Deformation Techniques for Efficient Polynomial Equation ... - RISC

76 HEINTZ ET AL.
holds (here *? &1 (t) denotes the cardinality of the set ? &1 (t) and
dim Q(T ) Q(T )[X](F 1 , ..., F n ) denotes the dimension of the Q(T )-vector
space Q(T )[X](F 1 , ..., F n )). In other words, *? &1 (t) equals the cardinality
D=deg ? of the generic fiber of ?. This fact may also be circumscribed as
the generic flatness of the morphism ?.
Moreover the given description of a geometric solution of the fiber
? &1 (t) provides an explicit description of the multiplication tensor of the
Q-algebra Q[X](F 1 (t, X), ..., F n (t, X)).
(iv) The assumption that G maps the fiber ? &1 (t) onto deg Y P
distinct points of A 1 is equivalent to the requirement that the specialized
polynomial P(t, Y) :=P(t 1 , ..., t m , Y)#Q[Y] is square-free, i.e., that
the discriminant of P(T, Y) as a polynomial in Y does not vanish on the
point t.
(v) Finally observe that by our assumptions the squarefreeness
of P(T, Y) and the generical unramifiedness of ? can be expressed as
consistent Zariski open conditions in A m . Therefore, there always exists a
point t satisfying our requirements.
Let us observe that in case D