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 <title>Bertini -- software for numerical algebraic geometry</title>
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o1 = Bertini

o1 : Package</code></pre>
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 Below is a simple example using the most popular function, a basic zero-dimensional solve with no special options.
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 <pre><code class="language-macaulay2">i2 : R = CC[x,y]

o2 = R

o2 : PolynomialRing</code></pre>
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 <pre><code class="language-macaulay2">i3 : F = {x^2-1,y^2-2}

2       2
o3 = {x  - 1, y  - 2}

o3 : List</code></pre>
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 <pre><code class="language-macaulay2">i4 : solns = bertiniZeroDimSolve(F)

o4 = {{1, 1.41421}, {1, -1.41421}, {-1, 1.41421}, {-1, -1.41421}}

o4 : List</code></pre>
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 <h2>Authors</h2>
 <ul>
 <li><a href="http://math.sjsu.edu/~egross">Elizabeth Gross</a> &lt;<a href="mailto:elizabeth.gross%40sjsu.edu">elizabeth.gross@sjsu.edu</a>></li>
 <li><a href="https://www.math.wisc.edu/~jose/">Jose Israel Rodriguez</a> &lt;<a href="mailto:Jose%40math.wisc.edu">Jose@math.wisc.edu</a>></li>
 <li><a href="http://www.math.colostate.edu/~bates">Dan Bates</a> &lt;<a href="mailto:bates%40math.colostate.edu">bates@math.colostate.edu</a>></li>
 <li><a href="http://www.math.gatech.edu/~leykin">Anton Leykin</a> &lt;<a href="mailto:leykin%40math.gatech.edu">leykin@math.gatech.edu</a>></li>
 </ul>
 </div>
 <div>
 <h2>Version</h2>
 This documentation describes version 2.1.2.3 of Bertini, released Feb 6 2024.
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 <div>
 <h2>Citation</h2>
 If you have used this package in your research, please cite it as follows:
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 <pre><code class="language-bib">@misc{BertiniSource,
 title = {{Bertini: interface to Bertini. Version~2.1.2.3}},
 author = {Elizabeth Gross and Jose Israel Rodriguez and Dan Bates and Anton Leykin},
 howpublished = {A \emph{Macaulay2} package available at
 \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}}
}
</code></pre>
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 <div>
 <h2>Exports</h2>
 <div class="exports">
 <ul>
 <li>Types <ul>
 <li><a title="a mutable hash table that gives information about a hyperplane used to slice a numerical variety." href="___B_sq__Section.html">B'Section</a> -- a mutable hash table that gives information about a hyperplane used to slice a numerical variety.</li>
 <li><a title="a mutable hash table that gives information about a linear space used to slice a numerical variety." href="___B_sq__Slice.html">B'Slice</a> -- a mutable hash table that gives information about a linear space used to slice a numerical variety.</li>
 </ul>
 </li>
 <li>Functions and commands <ul>
 <li><a title="a main method to test whether points lie on a given variety" href="_bertini__Component__Member__Test.html">bertiniComponentMemberTest</a> -- a main method to test whether points lie on a given variety</li>
 <li><a title="a main method to perform a parameter homotopy in Bertini" href="_bertini__Parameter__Homotopy.html">bertiniParameterHomotopy</a> -- a main method to perform a parameter homotopy in Bertini</li>
 <li><a title="a main method that is used to produce witness sets" href="_bertini__Pos__Dim__Solve.html">bertiniPosDimSolve</a> -- a main method that is used to produce witness sets</li>
 <li><a title="sharpen solutions to a prescribed number of digits" href="_bertini__Refine__Sols.html">bertiniRefineSols</a> -- sharpen solutions to a prescribed number of digits</li>
