Use Groebner's Bases, Krull's dimension, and Homotopy functions to compute local Dimension of an Algebraic Set at a Point

Abstract

The computation of the local dimension of an algebraic set at a numerical approximation to a point on it depends on the computation of the maximum dimension of irreducible components of the algebraic set, which pass through this point.  Depending on some theorems and properties in the algebraic geometry and in commutative algebra we  modified the algorithms in  and in  to compute local dimension by using less steps of Homotopy functions than it is in  by starting at a number less than the number of the variables and without the use of the concepts "Triangular Set" , Witness Point Sets, and the continuation of the Homotopy function as in .