The function WitnessPoints returns at least one point in each connected component of the real solutions of sys.

The system sys is either:

a list of polynomial equations with rational coefficients

or, a list of polynomial inequations of the form $p\ne q$ and strict inequalities of the form $p\>q$ with rational coefficients

The output is a list of points with rational coordinates.

A point is represented by a list of equations of the form $\mathrm{variable}=\mathrm{rational}$.

When sys is a list of equations, then each point of the output is a rational approximation of a solution of sys, the precision of which is controlled  by the option digits=d (default value: Digits). More precisely, if $a$ is a point returned by the command WitnessPoints, then there exists a solution $x$ of sys such that:

When the list of names vars is provided, it must contain all the indeterminates appearing in sys. In this case, the coordinates of the points returned by WitnessPoints are ordered according to vars.

Several points can belong to the same connected component.

For some inputs, repeated execution of the command WitnessPoints may return different outputs due to the use of randomization in the implementation. A way to keep the function deterministic is to use the command randomize with a fixed seed (see Examples below).

$\mathrm{with}\left(\mathrm{RootFinding}\right)\:$

$\mathrm{WitnessPoints}\left(\left[x^2+y+3=0\right]\right)$

$\left[\left[x=0\,y=\mathrm{-3}\right]\right]$

$\mathrm{WitnessPoints}\left(\left[{\left(x^2+{\left(y-1\right)}^2\right)}^2=0\right]\right)$

$\left[\left[x=0\,y=1\right]\right]$

$\mathrm{WitnessPoints}\left(\left[x^2+y^2\,z-x^3=0\right]\right)$

$\left[\left[x=0\,y=0\,z=0\right]\right]$

$\mathrm{evalf}\left(\mathrm{WitnessPoints}\left(\left[0<x^2+y+3\&comma;3y^3-x^5<10\right]\right)\right)$

$\left[\left[x=\mathrm{-3.}\&comma;y=\mathrm{-11.}\right]\&comma;\left[x=0.\&comma;y=\mathrm{-2.}\right]\right]$

$\mathrm{WitnessPoints}\left(\left[x^2+y^2-1\ne 0\&comma;x^2<1\right]\&comma;\left[x\&comma;y\right]\right)$

$\left[\left[x=0\&comma;y=\mathrm{-2}\right]\&comma;\left[x=-\frac{1423656585}{1525870529}\&comma;y=-\frac{1}{2}\right]\&comma;\left[x=0\&comma;y=-\frac{1}{2}\right]\&comma;\left[x=\frac{1423656585}{1525870529}\&comma;y=-\frac{1}{2}\right]\&comma;\left[x=-\frac{1423656585}{1525870529}\&comma;y=\frac{1}{2}\right]\&comma;\left[x=0\&comma;y=\frac{1}{2}\right]\&comma;\left[x=\frac{1423656585}{1525870529}\&comma;y=\frac{1}{2}\right]\&comma;\left[x=0\&comma;y=2\right]\right]$

$\mathrm{WitnessPoints}\left(\left[x^2+y^2-1\ne 0\&comma;x^2<-1\right]\&comma;\left[x\&comma;y\right]\right)$

$\mathrm{sols20}≔\mathrm{WitnessPoints}\left(\left[{\left(x-3\right)}^2+y^4-y=0\right]\&comma;\mathrm{digits}=20\right)\&colon;$

$\mathrm{sols30}≔\mathrm{WitnessPoints}\left(\left[{\left(x-3\right)}^2+y^4-y=0\right]\&comma;\mathrm{digits}=30\right)\&colon;$

$\mathrm{sols50}≔\mathrm{WitnessPoints}\left(\left[{\left(x-3\right)}^2+y^4-y=0\right]\&comma;\mathrm{digits}=50\right)\&colon;$

$\mathrm{A20}≔\mathrm{eval}\left(⟨x\&comma;y⟩\&comma;\mathrm{sols20}\left[1\right]\right)$

$\mathrm{A20}≔\left[\begin{array}{c}\frac{22366435058122870896193141}{9671406556917033397649408}\\ \frac{24370417406302138235346345}{38685626227668133590597632}\end{array}\right]$

$\mathrm{A30}≔\mathrm{eval}\left(⟨x\&comma;y⟩\&comma;\mathrm{sols30}\left[1\right]\right)$

$\mathrm{A30}≔\left[\begin{array}{c}\frac{96063107102745589648779753320559563}{41538374868278621028243970633760768}\\ \frac{104670145749936828015683713499914833}{166153499473114484112975882535043072}\end{array}\right]$

$X≔\mathrm{eval}\left(⟨x\&comma;y⟩\&comma;\mathrm{sols50}\left[1\right]\right)$

$X≔\left[\begin{array}{c}\frac{28352824826395170252212036438729579929395810572048135085}{12259964326927110866866776217202473468949912977468817408}\\ \frac{30893174252911395094184913620218023688036361519846797823}{49039857307708443467467104868809893875799651909875269632}\end{array}\right]$

$\mathrm{with}\left(\mathrm{LinearAlgebra}\right)\&colon;$

$\frac{\mathrm{Norm}\left(\mathrm{A20}-X\&comma;1.\right)}{\mathrm{Norm}\left(\mathrm{A20}\&comma;1.\right)}$

$3.639830510\times {10}^{\mathrm{-26}}$

$\frac{\mathrm{Norm}\left(\mathrm{A30}-X\&comma;1.\right)}{\mathrm{Norm}\left(\mathrm{A30}\&comma;1.\right)}$

$5.423126944\times {10}^{\mathrm{-36}}$

$\mathrm{WitnessPoints}\left(\left[yz\&comma;xy-z^2\right]\&comma;\left[x\&comma;y\&comma;z\right]\right)$

$\left[\left[x=0\&comma;y=0\&comma;z=0\right]\&comma;\left[x=18\&comma;y=0\&comma;z=0\right]\right]$

$\mathrm{WitnessPoints}\left(\left[yz\&comma;xy-z^2\right]\&comma;\left[x\&comma;y\&comma;z\right]\right)$

$\left[\left[x=0\&comma;y=0\&comma;z=0\right]\&comma;\left[x=75\&comma;y=0\&comma;z=0\right]\right]$

$\mathrm{seed}≔123456789$

$\mathrm{seed}≔123456789$

$\mathrm{randomize}\left(\mathrm{seed}\right)\&colon;$

$\mathrm{WitnessPoints}\left(\left[yz\&comma;xy-z^2\right]\&comma;\left[x\&comma;y\&comma;z\right]\right)$

$\left[\left[x=0\&comma;y=0\&comma;z=0\right]\&comma;\left[x=79\&comma;y=0\&comma;z=0\right]\right]$

$\mathrm{randomize}\left(\mathrm{seed}\right)\&colon;$

$\mathrm{WitnessPoints}\left(\left[yz\&comma;xy-z^2\right]\&comma;\left[x\&comma;y\&comma;z\right]\right)$

$\left[\left[x=0\&comma;y=0\&comma;z=0\right]\&comma;\left[x=79\&comma;y=0\&comma;z=0\right]\right]$

$\mathrm{WitnessPoints}\left(\left[x<0\&comma;x=3\right]\right)$

Error, (in RootFinding:-WitnessPoints) WitnessPoints does not handle mixed equations and inequalities/inequations