Compute the local Dimension of an Algebraic Set at an Approximated Point to a Point belonging to this Algebraic Set

  • 07 Dec 2020
  • Published Resarch - Mathematics



Shawki AL Rashed

Published in

Damascus University Journal of Basic Sciences, Volume 36, Issue 1, 2020.



In this paper we present an algorithm, which is used the standard bases  "Groebner's Bases", that considers one of the basic concepts of the commutative  algebra and algebraic geometry, to compute the local dimension of an algebraic set at an approximate point of a point belonging to this set.

This algorithm is an improvement of other algorithms [1,3,5]. Where the algorithm in [1] is an improvement of the algorithms in [3,5] with fewer steps in the Homotopy function, without the use of the Triangular set and Witness Point Sets, and the continuity of  the Homotopy function. But they are implemented in two computer algebra systems, in addition to that  all the algorithms in [1,3,5] require that the polynomial system that defines the algebraic set is square (the number of unknowns equals the number of polynomials). Therefore, they start to reduce the polynomial system to square system. While in the algorithm presented in this research there is no need for this step and this first improvement, and the number of steps is lower by using the theorem (3.3), this second improvement. In addition to the use of a theorem (3.2) showing that the local dimension of the algebraic set at a point greater than or equal to the local dimension of this algebraic set at points that sufficiently close to this point and dispensing with the homotopy function, which is allowing the implementation of the algorithm in the SINGULAR only one computer algebra system and this third improvement.

Keywords:  algebraic set, Krull's dimension, local dimension, Groebner Bases, rank of a polynomial system.

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