Polynomial-time computation of degrees of algebraic varieties in zero-characteristic and its applications

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.