s1 = 4x0 - (x1+x2+x3+x4)

s2 = (6 x0^2 + 2 x1^2 + 4 x1 x2 + 4 x1 x3 + 2 x2^2 + 4 x2 x4 + 2 x3^2 + 4 x3 x4 + 2 x4^2) - (3 x0 x1 + 3 x0 x2 + 3 x0 x3 + 3 x0 x4 + x1^2 + 2 x1 x2 + 2 x1 x3 + 3 x1 x4 + x2^2 + 3 x2 x3 + 2 x2 x4 + x3^2 + 2 x3 x4 + x4^2)
   = 6 x0^2 - 3 x0 x1 - 3 x0 x2 - 3 x0 x3 - 3 x0 x4 + x1^2 + 2 x1 x2 + 2 x1 x3 - 3 x1 x4 + x2^2 - 3 x2 x3 + 2 x2 x4 + x3^2 + 2 x3 x4 + x4^2

s3 = (4 x0^3 + 4 x0 x1^2 + 8 x0 x1 x2 + 8 x0 x1 x3 + 4 x0 x2^2 + 8 x0 x2 x4 + 4 x0 x3^2 + 8 x0 x3 x4 + 4 x0 x4^2 + 4 x1^2 x3 + 4 x1^2 x4 + 4 x1 x2^2 + 4 x1 x2 x3 + 4 x1 x2 x4 + 4 x1 x3 x4 + 4 x1 x4^2 + 4 x2^2 x3 + 4 x2 x3^2 + 4 x2 x3 x4 + 4 x2 x4^2 + 4 x3^2 x4) - (3 x0^2 x1 + 3 x0^2 x2 + 3 x0^2 x3 + 3 x0^2 x4 + 2 x0 x1^2 + 4 x0 x1 x2 + 4 x0 x1 x3 + 6 x0 x1 x4 + 2 x0 x2^2 + 6 x0 x2 x3 + 4 x0 x2 x4 + 2 x0 x3^2 + 4 x0 x3 x4 + 2 x0 x4^2 + x1^3 + 3 x1^2 x2 + 2 x1^2 x3 + 2 x1^2 x4 + 2 x1 x2^2 + 5 x1 x2 x3 + 5 x1 x2 x4 + 3 x1 x3^2 + 5 x1 x3 x4 + 2 x1 x4^2 + x2^3 + 2 x2^2 x3 + 3 x2^2 x4 + 2 x2 x3^2 + 5 x2 x3 x4 + 2 x2 x4^2 + x3^3 + 2 x3^2 x4 + 3 x3 x4^2 + x4^3)
   = 4 x0^3 - 3 x0^2 x1 - 3 x0^2 x2 - 3 x0^2 x3 - 3 x0^2 x4 + 2 x0 x1^2 + 4 x0 x1 x2 + 4 x0 x1 x3 - 6 x0 x1 x4 + 2 x0 x2^2 - 6 x0 x2 x3 + 4 x0 x2 x4 + 2 x0 x3^2 + 4 x0 x3 x4 + 2 x0 x4^2 - x1^3 - 3 x1^2 x2 + 2 x1^2 x3 + 2 x1^2 x4 + 2 x1 x2^2 - x1 x2 x3 - x1 x2 x4 - 3 x1 x3^2 - x1 x3 x4 + 2 x1 x4^2 - x2^3 + 2 x2^2 x3 - 3 x2^2 x4 + 2 x2 x3^2 - x2 x3 x4 + 2 x2 x4^2 - x3^3 + 2 x3^2 x4 - 3 x3 x4^2 - x4^3

