Difference between revisions of "ApCoCoA1:Hom.HSolve"
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Revision as of 09:39, 13 September 2019
Hom.HSolve
Solves a zero dimensional square homogeneous or nonhomogeneous polynomial system of equations.
Syntax
Hom.HSolve(P:LIST,HomTyp:INT):LIST
Description
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 can do two kinds of different computations depending on the input that you provide in ApCoCoAServer during execution. After passing the command Hom.HSolve(P) in ApCoCoA you need to interact with ApCoCoAServer. At this stage ApCoCoAServer asks you to enter 1 for the polyhedral homotopy and to enter 2 for the classical linear homotopy. As a result this function provides all isolated solutions of a zero dimensional system of polynomial equations. The system of polynomials may be homogeneous or nonhomogeneous.
@param P: List of polynomials of the given system.
@param HomTyp: set it to 1 for polyhedral homotopy and to 2 for classical linear homotopy.
@return A list of lists containing the finite solutions of the system P.
Example
 An example of zero dimensional NonHomogeneous Solving using the polyhedral homotopy.  We want to solve zero dimensional nonhomogeneous system x^2+y^25=0, xy2=0. Use S ::= QQ[x,y]; P := [x^2+y^25, xy2]; HomTyp:=1;  Then we compute the solution with Hom.HSolve(P,HomTyp);  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
Example
 An example of zero dimensional Homogeneous Solving using the classical linear homotopy.  We want to solve zero dimensional homogeneous system x^2y^2=0, xyy^2=0. Use S ::= QQ[x,y]; M := [x^2y^2, xyy^2]; HomTyp:=1;  Then we compute the solution with Hom.HSolve(M,HomTyp);  Now you have to interact with ApCoCoAServer  Enter 1 for the polyhedral homotopy and 2 for the classical linear homotopy.  If we enter 2 then the all finite solutions are:  [ [[20597740658111043/500000000000000000000000, 74403123210058537/1000000000000000000000000], [20597740658111043/500000000000000000000000, 74403123210058537/1000000000000000000000000]], [[0, 0], [0, 0]], [[0, 0], [0, 0]], [[60689558229793541/10000000000000000000000000, 245542879738863/2000000000000000000000], [3034482281801981/500000000000000000000000, 3069286290270979/25000000000000000000000]] ]  The smallest list represents a complex number. For example above system has 4 solutions the first solution is  [[20597740658111043/500000000000000000000000, 74403123210058537/1000000000000000000000000],  [20597740658111043/500000000000000000000000, 74403123210058537/1000000000000000000000000]]  and we read it as x=20597740658111043/500000000000000000000000  74403123210058537/1000000000000000000000000i,  y = 20597740658111043/500000000000000000000000  74403123210058537/1000000000000000000000000i 
Example
 An example of zero dimensional NonHomogeneous Solving using the polyhedral homotopy.  We want to solve zero dimensional nonhomogeneous system x[1]^21=0, x[1]x[2]1=0. Use QQ[x[1..2]]; P := [x[1]^21, x[1]x[2]1]; HomTyp:=1;  Then we compute the solution with Hom.HSolve(P,HomTyp);  Now you have to interact with ApCoCoAServer  Enter 1 for the polyhedral homotopy and 2 for the classical linear homotopy.  We enter 1 because we want to use polyhedral homotopy.  The all finite solutions are:  [ [[1, 0], [1, 0]], [[1, 0], [1, 0]] ]  The smallest list represents a complex number. For example above system has 2 solutions the first solution is [[1, 0], [1, 0]]  and we read it as x[1]=1+0i, x[2]=1+0i
See also