Small Terms

Q: I have a very large polynomial that has several "small" variables (substantially less than unity) and I would like to expand it to lowest order in those small variables. For example:

The lowest-order expansion keeps only , since is dominated by (when is small) and is dominated by (when is small). Note that although is of total order 5, compared with order 4 for , it is not known to be smaller than the second term because we don't know if x, y, z are small or large relative to each other. How can I find the lowest-order expansion?

Daniel Lichtblau (danl@wolfram.com) answers: You want to keep only those terms whose power-products are not divisible by those of other terms. These terms can be found using a Groebner basis of the set of monomials.

After recovering the coefficients,

the lowest-order expansion is