Algorithms for integral solutions of a class of diophantine equations

In 1970 a negative solution to the tenth Hilbert problem, concerning the determination of integral solutions of diophantine equations, has been published by Y. W. Matiyasevich (see Matiyasevich, 1970). Despite this result, we can present algorithms to compute integral solutions (roots) for a wide class of quadratic diophantine equations of the form q(x) = d, where q : Zⁿ → Z is a homogeneous quadratic form. We will focus on the roots of one (i.e., d = 1) of quadratic Euler forms of selected posets from Loupias list (see Loupias, 1975). In particular, we will describe the roots of positive definite quadratic forms and the roots of quadratic forms that are principal (see Simson, 2010a). The algorithms and results we present here are successfully used in the representation theory of finite groups and algebras.