// This solves the equations in Example 1.6 of:
// C. Hillar, L.-H. Lim, Most tensor problems are NP-hard, 2012.
// (the equations in Ex 1.6 are the "real" versions, produced using 
// Lemma 2.6, of the following equations from Definition 2.8)
//
// using the symbolic software SINGULAR
// http://www.singular.uni-kl.de/index.php

LIB "solve.lib";

ring R=0, (x(1..4),y(1..4),z), lp;
ideal I = z-1,x(1)*y(1)-z^2,x(2)*y(2)-z^2,x(3)*y(3)-z^2,x(4)*y(4)-z^2, y(1)*z-x(1)^2,y(2)*z-x(2)^2,y(3)*z-x(3)^2,y(4)*z-x(4)^2, x(1)*z-y(1)^2,x(2)*z-y(2)^2,x(3)*z-y(3)^2,x(4)*z-y(4)^2, x(1)^2+x(1)*x(2)+x(2)^2,x(2)^2+x(2)*x(3)+x(3)^2,x(1)^2+x(1)*x(3)+x(3)^2,x(1)^2+x(1)*x(4)+x(4)^2,x(3)^2+x(3)*x(4)+x(4)^2;

ideal sI = std(I);
def AC=solve(I,6,0); 
setring AC;
size(SOL);