ApCoCoA-1:Bertini.BMSolve
Bertini.BMSolve
Solves a zero dimensional non-homogeneous polynomial system using multi-homogenization and default configurations.
Syntax
Bertini.BMSolve(P:LIST):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 solves a polynomial system of equations using multihomogeneous homotopy. The polynomial system of equations must be quadratic. If the system has N variables then multihomogeneous homotopy will introduce N homogeneous variables to solve the system. It uses total degree homotopy to find all isolated solutions and default configurations provided by Bertini. The system of polynomials should be non-homogeneous. 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 polynomial system P.
Example
-- We want to solve the non-homogenous polynomial system x[1]^2+x[2]^2-5=0, x[1]x[2]-2=0, using multi-homogenization. Use S ::= QQ[x[1..2]]; P := [x[1]^2+x[2]^2-5, x[1]x[2]-2]; -- Then we compute the solution with Bertini.BMSolve(P); -- And we achieve a list of lists containing finite solutions. ---------------------------------------- [ [ Vector(1000000000000001/1000000000000000, -2305082859180703/100000000000000000000000000000), Vector(1999999999999971/1000000000000000, 4135565953005217/100000000000000000000000000000) ], [ Vector(1000000000000003/500000000000000, 2604577577014449/50000000000000000000000000000), Vector(500000000000001/500000000000000, -619892334722183/25000000000000000000000000000) ], [ Vector(-2, 1724810333092189/1000000000000000000000000000000), Vector(-500000000000001/500000000000000, -355984244774691/200000000000000000000000000000) ], [ Vector(-9999999999999971/10000000000000000, -4053926086793577/1000000000000000000000000000000), Vector(-1999999999999999/1000000000000000, -3669041992638223/5000000000000000000000000000000) ] ] --For Bertini output files refer to ApCoCoA directory/Bertini. ------------------------------------------
Example
-- We want to solve the non-homogenous polynomial system (29/16)z[1]^3 - 2z[1]z[2], z[2] - z[1]^2, using multi-homogenization. Use S ::= QQ[z[1..2]]; P := [(29/16)z[1]^3 - 2z[1]z[2], z[2] - z[1]^2]; -- Then we compute the solution with Bertini.BMSolve(P); -- And we achieve a list of lists containing finite solutions. ---------------------------------------- [ [ Vector(-1754775022937541/1000000000000000000000000000, -6761671559595563/10000000000000000000000000000), Vector(947843957587963/25000000000000000000000000000, 623113227620389/5000000000000000000000000000) ], [ Vector(-85573832182963743719/50000000000000000000000000000000, -89829012439528360233/250000000000000000000000000000000), Vector(-230164951873451072943/2500000000000000000000000000000000, 298328875801698252183/10000000000000000000000000000000000) ], [ Vector(-1479267029218781/1000000000000000000000000000, -5565180110034249/10000000000000000000000000000), Vector(-4881416330105221/50000000000000000000000000000, 856957743028027/5000000000000000000000000000) ] ] --For Bertini output files refer to ApCoCoA directory/Bertini. ------------------------------------------
See also
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