sage multivariate polynomial variable substitution

poly_sub.sage

def poly_sub(f, x, y):
    ret = f.parent().zero()
    for c, m in f:
        while m % x == 0:
            m //= x
            c *= y
        ret += c * m
    return ret


def poly_sub2(f, x, y):
    # https://ask.sagemath.org/question/25972/substitute-xy-by-u/#post-id-25996
    Q = f.parent().quotient(x - y)
    return Q(f).lift()


for R in [ZZ, Zmod(17 * 19), GF(2 ^ 8), CyclotomicField(17)]:
    PR = R["x, y, z"]
    x, y, z = PR.gens()
    assert poly_sub((x * y + 1) ^ 7, x * y, z) == (z + 1) ^ 7
    assert poly_sub2((x * y + 1) ^ 7, x * y, z) == (z + 1) ^ 7
    assert poly_sub((x ^ 2 * y + 1) ^ 3, x * y, z) == (z * x + 1) ^ 3
    assert poly_sub2((x ^ 2 * y + 1) ^ 3, x * y, z) == (z * x + 1) ^ 3