Efficient smooth stratification of an algebraic variety in zero characteristic and its applications

A. L. Chistov

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part V, Tome 266 (2000), pp. 254-311

Résumé

Let $V$ be an algebraic variety given by a system of homogeneous polynomials equations with degrees less than $d$ in $n+1$ variables. In zero-characteristic we prove that there is a smooth cover (smooth stratification) of $V$ with the number of strata at most $C(n)d^n$ (respectively $C(n)d^{n(n+1)/2}$) and degrees of strata at most $C(n)d^n$ where $C(n)>0$ depends only on $n$. Algorithms are suggested for constructing regular sequences and sequences of local parameters of irreducible components of $V$, computing dimension of a real algebraic variety with the complexity polynomial in $C(n)d^n$ and the size of input.