Let an algebraic variety over a zero--characteristic ground field be given as a set of 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 the given algebraic variety in the neighborhood of a given point, an algorithm for the computation of the multiplicity of a given point of the algebraic variety, an algorithm for the computation smooth points with their tangent spaces on each component of a given algebraic variety.