Construction of Gröbner bases for investigation of systems of polynomial equations
In the first part of this article, the authors give a quick introduction to Gröbner bases, limiting themselves to the definitions and facts necessary to understand how Gröbner bases can be employed to help find the roots of systems of multivariate polynomials over the rationals. The remainder of the article consists of an investigation of two small systems, the Katsura 3 and a set of 4 equations (3 quadratic and 1 linear) in 4 unknowns. With the help of the examples the authors make several points about how the ordering on the variables can affect the information obtainable from the resulting Gröbner basis. While some orderings may yield polynomials which can be completely solved using numerical methods, others may give polynomials which cannot be solved by known methods, polynomials such that the effects of numerical inaccuracies are greatly compounded as the values of successive variables are found, or polynomials yielding very little information about even the number of roots of the system. Very little is said about how to choose the best ordering on the variables.
scientific article; zbMATH DE number 1262961