Simple polynomials are known to most of us from basic school
algebra. They can be used to express geometrical identities such
as the Pythagorean Theorem a^{2} + b^{2} = c^{2}.
Algebraic manipulations can even mimic geometrical reasoning. Here
is a proof of the Pythagorean Theorem based on some areal
equations:

Picture
14: Pythagorean Theorem

In higher mathematics polynomials can be viewed as algebraic extensions of rings. An interesting operation is the evaluation of an algebraic expression in a polynomial ring. This is exactly what the following Prolog text does. The original Prolog text is an adaption of a Lisp program and can be found in the Aquarius test suite [3].

One test iteration computes (1-x+y-z)^{10}.