Decomposable Forms and Siegel's Lemma

Northern Illinois University, Dekalb IL

Investigators

Abstract

The investigator continues to study general decomposable forms. In particular, he seeks better bounds for the number of primitive integral solutions to non-degenerate decomposable form equations and also stronger asymptotic results for the number of integral solutions to the related inequality. In addition, the investigator studies the absolute Siegel's lemma and the related Hermite's constant over the algebraic numbers. The main goal here is an effective version of the former and improved upper bounds for the latter. The decomposable form equations dealt with here are examples of Diophantine equations. These have been studied for millennia. Long thought to be mere intellectual curiosities, it is now realized that knowledge in this area is crucial in our ``digital" age. Much of cryptography, coding, and data exchange rests on the ability, or lack thereof, to find solutions to such equations. Siegel's lemma and Hermite's constant are intimately connected with sphere packings, finite groups and error-correcting codes, and via these to many disparate areas of science.