# Bertini -- software for numerical algebraic geometry

## Description

Interfaces the functionality of the software Bertini to solve polynomial systems and perform calculations in numerical algebraic geometry. The software is available at http://bertini.nd.edu/. Bertini is under ongoing development by D. Bates, J. Hauenstein, A. Sommese, and C. Wampler.

The user may place the executable program bertini in the executation path. Alternatively, the path to the executable needs to be specified, for instance,

 i1 : needsPackage("Bertini", Configuration=>{"BERTINIexecutable"=>"/folder/subfolder/bertini"}) o1 = Bertini o1 : Package

Below is a simple example using the most popular function, a basic zero-dimensional solve with no special options.

 i2 : R = CC[x,y] o2 = R o2 : PolynomialRing i3 : F = {x^2-1,y^2-2} 2 2 o3 = {x - 1, y - 2} o3 : List i4 : solns = bertiniZeroDimSolve(F) o4 = {{1, 1.41421}, {1, -1.41421}, {-1, 1.41421}, {-1, -1.41421}} o4 : List

## Version

This documentation describes version 2.1.2.3 of Bertini.

## Source code

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

## Exports

• Types
• B'Section -- a mutable hash table that gives information about a hyperplane used to slice a numerical variety.
• Functions and commands
• Methods
• "bertiniComponentMemberTest(List,NumericalVariety)" -- see bertiniComponentMemberTest -- a main method to test whether points lie on a given variety
• "bertiniParameterHomotopy(List,List,List)" -- see bertiniParameterHomotopy -- a main method to perform a parameter homotopy in Bertini
• "bertiniPosDimSolve(Ideal)" -- see bertiniPosDimSolve -- a main method that is used to produce witness sets
• "bertiniPosDimSolve(List)" -- see bertiniPosDimSolve -- a main method that is used to produce witness sets
• "bertiniRefineSols(ZZ,List,List)" -- see bertiniRefineSols -- sharpen solutions to a prescribed number of digits
• "bertiniSample(ZZ,WitnessSet)" -- see bertiniSample -- a main method to sample points from an irreducible component of a variety
• "bertiniTrackHomotopy(RingElement,List,List)" -- see bertiniTrackHomotopy -- a main method to track using a user-defined homotopy
• "bertiniUserHomotopy(Thing,List,List,List)" -- see bertiniUserHomotopy -- a main method to track a user-defined homotopy
• "bertiniZeroDimSolve(Ideal)" -- see bertiniZeroDimSolve -- a main method to solve a zero-dimensional system of equations
• "bertiniZeroDimSolve(List)" -- see bertiniZeroDimSolve -- a main method to solve a zero-dimensional system of equations
• "importIncidenceMatrix(String)" -- see importIncidenceMatrix -- Imports an incidence matrix file after running makeMembershipFile.
• "importMainDataFile(String)" -- see importMainDataFile -- This function imports points from the main data file form a Bertini run.
• "importParameterFile(String)" -- see importParameterFile -- Imports parameters from a Bertini parameter file.
• "importSolutionsFile(String)" -- see importSolutionsFile -- Imports coordinates from a Bertini solution file.
• "makeB'InputFile(String)" -- see makeB'InputFile -- write a Bertini input file in a directory
• "makeB'Section(List)" -- see makeB'Section -- makeB'Section creates a hash table that represents a hyperplane.
• "makeB'Slice(Thing,List)" -- see makeB'Slice -- makeB'Slice creates a hash table that represents a linear slice.
• "moveB'File(String,String,String)" -- see moveB'File -- Move or copy files.
• "NumberToB'String(Thing)" -- see NumberToB'String -- Translates a number to a string that Bertini can read.
• "radicalList(List)" -- see radicalList -- A support function that removes multiplicities of numbers in a list up to a tolerance.
• "radicalList(List,Number)" -- see radicalList -- A support function that removes multiplicities of numbers in a list up to a tolerance.
• "subPoint(Thing,List,Thing)" -- see subPoint -- This function evaluates a polynomial or matrix at a point.
• "valueBM2(String)" -- see valueBM2 -- This function makes a number in CC from a string.
• "writeStartFile(String,List)" -- see writeStartFile -- Writes the list of list of coordinates to a file that Bertini can read.
• Symbols
• Other things
• storeBM2Files (missing documentation)

## For the programmer

The object Bertini is .