Deformation Techniques for Efficient Polynomial Equation ... - RISC

84 HEINTZ ET AL.
The zero dimensionality and the unramifiedness of the fiber ? &1 (t) can be
expressed algebraically by the fact that Q[X](F 1 (t, X), ..., F n (t, X)) is a
finite-dimensional, reduced Q-algebra of dimension D=deg ?.
Hence, the same argumentation can be applied to the generic fiber of ?,
which is scheme-theoretically represented by the finite-dimensional reduced
Q(T )-algebra Q(T )[X](F 1 , ..., F n ) of dimension D. Therefore we conclude
that the polynomial P is the minimal polynomial of the homothety ' G in
Q(T )[X](F 1 , ..., F n ) induced by the multiplication by the residue class of
G (observe that the finiteness and the surjectivity of the morphism ? and
the fact that Q[T] is integrally closed in its fraction field Q(T ) implies that
the minimal polynomial of ' G belongs, in fact, to the Q-algebra Q[T, Y]).
The question is now: How can we compute the minimal polynomial of
the homothety ' G without determining first a matrix form of the multiplication
tensor of the Q(T )-algebra Q(T )[X](F 1 , ..., F n )?
This computation will be done by means of an approximation method
based on the NewtonHensel procedure.
Our first aim is to ``lift'' algorithmically each point !=(! 1 , ..., ! n )ofV t =
? &1 (t) toan-tuple R (!) :=(R (!)
1
, ..., R(!)
n
) of power series of CT&t such
that the following two conditions hold:
v F 1 (T, R (!) )=0, ..., F n (T, R (!) )=0
v R (!) (t) :=(R (!)
1
(t), ..., R(!) (t))=(! n 1, ..., ! n )=!.
We will show that, by our unramifiedness assumptions on ? and ? &1 (t)
and Lemma 3 below, such a lifting procedure always exists.
Thus, the D distinct points R (! (1)) , ..., R (! (D)) of CT&t n have coordinates
which are algebraic over Q(T ) and they form a complete solution set of the
zero-dimensional equation system
[F 1 (T, X)=0, ..., F n (T, X)=0]
over the ground field Q(T ).
From our previous argumentation one deduces immediately that P(T, Y)
is the minimal polynomial of the diagonal matrix
G(T, R (! (1))
\
) }}} 0
.
b . . b
+
.
0 }}} G(T, R (! (D)) )
This matrix represents the homothety ' G in a suitable C(T )-vector space
basis of C(T )[X](F 1 , ..., F n ).
Let M :=(T 1 &t 1 , ..., T m &t m ) be the maximal ideal of C[T] corresponding
to the parameter point t.