 <li><a title="a main method to sample points from an irreducible component of a variety" href="_bertini__Sample.html">bertiniSample</a> -- a main method to sample points from an irreducible component of a variety</li>
 <li><a title="a main method to track using a user-defined homotopy" href="_bertini__Track__Homotopy.html">bertiniTrackHomotopy</a> -- a main method to track using a user-defined homotopy</li>
 <li><a title="a main method to track a user-defined homotopy" href="_bertini__User__Homotopy.html">bertiniUserHomotopy</a> -- a main method to track a user-defined homotopy</li>
 <li><a title="a main method to solve a zero-dimensional system of equations" href="_bertini__Zero__Dim__Solve.html">bertiniZeroDimSolve</a> -- a main method to solve a zero-dimensional system of equations</li>
 <li><a title="Imports an incidence matrix file after running makeMembershipFile." href="_import__Incidence__Matrix.html">importIncidenceMatrix</a> -- Imports an incidence matrix file after running makeMembershipFile.</li>
 <li><a title="This function imports points from the main data file form a Bertini run." href="_import__Main__Data__File.html">importMainDataFile</a> -- This function imports points from the main data file form a Bertini run.</li>
 <li><a title="Imports parameters from a Bertini parameter file." href="_import__Parameter__File.html">importParameterFile</a> -- Imports parameters from a Bertini parameter file.</li>
 <li><a title="Imports coordinates from a Bertini solution file." href="_import__Solutions__File.html">importSolutionsFile</a> -- Imports coordinates from a Bertini solution file.</li>
 <li><a title="write a Bertini input file in a directory" href="_make__B_sq__Input__File.html">makeB'InputFile</a> -- write a Bertini input file in a directory</li>
 <li><a title="makeB'Section creates a hash table that represents a hyperplane." href="_make__B_sq__Section.html">makeB'Section</a> -- makeB'Section creates a hash table that represents a hyperplane.</li>
 <li><a title="makeB'Slice creates a hash table that represents a linear slice." href="_make__B_sq__Slice.html">makeB'Slice</a> -- makeB'Slice creates a hash table that represents a linear slice.</li>
 <li><a title="Move or copy files." href="_move__B_sq__File.html">moveB'File</a> -- Move or copy files.</li>
 <li><a title="Translates a number to a string that Bertini can read." href="___Number__To__B_sq__String.html">NumberToB'String</a> -- Translates a number to a string that Bertini can read.</li>
 <li><a title="A support function that removes multiplicities of numbers in a list up to a tolerance." href="_radical__List.html">radicalList</a> -- A support function that removes multiplicities of numbers in a list up to a tolerance.</li>
 <li><a title="This function evaluates a polynomial or matrix at a point." href="_sub__Point.html">subPoint</a> -- This function evaluates a polynomial or matrix at a point.</li>
 <li><a title="This function makes a number in CC from a string." href="_value__B__M2.html">valueBM2</a> -- This function makes a number in CC from a string.</li>
 <li><a title="Writes the list of list of coordinates to a file that Bertini can read." href="_write__Start__File.html">writeStartFile</a> -- Writes the list of list of coordinates to a file that Bertini can read.</li>
 </ul>
 </li>
 <li>Methods <ul>
 <li><kbd>bertiniComponentMemberTest(List,NumericalVariety)</kbd> -- see <a title="a main method to test whether points lie on a given variety" href="_bertini__Component__Member__Test.html">bertiniComponentMemberTest</a> -- a main method to test whether points lie on a given variety</li>
 <li><kbd>bertiniParameterHomotopy(List,List,List)</kbd> -- see <a title="a main method to perform a parameter homotopy in Bertini" href="_bertini__Parameter__Homotopy.html">bertiniParameterHomotopy</a> -- a main method to perform a parameter homotopy in Bertini</li>
