Gröbner base.

Abstract

Suppose we are given a set of polynomials gl,…,gs and we wish to know whether another polynomial can be expressed in the form glh,+...+,gshs for polynomials hl,...,hs. This is often called the ideal membership problem. If gl, ,..., gs. are polynomials in one variable, then there is really no difficulty at all. If, however, the polynomials are in n variables, then the problem becomes much more difficult. Now suppose we have a system of polynomial equations and are looking for the solutions to fl(x, ,..., x,) = ... = fs(x, ,..., x,) = O. If all the equations are linear, we can use Gaussian elimination on the matrix of coefficients and backsubstitution. The problem arises when the polynomials are nonlinear. Both of these problems can be simplified by considering the theory of Grobner bases. The ideal membership problem can be solved for polynomials in n variables similar to the case of polynomials in one variable by using a general form of the division algorithm and a Grobner basis for the ideal. Also we can create a corresponding system of polynomial equations from any system of polynomial equations with a reduction of variables that will at least simplify the work of finding solutions to the original system.