On The Spatial Distribution Of Solutions Of Decomposable Form Equations

. We study the distribution in space of the integral solutions to an integral decomposable form equation, by considering the images of these solutions under central projection onto a unit ball. If we think of the solutions as stars in the night sky, we ask what constellations are visible from the earth (the unit ball). Answers are given for a large class of examples which are then illustrated using the software packages KANT and Maple. These pictures highlight the accuracy of our predictions and arouse interest in cases not covered by our results. Within the range of applicability of our results lie solutions to norm form equations and units in abelian group rings. Thus our theory has a lot to say about where these interesting objects can be found and what they look like. Introduction Let F (x) = F (x 1 ; : : : ; xn ) 2 Z[x 1 ; : : : ; xn ] denote a decomposable form. This is a homogeneous polynomial with coefficients in Z which factorises over C as a product of linear forms. We assum..