 <li><kbd>bertiniPosDimSolve(Ideal)</kbd> -- see <a title="a main method that is used to produce witness sets" href="_bertini__Pos__Dim__Solve.html">bertiniPosDimSolve</a> -- a main method that is used to produce witness sets</li>
 <li><kbd>bertiniPosDimSolve(List)</kbd> -- see <a title="a main method that is used to produce witness sets" href="_bertini__Pos__Dim__Solve.html">bertiniPosDimSolve</a> -- a main method that is used to produce witness sets</li>
 <li><kbd>bertiniRefineSols(String,ZZ,List,String)</kbd> -- see <a title="sharpen solutions to a prescribed number of digits" href="_bertini__Refine__Sols.html">bertiniRefineSols</a> -- sharpen solutions to a prescribed number of digits</li>
 <li><kbd>bertiniRefineSols(ZZ,List)</kbd> -- see <a title="sharpen solutions to a prescribed number of digits" href="_bertini__Refine__Sols.html">bertiniRefineSols</a> -- sharpen solutions to a prescribed number of digits</li>
 <li><kbd>bertiniRefineSols(ZZ,List,String)</kbd> -- see <a title="sharpen solutions to a prescribed number of digits" href="_bertini__Refine__Sols.html">bertiniRefineSols</a> -- sharpen solutions to a prescribed number of digits</li>
 <li><kbd>bertiniSample(ZZ,WitnessSet)</kbd> -- see <a title="a main method to sample points from an irreducible component of a variety" href="_bertini__Sample.html">bertiniSample</a> -- a main method to sample points from an irreducible component of a variety</li>
 <li><kbd>bertiniTrackHomotopy(RingElement,List,List)</kbd> -- see <a title="a main method to track using a user-defined homotopy" href="_bertini__Track__Homotopy.html">bertiniTrackHomotopy</a> -- a main method to track using a user-defined homotopy</li>
 <li><kbd>bertiniUserHomotopy(Thing,List,List,List)</kbd> -- see <a title="a main method to track a user-defined homotopy" href="_bertini__User__Homotopy.html">bertiniUserHomotopy</a> -- a main method to track a user-defined homotopy</li>
 <li><kbd>bertiniZeroDimSolve(Ideal)</kbd> -- see <a title="a main method to solve a zero-dimensional system of equations" href="_bertini__Zero__Dim__Solve.html">bertiniZeroDimSolve</a> -- a main method to solve a zero-dimensional system of equations</li>
 <li><kbd>bertiniZeroDimSolve(List)</kbd> -- see <a title="a main method to solve a zero-dimensional system of equations" href="_bertini__Zero__Dim__Solve.html">bertiniZeroDimSolve</a> -- a main method to solve a zero-dimensional system of equations</li>
 <li><kbd>importIncidenceMatrix(String)</kbd> -- see <a title="Imports an incidence matrix file after running makeMembershipFile." href="_import__Incidence__Matrix.html">importIncidenceMatrix</a> -- Imports an incidence matrix file after running makeMembershipFile.</li>
 <li><kbd>importMainDataFile(String)</kbd> -- see <a title="This function imports points from the main data file form a Bertini run." href="_import__Main__Data__File.html">importMainDataFile</a> -- This function imports points from the main data file form a Bertini run.</li>
 <li><kbd>importParameterFile(String)</kbd> -- see <a title="Imports parameters from a Bertini parameter file." href="_import__Parameter__File.html">importParameterFile</a> -- Imports parameters from a Bertini parameter file.</li>
 <li><kbd>importSolutionsFile(String)</kbd> -- see <a title="Imports coordinates from a Bertini solution file." href="_import__Solutions__File.html">importSolutionsFile</a> -- Imports coordinates from a Bertini solution file.</li>
 <li><kbd>makeB'InputFile(String)</kbd> -- see <a title="write a Bertini input file in a directory" href="_make__B_sq__Input__File.html">makeB'InputFile</a> -- write a Bertini input file in a directory</li>
 <li><kbd>makeB'Section(List)</kbd> -- see <a title="makeB'Section creates a hash table that represents a hyperplane." href="_make__B_sq__Section.html">makeB'Section</a> -- makeB'Section creates a hash table that represents a hyperplane.</li>
