**Akira Terui
and Tateaki Sasaki
Doctoral Program in Mathematics
Institute of Mathematics
University of Tsukuba
Tsukuba-shi, Ibaraki 305-8571, Japan
**

Let *P*(*x*) be a given univariate polynomial and
,
where
is a
sum of error terms, i.e., a polynomial with small unknown but bounded
coefficients. We first consider specifying small domains in which the
zero-points of
exist, by the (approximate) zero-points
of *P*(*x*) and the coefficient bounds of
.
The
existential domains are determined fairly small by applying Smith's
theorem which specify the error bounds of the approximations for the
zero-points. This leads us to a concept of ``approximate
zero-points'' of an approximate polynomial. We next consider counting
the number of approximate real zero-points of
,
by
investigating the Sturm sequence. For polynomials with small error
terms, calculating the Sturm sequence imposes us two problems: 1) some
leading term may become very small, and 2) it is not easy to count the
approximately multiple zero-points because the termination criterion
is unclear. For the first problem, we apply the subresultant theory
to the sequence and show that very small leading terms may be
discarded. For the second problem, we propose a method of counting
the number of approximately multiple zero-points by using approximate
square-free decomposition and calculating existential domains of the
zero-points of the approximate polynomial as the above.