require "algebra"

P = MPolynomial(Rational)
x, y, z = P.vars "xyz"
f1 = x**2 + y**2 + z**2 -1
f2 = x**2 + z**2 - y
f3 = x - z

coeff, basis = Groebner.basis_coeff([f1, f2, f3])
basis.each_with_index do |b, i|
  p [coeff[i].inner_product([f1, f2, f3]), b]
  p coeff[i].inner_product([f1, f2, f3]) == b #=> true
end