I'm writing a program where I have a 3-dimensional polynomial (3 variables) of which I have to check if it is positive in a given product of 3 intervals (a volume).
I found a paper which solved this problem but I didn't really understand it, nor am I able to extract an algorithm out of it: http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1084864
So can this approach be formulated as an algorithm for a computer?
Thanks in advance.
Edit:
To make things more specific.
My polynomial is always:
z^3*(z*(6*z-15)+10)(-y^3(y*(6*y-15)+10)(x^3(x*(6*x-15)+10)*(g3z*z-g2z*z+g3y*(y-1)-g2y*(y-1)-g2x*x+g3x*(x-1))-x^3*(x*(6*x-15)+10)*(g1z*z-g0z*z+g1y*y-g0y*y-g0x*x+g1x*(x-1))+g2z*z-g0z*z-g0y*y+g2y*(y-1)+g2x*x-g0x*x)-x^3*(x*(6*x-15)+10)*(g1z*z-g0z*z+g1y*y-g0y*y-g0x*x+g1x*(x-1))-g0z*z+g4z*(z-1)+y^3*(y*(6*y-15)+10)(g6z(z-1)-g4z*(z-1)+x^3*(x*(6*x-15)+10)(g7z(z-1)-g6z*(z-1)+g7y*(y-1)-g6y*(y-1)-g6x*x+g7x*(x-1))-x^3*(x*(6*x-15)+10)(g5z(z-1)-g4z*(z-1)+g5y*y-g4y*y-g4x*x+g5x*(x-1))-g4y*y+g6y*(y-1)+g6x*x-g4x*x)+x^3*(x*(6*x-15)+10)(g5z(z-1)-g4z*(z-1)+g5y*y-g4y*y-g4x*x+g5x*(x-1))+g4y*y-g0y*y+g4x*x-g0x*x)+y^3*(y*(6*y-15)+10)(x^3(x*(6*x-15)+10)*(g3z*z-g2z*z+g3y*(y-1)-g2y*(y-1)-g2x*x+g3x*(x-1))-x^3*(x*(6*x-15)+10)*(g1z*z-g0z*z+g1y*y-g0y*y-g0x*x+g1x*(x-1))+g2z*z-g0z*z-g0y*y+g2y*(y-1)+g2x*x-g0x*x)+x^3*(x*(6*x-15)+10)*(g1z*z-g0z*z+g1y*y-g0y*y-g0x*x+g1x*(x-1))+g0z*z+g0y*y+g0x*x
where all variables starting with $g$ are constant. The given intervals for $x,y,z$ are limited to $[0,1]$.