-- MATH 531: Nov 8. -- u-resultants and hidden variables.
restart
M2dir = "/home/x/leykin/M2/" 
load (M2dir|"lab9.m2")

-- Res_{1,1,2} (from previous session)
R = QQ[x,y,z];
degs = {1,2,2};
D = DN(R,degs) ;
D' = MacMinor(R,degs);
Res122 = D//sub(D',ring D);

-- To evaluate the resultant substitute the coefficients of polynomials
A = ring Res122
numgens A
vars A
sub(Res122, matrix{{1,0,0,0,0,1,0,0,0,0,0,0,0,0,1}}) -- sub coefficients of F_0=x, F_1=y, F_2=z^2

------------- Problem 1 ------------------
-- Does the homogeneous system 
-- F_0 = x + y + z;
-- F_1 = -5*x^2+y^2+z^2;
-- F_2 = 2*x^2+y*z
-- have a nontrivial solution? (Compute the resultant)
sub(Res122, matrix{{1,1,1,-5,0,1,0,0,1,2,0,0,0,1,0}}) 

------------- Problem 2 ------------------
-- Let (*) be the system
-- f_1 = y^2+z^2-5;
-- f_2 = 2+y*z.
-- Solve (*) using u-resultant
-- Hint: use
M = map(QQ[u_0..u_2], A,  matrix{{u_0,u_1,u_2,...

-- Now use the univariate u-resultant to recover the possible y and z coordinates of the solutions of (*).

------------- Problem 3 ------------------
-- Let (**) be the system
-- f_0 = 1+y+z
-- f_1 = -5*x^2+y^2+1;
-- f_2 = 2*x^2+y*z
-- Solve (**) via "hidden variable" method.
--
-- For example, to get the y-coordinates, "hide" y (replace it with Y) and then homogenize w.r.t. "new y".
-- F_0 = (1+Y)*y+z
-- F_1 = -5*x^2+(Y^2+1)y^2
-- F_2 = 2*x^2+Y*y*z
My = map(QQ[Y], A,  matrix{{0,Y+1,1,
	       -5,0,Y^2+1,0,0,0, 
	       2,0,0,0,Y,0}})
factor My Res122	       

-- Using the obtained results, construct all solutions to (**). (by hand)