# bertiniComponentMemberTest -- a main method to test whether points lie on a given variety

## Synopsis

• Usage:
L = bertiniComponentMemberTest (pts, NV)
• Inputs:
• pts, a list, a list of points to test
• NV, , a numerical variety to test if points lie on a given irreducible component
• Optional inputs:
• BertiniInputConfiguration (missing documentation) => ..., default value {},
• IsProjective => ..., default value -1, optional argument to specify whether to use homogeneous coordinates
• Verbose => ..., default value false, Option to silence additional output
• Outputs:
• L, a list, entries are lists of witness sets containing the test point

## Description

This method checks whether the test points pts lie on NV using Bertini by (1) writing the witness set information of NV and the test points to temporary files, (2) invokes Bertini's solver with option TRACKTYPE => 3, (3) stores output of Bertini in temporary file, (4) parses and outputs the solutions.

 i1 : R = CC[x,y,z]; i2 : F = {(y^2+x^2+z^2-1)*x,(y^2+x^2+z^2-1)*y}; i3 : NV = bertiniPosDimSolve(F) o3 = NV o3 : NumericalVariety i4 : pts = {{0,0,0}} --z-axis o4 = {{0, 0, 0}} o4 : List i5 : bertiniComponentMemberTest(pts, NV) o5 = {{(dim=1,deg=1)}} o5 : List

In the current implementation, at most one witness set is listed for each test point although the point may lie on more than one component.

## Ways to use bertiniComponentMemberTest :

• "bertiniComponentMemberTest(List,NumericalVariety)"

## For the programmer

The object bertiniComponentMemberTest is .