In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Spijker's lemma is a result in the theory of
rational mapping
In mathematics, in particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties. This article uses the convention that varieties are Irreducible component, irreducible ...
s of the
Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann,
is a Mathematical model, model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents ...
. It states that the
image
An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
of a
circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
under a complex rational map with
numerator
A fraction (from , "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, thre ...
and
denominator
A fraction (from , "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, thre ...
having
degree at most ''n'' has length at most 2''nπ''.
Applications
Spijker's lemma can be used to derive a sharp bound version of
Kreiss matrix theorem In matrix analysis, Kreiss matrix theorem relates the so-called Kreiss constant of a matrix with the power iterates of this matrix. It was originally introduced by Heinz-Otto Kreiss to analyze the stability of finite difference methods for partia ...
.
See also
*
Buffon's needle
External links
*
References
*{{cite journal, last = Wegert, first = Elias, author2=Trefethen, Lloyd N. , title = From the Buffon Needle Problem to the Kreiss Matrix Theorem, journal = The American Mathematical Monthly, volume = 101, issue = 2, pages = 132–139, date=February 1994, doi = 10.2307/2324361, jstor=2324361, url = https://ecommons.cornell.edu/bitstream/1813/7113/1/92-1273.pdf, hdl = 1813/7113, hdl-access = free
Theorems in complex analysis
Lemmas in mathematical analysis