From ApCoCoAWiki
Revision as of 09:59, 22 July 2010 by (talk)


Solves a zero dimensional homogeneous or non-homogeneous polynomial system of equations.




Please note: The function(s) explained on this page is/are using the ApCoCoAServer. You will have to start the ApCoCoAServer in order to use it/them.

This function uses total degree homotopy to find all isolated solutions of a zero dimensional system of polynomial equations. It uses default configurations provided by Bertini. The system of polynomials may be homogeneous or nonhomogeneous. For homogeneous polynomial system the output will be the list of all real solutions and for nonhomogeneous system the output will be the list of all finite solutions.

  • @param P: List of polynomials of the given system.

  • @return A list of lists containing the finite solutions of the system P.


-- An example of zero dimensional Non-Homogeneous Solving.
-- We want to solve zero dimensional non-homogeneous system x^2+y^2-5=0, xy-2=0. 

Use S ::= QQ[x,y];              
P := [x^2+y^2-5, xy-2];

-- Then we compute the solution with

-- Now you have to interact with ApCoCoAServer
-- Enter 1 for the polyhedral homotopy and 2 for the classical linear homotopy.
-- If we enter 1 then the all finite solutions are:

[[2, 0], [1, 0]],
 [[-1, 0], [-2, 0]],
 [[-2, 0], [-1, 0]],
 [[1, 0], [2, 0]]

-- The smallest list represents a complex number. For example above system has 4 solutions the first solution is [[2, 0], [1, 0]] 
-- and we read it as  x=2+0i, y=1+0i 


-- An example of zero dimensional Homogeneous Solving
-- We want to solve zero dimensional homogeneous system x^2-z^2=0, xy-z^2=0.

Use S ::= QQ[x,y,z];            
M := [x^2-z^2, xy-z^2];
SysTyp := <quotes>hom</quotes>;

-- Then we compute the solution with

-- And we achieve a list of lists containing all real solutions.
[2190685167348543/5000000000000000, 2190685167348543/5000000000000000, 2190685167348543/5000000000000000],
[1237092982347763/5000000000000000, 1237092982347763/5000000000000000, -1237092982347763/5000000000000000],
[3235177805819999/100000000000000000000000000000, 9932123317905381/10000000000000000,621807549382663/5000000000000000000000000000], 
[3006769352985381/100000000000000000000000000000,1241515414738241/1250000000000000, 555981798431817/5000000000000000000000000000]

--These are the real solutions of the system
--For Bertini output files please refer to ApCoCoA directory/Bertini.

See also

Introduction to CoCoAServer