Background Material
In this chapter, we will introduce some tools used to prove the Desingularization
Theorem. The desingularization method for two-dimensional families is basically
described in [D-R]. Here, we have chosen to place the discussion in the general set-
ting of n-dimensional families, where the concepts of monomial fields and foliations
tangent to a monomial field will appear rather naturally. Some results (mainly, the
ones in the appendices) will not be explicitly used in the next chapters, but they
seem to be of interest in the applications of the desingularization method.
2.1. Monomia l fields
A rational monomial on E ^ is a symbol m x 7 , where
J = l
To such symbol, one uniquely associates an analytic function on the domain U =
M j V \ U { ^ = 0}J given by
(xu x
, . . . , xN) —+ \Xl\^ | x
p \xN\^N.
Let us denote such function simply by ra(|x|).
Given a set of non-zero rational monomials rai,...,ran on R^ , we define
the monomia l field generate d by { m j to be the smaller field extension F =
F(mi,..., mn) of R which contains all monomials of the form
(2.1) A m f - . - m ^
with rational exponents Q G Q and a constant A G R. Clearly, each element / G F
can be written in the form / = P/Q, where P and Q are finite sums of monomials
like (2.1) and Q is nonzero. We shall simply say that F is a monomia l field if it
is the monomial field generated by some finite set of rational monomials.
A sub-collection of rational monomials {rrii,..., m^} of a monomial field F
will be called independent if no monomial rrii can be generated as an element of
F ( r a i , . . . , m,_i , m ,
i , . . . , mn).
L E M M A 2.1. Write rrii = x T *
fori 1 , . . . ,n. Then, {mi,... , m
} is indepen-
dent if and only if the set of vectors {/y1,..., 7
} C Q ^ is Q-linearly independent.
Proof: Notice that ^yi = ^2j^i ajlj, (f° r some constants dj G Q) if and only if
(2.2) mi =
On the other hand, if rrii is generated in F(m\,..., ra^-i, m
i +
i , . . . , m
) , a simple
computation using the fact that rrii is a monomial shows that it necessarily can be
expressed in the form (2.2). This proves the lemma. {
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