 <li><kbd>makeB'Slice(Thing,List)</kbd> -- see <a title="makeB'Slice creates a hash table that represents a linear slice." href="_make__B_sq__Slice.html">makeB'Slice</a> -- makeB'Slice creates a hash table that represents a linear slice.</li>
 <li>makeMembershipFile(String) (missing documentation)
</li>
 <li>makeSampleSolutionsFile(String,Number) (missing documentation)
</li>
 <li><kbd>moveB'File(String,String,String)</kbd> -- see <a title="Move or copy files." href="_move__B_sq__File.html">moveB'File</a> -- Move or copy files.</li>
 <li><kbd>NumberToB'String(Thing)</kbd> -- see <a title="Translates a number to a string that Bertini can read." href="___Number__To__B_sq__String.html">NumberToB'String</a> -- Translates a number to a string that Bertini can read.</li>
 <li><kbd>radicalList(List)</kbd> -- see <a title="A support function that removes multiplicities of numbers in a list up to a tolerance." href="_radical__List.html">radicalList</a> -- A support function that removes multiplicities of numbers in a list up to a tolerance.</li>
 <li><kbd>radicalList(List,Number)</kbd> -- see <a title="A support function that removes multiplicities of numbers in a list up to a tolerance." href="_radical__List.html">radicalList</a> -- A support function that removes multiplicities of numbers in a list up to a tolerance.</li>
 <li>readFile(String) (missing documentation)
</li>
 <li>readFile(String,Number) (missing documentation)
</li>
 <li>readFile(String,String,Number) (missing documentation)
</li>
 <li>runBertini(String) (missing documentation)
</li>
 <li>sortMainDataComponents(List) (missing documentation)
</li>
 <li><kbd>subPoint(Thing,List,Thing)</kbd> -- see <a title="This function evaluates a polynomial or matrix at a point." href="_sub__Point.html">subPoint</a> -- This function evaluates a polynomial or matrix at a point.</li>
 <li><kbd>valueBM2(String)</kbd> -- see <a title="This function makes a number in CC from a string." href="_value__B__M2.html">valueBM2</a> -- This function makes a number in CC from a string.</li>
 <li>writeParameterFile(String,List) (missing documentation)
</li>
 <li><kbd>writeStartFile(String,List)</kbd> -- see <a title="Writes the list of list of coordinates to a file that Bertini can read." href="_write__Start__File.html">writeStartFile</a> -- Writes the list of list of coordinates to a file that Bertini can read.</li>
 </ul>
 </li>
 <li>Symbols <ul>
 <li>AdditionalFiles (missing documentation)
</li>
 <li><a title="an option to designate the constants for a Bertini Input file" href="___B_sq__Constants.html">B'Constants</a> -- an option to designate the constants for a Bertini Input file</li>
 <li><kbd>CoeffBound</kbd> -- see <a title="a configuration option for a Bertini input file" href="___Bertini_spinput_spconfiguration.html">Bertini input configuration</a> -- a configuration option for a Bertini input file</li>
 <li><kbd>CondNumThreshold</kbd> -- see <a title="a configuration option for a Bertini input file" href="___Bertini_spinput_spconfiguration.html">Bertini input configuration</a> -- a configuration option for a Bertini input file</li>
 <li><kbd>DegreeBound</kbd> -- see <a title="a configuration option for a Bertini input file" href="___Bertini_spinput_spconfiguration.html">Bertini input configuration</a> -- a configuration option for a Bertini input file</li>
 <li><kbd>EndGameNum</kbd> -- see <a title="a configuration option for a Bertini input file" href="___Bertini_spinput_spconfiguration.html">Bertini input configuration</a> -- a configuration option for a Bertini input file</li>
 <li><kbd>FinalTol</kbd> -- see <a title="a configuration option for a Bertini input file" href="___Bertini_spinput_spconfiguration.html">Bertini input configuration</a> -- a configuration option for a Bertini input file</li>
