全学共通科目講義(1回生~4回生対象)
現代の数学と数理解析 |
―― 基礎概念とその諸科学への広がり |
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日時: | 2018年5月11日(金) 16:30-18:00 |
場所: | 数理解析研究所 420号室 |
講師: | Helmke, Stefan 助教 |
題目: |
On the Newton-Puiseux Series
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要約: |
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. References: The following two books include an elementary introduction to plane algebraic curves, in particular the Newton-Puiseux series.
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"http://www.kurims.kyoto-u.ac.jp/ja/special-02.html" |