Chistov, Alexander L.

 Efficient Smooth Stratification of an Algebraic Variety in
 Zero--Characteristic and its Applications.
 (English, Russian)

 Email: sliss@iias.spb.su

 St. Petersburg Mathematical Society Preprint

 Submitted: 1999.04.13

 MSC: 14Q15

 Abstract:
Consider a projective algebraic variety $V$  which is given as a set of common
zeros of a family of homogeneous polynomials of degrees less than $d$ in $n+1$
variables with coefficients from a field $k$ of zero--characteristic. We prove
that $V$ can be represented as a union (respectively disjoint union) of at
most $C(n)d^n$ (respectively $C(n)d^{n(n+1)/2}$) smooth quasiprojective
algebraic varieties with degrees (herewith degree of a quasiprojective
algebraic variety is degree of its closure in the Zariski topology) bounded
from above by $C(n)d^n$ where $C(n)<2^{2^{n^C}}$ for an absolute constant
$C>0$. Further, in the case of zero--characteristic algorithms for
constructing regular sequences and sequences of local parameters for
irreducible components of $V$ are described. The complexity of these
algorithms is polynomial in the size of input and $C(n)d^n$. Finally, in the
case when $k$ is a subfield of ${\mathbb R}$ an algorithm is suggested to
compute the dimension of the set $V({\mathbb R})$ of ${\mathbb R}$-points of
$V$ with the complexity polynomial in the size of input and $C(n)d^n$.