Polynomial-Time Computation of the Degree of Algebraic Varieties in
Zero-Characteristic and its Applications

Chistov, Alexander L.

sliss@mail.iias.spb.su

Consider an algebraic variety over a zero--characteristic ground field which
is  given as a set of all common zeros of a family of polynomials of the
degree less than $d$ in $n$ variables. In this paper the following algorithms
with the  working time polynomial in the size of input and $d^n$ are
constructed: an algorithm for the computation of the degrees of algebraic
varieties, an algorithm for the computation of the dimension of a given
algebraic variety in the neighbourhood of a given point, an algorithm for the
computation of the multiplicity of a given point of an algebraic variety, an
algorithm for the computation of a representative system of smooth points
with their tangent spaces on each component of a given algebraic variety, an
algorithm for deciding whether a given morphism of algebraic varieties is
dominant.


1991 Math. Subject Class. 14Q15