Classical Diophantine Equations (lecture Notes In Mathematics)
by Vladimir G. Sprindzuk /
1994 / English / PDF
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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in Russian in 1982, is mostly superseded.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.
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in Russian in 1982, is mostly superseded.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.