR
All singularities are analyzed cOITectly and the methods presented find application in solid modeling and robotics.
Principles of Algebraic Geometry
A comprehensive, self-contained treatment presenting general results of the theory. Establishes a geometric intuition and a working facility with specific geometric practices. Emphasizes applications…
The numerical solution of systems of polynomials - arising in engineering and science
Background: Polynomial Systems Homotopy Continuation Projective Spaces Probability One Polynomials of One Variable Other Methods Isolated Solutions: Coefficient-Parameter Homotopy Polynomial…
A geometric-numeric algorithm for absolute factorization of multivariate polynomials
TLDR
A new semi-numerical algorithmic method for factoring multivariate polynomials absolutely based on algebraic and geometric properties after reduction to the bivariate case in a generic system of coordinates is proposed.
A Convex Geometric Approach to Counting the Roots of a Polynomial System
Numerical Solution of Polynomial Systems by Homotopy Continuation Methods
Semi-numerical determination of irreducible branches of a reduced space curve
TLDR
A semi-numerical algorithm for computing all irreducible branches of a curve in C3 defined by polynomials with rational coefficients is proposed, based on some properties appearing after a generic change of coordinate.
Computing Riemann matrices of algebraic curves
...
...