The Hardy-Littlewood circle method was invented over a century ago
to study integer solutions to special Diophantine equations, but it
has since proven to be one of the most successful all-purpose tools
available to number theorists. Not only is it capable of handling
remarkably general systems of polynomial equations defined over
arbitrary global fields, but it can also shed light on the space of
rational curves that lie on algebraic varieties. This book, in
which the arithmetic of cubic polynomials takes centre stage, is
aimed at bringing beginning graduate students into contact with
some of the many facets of the circle method, both classical and
modern. This monograph is the winner of the 2021 Ferran Sunyer i
Balaguer Prize, a prestigious award for books of expository nature
presenting the latest developments in an active area of research in
mathematics.
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