The expansion base algorithm, which was devised by Abhyankar, Kuo and
McCallum is very efficient for analytic factorization of bivariate
polynomials.

The author had extended it to more than two variables but it was only
for polynomials with non-vanishing leading coefficient at the expansion
point.

Here, we improve it to be able to apply to polynomials including the
case of vanishing leading coefficient, that is, singular leading
coefficient, which comes to a specific problem only for more than two
variables.

We apply a simple transformation to multivariate expansion base method
and see that the analytic factorization for polynomials with singular
and non singular leading coefficients can be done in a unified way, too.
This breakthrough comes from the identification of "weight of expansion
base" and "slope of Newton's line". Note that weights of bivariate
expansion base method take only positive values, whereas weights of
multivariate expansion base method can be negative, 0 or positive
(slope of Newton's line can be positive, 0 or negative, respectively).
We can say this unified method is a blend of techniques of "multivariate
expansion base" and "the extended Hensel construction".