R.<a,b,c,d,e,A,B,C,D,E,F> = PolynomialRing(QQ)

I = ideal(-A+a^2+b^2+c^2+d^2+e^2,-B+a^2*b^2+a^2*c^2+a^2*d^2+a^2*e^2+b^2*c^2+b^2*d^2+b^2*e^2+c^2*d^2+c^2*e^2+d^2*e^2,-C+a^2*b^2*c^2+a^2*b^2*d^2+a^2*b^2*e^2+a^2*c^2*d^2+a^2*c^2*e^2+a^2*d^2*e^2+b^2*c^2*d^2+b^2*c^2*e^2+b^2*d^2*e^2+c^2*d^2*e^2,-D+a^2*b^2*c^2*d^2+a^2*b^2*c^2*e^2+a^2*c^2*d^2*e^2+a^2*b^2*d^2*e^2+b^2*c^2*d^2*e^2,-E+a^2*b^2*c^2*d^2*e^2,-F+a+b+c+d+e)

import itertools

z = [a,b,c,d,e]

Q = prod(F-sum(zi*(-Integer(1))**ei for zi, ei in zip(z, e)) for e in itertools.product((Integer(0),Integer(1)),repeat=Integer(5)))

qs = [Q.coefficient({F:k}) for k in range(32+1)]

assert Q == sum(qk*F**k for k, qk in enumerate(qs))

rs = [I.reduce(qk) for qk in qs]

print('P(A,B,C,D,E,F) = r0 + r1*F + ... + r32*F^32\n  where')
for k, rk in enumerate(rs):
    print('    r{} = {}'.format(k, rk))