|講師：||Helmke, Stefan 助教|
On the Newton-Puiseux Series
A plane algebraic curve is the set of pairs (x,y) satisfying
a relation f(x,y)=0, where f is a polynomial in two variables.
In 1669, Isaac Newton (1642--1727) published an interesting
method to describe such a curve locally around one of its points,
in which y is expressed as a fractional power series in x,
(if f contains some power of y.) This series is now called the
Newton-Puiseux series, named after Newton himself of course,
and the French mathematician Victor Alexandre Puiseux (1820--1883),
who in 1850 wrote the first more formal treatise on this subject.
For practical purposes -- certainly Newton's original motivation --
the series had never been of great importance, but theoretically
it played an important role in the foundations of algebraic
geometry at least until the 1960's.
In this lecture, I will explain Newton's original idea with some examples and how his method developed over the centuries.
The following two books include an elementary introduction to plane algebraic curves, in particular the Newton-Puiseux series.