Deformation Techniques for Efficient Polynomial Equation ... - RISC

96 HEINTZ ET AL.
Taking into account M 1 (T ){0 we conclude
R(T, Y) := 1
M 1 (T )
:
0 jr
(&1) r& j M r+1&j (T ) Y j .
Now we are going to construct our approximation polynomial R (T, Y).
Let M (T ) be the 2r_(2r+1) submatrix of the Sylvester matrix of Q and
Q $:=Q Y determined by the same choice of rows and columns as the
matrices M t and M(T ) (this makes sense because we have, by assumption,
deg Y Q =deg Y Q).
Again, for 1 jr+1, denote by M j(T ) the minor of M(T ) obtained
by deleting the j th column. As by assumption Q (t, Y)=Q(t, Y) holds we
deduce easily the identities M j(t)=M j (t) for 1 jr+1. In particular we
have M 1(t){0 and M 1(T ){0. More precisely, M 1 (t){0 implies that
M 1(T ) is a unit of the local Q-algebra Q[T] M . In particular, for any
1 jr+1 the rational function M j(T )M 1(T ) belongs to Q[T] M .We
would like to use the expression
1
M 1(T )
:
0 jr
(&1) r& j M r+1&j(T ) Y j (16)
for the definition of the approximation polynomial R (T, Y).
However, the expression (16) contains terms which are rational functions
in T=(T 1 , ..., T m ). In order to avoid this difficulty we replace the rational
functions M r+1&j(T )M 1(T ) occurring in (16) by suitable approximation
polynomials of Q[T]. For this purpose we proceed as follows.
Without loss of generality we may assume that M 1 (t)=M 1(t)=1 holds.
Let us write H :=1&M 1 and H :=1&M 1. Hence, H(t)=H (t)=0 and we
obtain the following identities in the power series ring QT&t :
1
= : H l
M 1 l0
and
1
= : H l
.
M 1
l0
Thus, in order to approximate R(T, Y) with precision s in QT&t, we
define the polynomial R (T, Y)#Q[T, Y] as
R (T, Y):= :
0ls
H
l (T ) :
0 jr
(&1) r& j M r+1&j(T ) Y j .
Let us verify that R satisfies the requirements of the statement of Lemma 7:
v Obviously we have R {0 and from M 1 {0 we deduce that
deg Y R =r=deg Y R holds.