|Title||Algebraic Solution of Systems of Polynomial Equations Using Groebner Bases|
|Author(s)|| Patrizia M. Gianni, Ferdinando Mora|
|Type||Article in Conference Proceedings|
|Abstract||This paper, together with the application to the present problem of an algorithm by Gianni that computes the radical of a O-dimensional ideal after a "generic" change of coordinates, a different approach, based on her "splitting algorithm", to compute solutions of systems of polynomial equations without the need of polynomial factorisations has been proposed by D,Duval ([DUV]); also her algorithm should be simplified by a "generic" change of coordinates.|
The algorithms discussed in this paper are implemented in SCRATCHPAD TT.
In the first section we recall some well-known properties of Gröbner bases and properties on the structure of Gröbner bases of zero-dimensional ideals from [GTZ]; in the second section we recall the Gröbner basis algorithm for solving systems of algebraic equations.
The original results are contained in Sections 3 to 5; in Section 3 we take advantage of the obvious fact that density can be controlled performing "small" changes of coordinates: we show that such approach is possible during a Gröbner basis computation, in such a way that computations done before a change of coordinates are valid also after it; in Section 4 we propose a "linear algebra" approach to obtain the Gröbner basis w.r.t. the lexicographical ordering from the one w.r.t. the total-degree ordering; in Section 5, we present a zero-dimensional radical algorithm and show how to apply it to the present problem.