In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, ramification is 'branching out', in the way that the
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...
function, for
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as when a
covering map A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
degenerates at a point of a space, with some collapsing of the fibers of the mapping.
In complex analysis
In
complex analysis, the basic model can be taken as the ''z'' → ''z''
''n'' mapping in the complex plane, near ''z'' = 0. This is the standard local picture in
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
theory, of ramification of order ''n''. It occurs for example in the
Riemann–Hurwitz formula In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ''ramified covering'' of the other. It therefore connects ramif ...
for the effect of mappings on the
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
. See also
branch point
In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point, ...
.
In algebraic topology
In a covering map the
Euler–Poincaré characteristic should multiply by the number of sheets; ramification can therefore be detected by some dropping from that. The ''z'' → ''z''
''n'' mapping shows this as a local pattern: if we exclude 0, looking at 0 < , ''z'', < 1 say, we have (from the
homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defor ...
point of view) the
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
mapped to itself by the ''n''-th power map (Euler–Poincaré characteristic 0), but with the whole
disk the Euler–Poincaré characteristic is 1, ''n'' – 1 being the 'lost' points as the ''n'' sheets come together at ''z'' = 0.
In geometric terms, ramification is something that happens in ''codimension two'' (like
knot theory, and
monodromy); since ''real'' codimension two is ''complex'' codimension one, the local complex example sets the pattern for higher-dimensional
complex manifolds. In complex analysis, sheets can't simply fold over along a line (one variable), or codimension one subspace in the general case. The ramification set (branch locus on the base, double point set above) will be two real dimensions lower than the ambient
manifold, and so will not separate it into two 'sides', locally―there will be paths that trace round the branch locus, just as in the example. In
algebraic geometry over any
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
, by analogy, it also happens in algebraic codimension one.
In algebraic number theory
In algebraic extensions of
Ramification in
algebraic number theory means a prime ideal factoring in an extension so as to give some repeated prime ideal factors. Namely, let
be the
ring of integers of an
algebraic number field , and
a
prime ideal of
. For a field extension
we can consider the
ring of integers
(which is the
integral closure of
in
), and the ideal
of
. This ideal may or may not be prime, but for finite