Efficient Smooth Stratification of an Algebraic Variety in Zero Characteristic and Its Applications

Abstract

Consider a projective algebraic variety V defined as the set of common zeros of a family of homogeneous polynomials of degree 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, a disjoint union) of at most $$C(n)d^n $$ (respectively, $$C(n)d^{n{\text{(}}n{\text{ + 1)/2}}} $$ ) smooth quasiprojective algebraic varieties such that the degrees of these varieties are bounded from above by $$C(n)d^{ n} $$ , where $$C(n){\text{ < 2}}$$ depends only on n. We propose algorithms for constructing regular sequences and sequences of local parameters for irreducible components of V and for computing the dimension of a real variety. The complexity of these algorithms is polynomial in the size of the input and in $$C(n)d^{ n} $$ . Bibliography: 15 titles.