 <li><kbd>ImagThreshold</kbd> -- see <a title="a configuration option for a Bertini input file" href="___Bertini_spinput_spconfiguration.html">Bertini input configuration</a> -- a configuration option for a Bertini input file</li>
 <li><kbd>MaxCycleNum</kbd> -- see <a title="a configuration option for a Bertini input file" href="___Bertini_spinput_spconfiguration.html">Bertini input configuration</a> -- a configuration option for a Bertini input file</li>
 <li><kbd>MaxNewtonIts</kbd> -- see <a title="a configuration option for a Bertini input file" href="___Bertini_spinput_spconfiguration.html">Bertini input configuration</a> -- a configuration option for a Bertini input file</li>
 <li><kbd>MaxNorm</kbd> -- see <a title="a configuration option for a Bertini input file" href="___Bertini_spinput_spconfiguration.html">Bertini input configuration</a> -- a configuration option for a Bertini input file</li>
 <li><kbd>MaxNumberSteps</kbd> -- see <a title="a configuration option for a Bertini input file" href="___Bertini_spinput_spconfiguration.html">Bertini input configuration</a> -- a configuration option for a Bertini input file</li>
 <li><kbd>MaxStepSize</kbd> -- see <a title="a configuration option for a Bertini input file" href="___Bertini_spinput_spconfiguration.html">Bertini input configuration</a> -- a configuration option for a Bertini input file</li>
 <li><kbd>MinStepSizeBeforeEG</kbd> -- see <a title="a configuration option for a Bertini input file" href="___Bertini_spinput_spconfiguration.html">Bertini input configuration</a> -- a configuration option for a Bertini input file</li>
 <li><kbd>MinStepSizeDuringEG</kbd> -- see <a title="a configuration option for a Bertini input file" href="___Bertini_spinput_spconfiguration.html">Bertini input configuration</a> -- a configuration option for a Bertini input file</li>
 <li><kbd>MPType</kbd> -- see <a title="a configuration option for a Bertini input file" href="___Bertini_spinput_spconfiguration.html">Bertini input configuration</a> -- a configuration option for a Bertini input file</li>
 <li><kbd>ODEPredictor</kbd> -- see <a title="a configuration option for a Bertini input file" href="___Bertini_spinput_spconfiguration.html">Bertini input configuration</a> -- a configuration option for a Bertini input file</li>
 <li><kbd>OutputLevel</kbd> -- see <a title="a configuration option for a Bertini input file" href="___Bertini_spinput_spconfiguration.html">Bertini input configuration</a> -- a configuration option for a Bertini input file</li>
 <li><kbd>PRECISION</kbd> -- see <a title="a configuration option for a Bertini input file" href="___Bertini_spinput_spconfiguration.html">Bertini input configuration</a> -- a configuration option for a Bertini input file</li>
 <li><kbd>RandomSeed</kbd> -- see <a title="a configuration option for a Bertini input file" href="___Bertini_spinput_spconfiguration.html">Bertini input configuration</a> -- a configuration option for a Bertini input file</li>
 <li><kbd>RegenStartLevel</kbd> -- see <a title="a configuration option for a Bertini input file" href="___Bertini_spinput_spconfiguration.html">Bertini input configuration</a> -- a configuration option for a Bertini input file</li>
 <li><kbd>ScreenOut</kbd> -- see <a title="a configuration option for a Bertini input file" href="___Bertini_spinput_spconfiguration.html">Bertini input configuration</a> -- a configuration option for a Bertini input file</li>
 <li><kbd>SecurityLevel</kbd> -- see <a title="a configuration option for a Bertini input file" href="___Bertini_spinput_spconfiguration.html">Bertini input configuration</a> -- a configuration option for a Bertini input file</li>
 <li><kbd>SingValZeroTol</kbd> -- see <a title="a configuration option for a Bertini input file" href="___Bertini_spinput_spconfiguration.html">Bertini input configuration</a> -- a configuration option for a Bertini input file</li>