s4 = (x0^4 + 2 x0^2 x1^2 + 4 x0^2 x1 x2 + 4 x0^2 x1 x3 + 2 x0^2 x2^2 + 4 x0^2 x2 x4 + 2 x0^2 x3^2 + 4 x0^2 x3 x4 + 2 x0^2 x4^2 + 4 x0 x1^2 x3 + 4 x0 x1^2 x4 + 4 x0 x1 x2^2 + 4 x0 x1 x2 x3 + 4 x0 x1 x2 x4 + 4 x0 x1 x3 x4 + 4 x0 x1 x4^2 + 4 x0 x2^2 x3 + 4 x0 x2 x3^2 + 4 x0 x2 x3 x4 + 4 x0 x2 x4^2 + 4 x0 x3^2 x4 + x1^4 + 2 x1^2 x2^2 + 4 x1^2 x2 x3 + 4 x1^2 x2 x4 + 2 x1^2 x3^2 + 2 x1^2 x4^2 + 4 x1 x2^2 x4 + 4 x1 x2 x3^2 + 4 x1 x2 x3 x4 + 4 x1 x3^2 x4 + 4 x1 x3 x4^2 + x2^4 + 2 x2^2 x3^2 + 4 x2^2 x3 x4 + 2 x2^2 x4^2 + 4 x2 x3 x4^2 + x3^4 + 2 x3^2 x4^2 + x4^4) - (x0^3 x1 + x0^3 x2 + x0^3 x3 + x0^3 x4 + x0^2 x1^2 + 2 x0^2 x1 x2 + 2 x0^2 x1 x3 + 3 x0^2 x1 x4 + x0^2 x2^2 + 3 x0^2 x2 x3 + 2 x0^2 x2 x4 + x0^2 x3^2 + 2 x0^2 x3 x4 + x0^2 x4^2 + x0 x1^3 + 3 x0 x1^2 x2 + 2 x0 x1^2 x3 + 2 x0 x1^2 x4 + 2 x0 x1 x2^2 + 5 x0 x1 x2 x3 + 5 x0 x1 x2 x4 + 3 x0 x1 x3^2 + 5 x0 x1 x3 x4 + 2 x0 x1 x4^2 + x0 x2^3 + 2 x0 x2^2 x3 + 3 x0 x2^2 x4 + 2 x0 x2 x3^2 + 5 x0 x2 x3 x4 + 2 x0 x2 x4^2 + x0 x3^3 + 2 x0 x3^2 x4 + 3 x0 x3 x4^2 + x0 x4^3 + x1^3 x2 + x1^3 x3 + x1^3 x4 + x1^2 x2^2 + 2 x1^2 x2 x3 + 2 x1^2 x2 x4 + x1^2 x3^2 + 3 x1^2 x3 x4 + x1^2 x4^2 + x1 x2^3 + 3 x1 x2^2 x3 + 2 x1 x2^2 x4 + 2 x1 x2 x3^2 + 5 x1 x2 x3 x4 + 3 x1 x2 x4^2 + x1 x3^3 + 2 x1 x3^2 x4 + 2 x1 x3 x4^2 + x1 x4^3 + x2^3 x3 + x2^3 x4 + x2^2 x3^2 + 2 x2^2 x3 x4 + x2^2 x4^2 + x2 x3^3 + 3 x2 x3^2 x4 + 2 x2 x3 x4^2 + x2 x4^3 + x3^3 x4 + x3^2 x4^2 + x3 x4^3)
   = x0^4 - x0^3 x1 - x0^3 x2 - x0^3 x3 - x0^3 x4 + x0^2 x1^2 + 2 x0^2 x1 x2 + 2 x0^2 x1 x3 - 3 x0^2 x1 x4 + x0^2 x2^2 - 3 x0^2 x2 x3 + 2 x0^2 x2 x4 + x0^2 x3^2 + 2 x0^2 x3 x4 + x0^2 x4^2 - x0 x1^3 - 3 x0 x1^2 x2 + 2 x0 x1^2 x3 + 2 x0 x1^2 x4 + 2 x0 x1 x2^2 - x0 x1 x2 x3 - x0 x1 x2 x4 - 3 x0 x1 x3^2 - x0 x1 x3 x4 + 2 x0 x1 x4^2 - x0 x2^3 + 2 x0 x2^2 x3 - 3 x0 x2^2 x4 + 2 x0 x2 x3^2 - x0 x2 x3 x4 + 2 x0 x2 x4^2 - x0 x3^3 + 2 x0 x3^2 x4 - 3 x0 x3 x4^2 - x0 x4^3 + x1^4 - x1^3 x2 - x1^3 x3 - x1^3 x4 + x1^2 x2^2 + 2 x1^2 x2 x3 + 2 x1^2 x2 x4 + x1^2 x3^2 - 3 x1^2 x3 x4 + x1^2 x4^2 - x1 x2^3 - 3 x1 x2^2 x3 + 2 x1 x2^2 x4 + 2 x1 x2 x3^2 - x1 x2 x3 x4 - 3 x1 x2 x4^2 - x1 x3^3 + 2 x1 x3^2 x4 + 2 x1 x3 x4^2 - x1 x4^3 + x2^4 - x2^3 x3 - x2^3 x4 + x2^2 x3^2 + 2 x2^2 x3 x4 + x2^2 x4^2 - x2 x3^3 - 3 x2 x3^2 x4 + 2 x2 x3 x4^2 - x2 x4^3 + x3^4 - x3^3 x4 + x3^2 x4^2 - x3 x4^3 + x4^4