This paper describes the use of Groebner bases for solving systems of polynomial equations. In particular the paper considers Groebner bases applied to the systems of equations associated with constrained structural optimisation problems.

The paper introduces the key concepts of varieties, bases and ideals and describes how they are related. The paper then introduces Groebner, or standard, bases and describes how they can be applied to constrained optimisation problems. Groebner bases offer the advantage that once formulated they can be used to determine whether a series of polynomial equations has either no solutions, a finite number of solutions or an infinite number of solutions. Where there are a finite number of solutions Groebner bases offer a method of systematically identifying them all.

The paper illustrates the use of Groebner bases by applying to method to a simple structural engineering optimisation problem; the design of a minimum weight plate girder section.

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