In his 1999 paper D. W. Masser talks about effective search bounds for polynomial equations over integers and rationals. This discussion can also be extended over number fields. Unfortunately, as illustrated by Matiasevich's negative answer to Hilbert's 10-th problem, search bounds in general probably do not exist. Some special cases are understood, but in general very little is known. I will talk about effective search bounds for solutions of polynomial equations over Q-bar with some additional arithmetic conditions. This discussion also naturaly ties into the realm of "absolute" diophantine results, like Siegel's lemma of Roy and Thunder. I will try to talk about an analogous result for quadratic spaces.
© 2005 Lenny Fukshansky
Fukshansky, Lenny, "Some Effective Diophantine Results Over Q-bar." XXIVth Journées Arithmétiques, Marseilles, France. July 2005.