 <li><kbd>StepsForIncrease</kbd> -- see <a title="a configuration option for a Bertini input file" href="___Bertini_spinput_spconfiguration.html">Bertini input configuration</a> -- a configuration option for a Bertini input file</li>
 <li><kbd>TrackTolBeforeEG</kbd> -- see <a title="a configuration option for a Bertini input file" href="___Bertini_spinput_spconfiguration.html">Bertini input configuration</a> -- a configuration option for a Bertini input file</li>
 <li><kbd>TrackTolDuringEG</kbd> -- see <a title="a configuration option for a Bertini input file" href="___Bertini_spinput_spconfiguration.html">Bertini input configuration</a> -- a configuration option for a Bertini input file</li>
 <li><kbd>RandomComplex</kbd> -- see <a title="an option which designates symbols/strings/variables that will be set to be a random real number or random complex number" href="___Bertini_spinput_spfile_spdeclarations_co_sprandom_spnumbers.html">Bertini input file declarations: random numbers</a> -- an option which designates symbols/strings/variables that will be set to be a random real number or random complex number</li>
 <li><kbd>RandomReal</kbd> -- see <a title="an option which designates symbols/strings/variables that will be set to be a random real number or random complex number" href="___Bertini_spinput_spfile_spdeclarations_co_sprandom_spnumbers.html">Bertini input file declarations: random numbers</a> -- an option which designates symbols/strings/variables that will be set to be a random real number or random complex number</li>
 <li><a title="an optional argument to specify whether make a copy of the file." href="___Copy__B_sq__File.html">CopyB'File</a> -- an optional argument to specify whether make a copy of the file.</li>
 <li><a title="optional argument to specify whether to use homogeneous coordinates" href="___Is__Projective.html">IsProjective</a> -- optional argument to specify whether to use homogeneous coordinates</li>
 <li>MainDataDirectory (missing documentation)
</li>
 <li><kbd>NameB'Slice</kbd> -- see <a title="makeB'Slice creates a hash table that represents a linear slice." href="_make__B_sq__Slice.html">makeB'Slice</a> -- makeB'Slice creates a hash table that represents a linear slice.</li>
 <li>PathList (missing documentation)
</li>
 <li><a title="Option to change directory for file storage." href="___Top__Directory.html">TopDirectory</a> -- Option to change directory for file storage.</li>
 <li><a title="an option specifying when to use regeneration" href="___Use__Regeneration.html">UseRegeneration</a> -- an option specifying when to use regeneration</li>
 <li><kbd>AffVariableGroup</kbd> -- see <a title="an option to group variables and use multihomogeneous homotopies" href="___Variable_spgroups.html">Variable groups</a> -- an option to group variables and use multihomogeneous homotopies</li>
 <li><kbd>HomVariableGroup</kbd> -- see <a title="an option to group variables and use multihomogeneous homotopies" href="___Variable_spgroups.html">Variable groups</a> -- an option to group variables and use multihomogeneous homotopies</li>
 </ul>
 </li>
 <li>Mutable symbols <ul>
 <li>storeBM2Files (missing documentation)
</li>
 </ul>
 </li>
 <li>Other things <ul>
 <li>storeBM2Files (missing documentation)
</li>
 </ul>
 </li>
 </ul>
 </div>
 </div>
 </div>
 <div class="waystouse">
 <h2>For the programmer</h2>
 The object <a title="software for numerical algebraic geometry" href="index.html">Bertini</a> is a <a title="the class of all packages" href="../../Macaulay2Doc/html/___Package.html">package</a>, defined in Bertini.m2, with auxiliary files in Bertini/.
 </div>
 <hr>
 <div class="waystouse">
 The source of this document is in Bertini.m2:2624:0.
 </div>
 </div>
 </div>
 </body>

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