# NumericalAlgebraicGeometry -- Numerical Algebraic Geometry

## Description

The package NumericalAlgebraicGeometry, also known as NAG4M2 (Numerical Algebraic Geometry for Macaulay2), implements methods of polynomial homotopy continuation to solve systems of polynomial equations,
 `i1 : R = CC[x,y,z];` `i2 : F = {x^2+y^2+z^2-1, y-x^2, z-x^3};` ```i3 : s = solveSystem F o3 = {{.540536+1.03152*ii, -.771845+1.11514*ii, -1.5675-.193395*ii}, ------------------------------------------------------------------------ {-.737353, .543689, -.400891}, {-.540536-1.03152*ii, ------------------------------------------------------------------------ -.771845+1.11514*ii, 1.5675+.193395*ii}, {-.540536+1.03152*ii, ------------------------------------------------------------------------ -.771845-1.11514*ii, 1.5675-.193395*ii}, {.737353, .543689, .400891}, ------------------------------------------------------------------------ {.540536-1.03152*ii, -.771845-1.11514*ii, -1.5675+.193395*ii}} o3 : List``` ```i4 : realPoints s o4 = {{-.737353, .543689, -.400891}, {.737353, .543689, .400891}} o4 : List```
and describe positive-dimensional complex algebraic varieties,
 `i5 : R = CC[x,y,z];` `i6 : sph = x^2+y^2+z^2-1; ` ```i7 : I = ideal {x*sph*(y-x^2), sph*(z-x^3)}; o7 : Ideal of R``` ```i8 : numericalIrreducibleDecomposition I o8 = a numerical variety with components in dim 1: (dim=1,deg=1) (dim=1,deg=3) dim 2: (dim=2,deg=2) o8 : NumericalVariety```

Basic types (such as Point and WitnessSet) are defined in the package NAGtypes.

### Basic functions:

Optionally, the user may outsource some basic routines to Bertini and PHCpack (look for Software option).

### Functions related to scheme analysis:

• isPointEmbedded -- determine if the point is an embedded component of the scheme
• isPointEmbeddedInCurve -- determine if the point is an embedded component of a 1-dimensional scheme
• colon -- colon of a (truncated) dual space

### References:

• A.J. Sommese, J. Verschelde, and C.W. Wampler, "Introduction to numerical algebraic geometry", in "Solving polynomial equations" (2005), 301--338
• A.J. Sommese and C.W. Wampler, "The numerical solution of systems of polynomials", World Scientific Publishing (2005)
• C. Beltran and A. Leykin, "Certified numerical homotopy tracking", Experimental Mathematics 21(1): 69-83 (2012)
• R. Krone and A. Leykin, "Numerical algorithms for detecting embedded components.", arXiv:1405.7871

## Certification Version 1.4 of this package was accepted for publication in volume 3 of the journal The Journal of Software for Algebra and Geometry: Macaulay2 on 2011-05-20, in the article Numerical Algebraic Geometry. That version can be obtained from the journal or from the Macaulay2 source code repository, svn://svn.macaulay2.com/Macaulay2/trunk/M2/Macaulay2/packages/NumericalAlgebraicGeometry.m2, release number 13254.

## Version

This documentation describes version 1.14 of NumericalAlgebraicGeometry.

## Source code

The source code from which this documentation is derived is in the file NumericalAlgebraicGeometry.m2. The auxiliary files accompanying it are in the directory NumericalAlgebraicGeometry/.