Classical Diophantine Equations / Lecture Notes in Mathematics Bd.1559 (PDF)

The author had initiated a revision and translation of
"Classical Diophantine Equations" prior to his death.
Given the rapid advances in transcendence theory and
diophantine approximation over recent years, one might fear
that the present work, originally published in Russian in
1982, is mostly superseded. That is not so. A certain amount
of updating had been prepared by the author himself before
his untimely death. Some further revision was prepared by
close colleagues.
The first seven chapters provide a detailed, virtually
exhaustive, discussion of the theory of lower bounds for
linear forms in the logarithms of algebraic numbers and its
applications to obtaining upper bounds for solutions to the
eponymous classical diophantine equations. The detail may
seem stark--- the author fears that the reader may react
much as does the tourist on first seeing the centre
Pompidou; notwithstanding that, Sprind zuk maintainsa
pleasant and chatty approach, full of wise and interesting
remarks. His emphases well warrant, now that the book
appears in English, close studyand emulation. In particular
those emphases allow him to devote the eighth chapter to an
analysis of the interrelationship of the class number of
algebraic number fields involved and the bounds on the
heights of thesolutions of the diophantine equations. Those
ideas warrant further development. The final chapter deals
with effective aspects of the Hilbert Irreducibility
Theorem, harkening back to earlier work of the author. There
is no other congenial entry point to the ideas of the last
two chapters in the literature.