Using Gröbner bases to reason about geometry problems

Abstract

The use of Gröbner basis computation for reasoning about geometry problems is demonstrated. Two kinds of geometry problems are considered: (i) Given a finite set of geometry relations expressed as polynomial equations, in conjunction with a finite set of subsidiary conditions stated as negations of polynomial equations to rule out certain degenerate eases, check whether another geometry relation expressed as a polynomial equation and given as a conclusion, holds. (ii) Given a finite set of geometry relations expressed as polynomial equations, find a finite set of subsidiary conditions, if any, stated as negations of polynomial equations which rule out certain values of variables, such that another geometry relation expressed as a polynomial equation and given as a conclusion, holds under these conditions. Using a refutational approach for theorem proving, both kinds of problems are converted into reasoning about a finite set of polynomial equations. The first problem is shown to be equivalent to checking whether a set of polynomial equations does not have a solution; this can be decided by computing a Gröbner basis of these polynomials and checking whether I is included in such a basis. In addition, it is shown that the second problem can also be solved by computing a Gröbner basis and appropriately picking polynomials from it. A number of geometry problems of both kinds have been solved using this approach.