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:

[Graphics:inoutgr3.gif][Graphics:inoutgr72.gif]

The lowest-order expansion keeps only [Graphics:inoutgr73.gif], since [Graphics:inoutgr74.gif] is dominated by [Graphics:inoutgr75.gif] (when [Graphics:inoutgr76.gif] is small) and [Graphics:inoutgr77.gif] is dominated by [Graphics:inoutgr78.gif] (when [Graphics:inoutgr79.gif] is small). Note that although [Graphics:inoutgr80.gif] is of total order 5, compared with order 4 for [Graphics:inoutgr81.gif], 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.

[Graphics:inoutgr3.gif][Graphics:inoutgr82.gif]
[Graphics:inoutgr3.gif][Graphics:inoutgr83.gif]

After recovering the coefficients,

[Graphics:inoutgr3.gif][Graphics:inoutgr84.gif]
[Graphics:inoutgr3.gif][Graphics:inoutgr85.gif]

the lowest-order expansion is

[Graphics:inoutgr3.gif][Graphics:inoutgr86.gif]
[Graphics:inoutgr3.gif][Graphics:inoutgr87.gif]