Generic property
Encyclopedia
In mathematics
, properties that hold for "typical" examples are called generic properties. For instance, a generic property of a class of functions is one that is true of "almost all" of those functions, as in the statements, "A generic polynomial
does not have a root at zero," or "A generic matrix
is invertible." As another example, a generic property of a space is a property that holds at "almost all" points of the space, as in the statement, "If f : M → N is a smooth function between smooth manifolds, then a generic point of N is not a critical value of f." (This is by Sard's theorem.)
There are many different notions of "generic" (what is meant by "almost all") in mathematics, with corresponding dual notions
of "almost none" (negligible set
); the two main classes are:
, meaning "with probability 1", with the dual concept being null set
, meaning "with probability 0".
for "with probability 1". For example, the Law of large numbers
states that the sample mean converges almost surely to the population mean.
, one uses the term almost all
to mean cofinite
(all but finitely many), cocountable
(all but countably many), for sufficiently large numbers, or, sometimes, asymptotically almost surely. The concept is particularly important in the study of random graph
s.
and algebraic geometry
, a generic property is one that holds on a dense
open set, or more generally on a residual set (a countable intersection of dense open sets), with the dual concept being a closed nowhere dense set
, or more generally a meagre set
(a countable union of nowhere dense closed sets).
However, density alone is not sufficient to characterize a generic property. This can be seen even in the real numbers, where both the rational numbers and their complement, the irrational numbers, are dense. Since it does not make sense to say that both a set and its complement exhibit typical behavior, both the rationals and irrationals cannot be examples of sets large enough to be typical. Consequently we rely on the stronger definition above which implies that the irrationals are typical and the rationals are not.
For applications, if a property holds on a residual set, it may not hold for every point, but perturbing it slightly will generally land one inside the residual set (by nowhere density of the components of the meagre set), and these are thus the most important case to address in theorems and algorithms.
if the set holding this property contains a residual subset in the Cr topology
. Here Cr is the function space
whose members are continuous functions with r continuous derivatives from a manifold M to a manifold N.
The space Cr(M, N), of Cr mappings between M and N, is a Baire space
, hence any residual set is dense
. This property of the function space is what makes generic properties typical.
X is said to be true generically if it holds except on a proper Zariski-closed
subset of X, in other words, if it holds on a non-empty Zariski-open subset. By the definition of algebraic varieties, they are irreducible spaces, hence any non-empty open set is dense.
For example, by the Jacobian criterion for regularity, a generic point of a variety over a field of characteristic zero is smooth. (This statement is known as generic smoothness.) This is true because the Jacobian criterion can be used to find equations for the points which are not smooth: They are exactly the points where the Jacobian matrix of a point of X does not have full rank. In characteristic zero, these equations are non-trivial, so they cannot be true for every point in the variety. Consequently, the set of all non-regular points of X is a proper Zariski-closed subset of X.
Here is another example. Let f : X → Y be a regular map between two algebraic varieties. For every point y of Y, consider the dimension of the fiber of f over y, that is, dim f−1(y). Generically, this number is constant. It is not necessarily constant everywhere. If, say, X is the blowup of Y at a point and f is the natural projection, then the relative dimension of f is zero except at the point which is blown up, where it is dim Y - 1.
Some properties are said to hold very generically. Frequently this means that the ground field
is uncountable and that the property is true except on a countable union of proper Zariski-closed subsets (i.e., the property holds on a dense Gδ set). For instance, this notion of very generic occurs when considering rational connectedness. However, other definitions of very generic can and do occur in other contexts.
, one formalizes the notion of a generic property by adding additional points for each subvariety, called the "generic point
" of the subvariety. Then a generic property is a property of the generic point. For any reasonable property, it turns out that the property is true generically on the subvariety (in the sense of being true on an open dense subset) if and only if the property is true at the generic point. Such results are frequently proved using the methods of limit
s of affine schemes developed in EGA
IV 8.
– points are in general position if they satisfy no more equations than are necessary. For instance, no 3 points are collinear
. This is a notion of a generic property of a configuration space
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, properties that hold for "typical" examples are called generic properties. For instance, a generic property of a class of functions is one that is true of "almost all" of those functions, as in the statements, "A generic polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
does not have a root at zero," or "A generic matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
is invertible." As another example, a generic property of a space is a property that holds at "almost all" points of the space, as in the statement, "If f : M → N is a smooth function between smooth manifolds, then a generic point of N is not a critical value of f." (This is by Sard's theorem.)
There are many different notions of "generic" (what is meant by "almost all") in mathematics, with corresponding dual notions
Duality (mathematics)
In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have...
of "almost none" (negligible set
Negligible set
In mathematics, a negligible set is a set that is small enough that it can be ignored for some purpose.As common examples, finite sets can be ignored when studying the limit of a sequence, and null sets can be ignored when studying the integral of a measurable function.Negligible sets define...
); the two main classes are:
- In measure theory, a generic property is one that holds almost everywhereAlmost everywhereIn measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
, meaning "with probability 1", with the dual concept being null setNull setIn mathematics, a null set is a set that is negligible in some sense. For different applications, the meaning of "negligible" varies. In measure theory, any set of measure 0 is called a null set...
, meaning "with probability 0". - In topologyTopologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
and algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, a generic property is one that holds on a denseDense setIn topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
open setOpen setThe concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
, or more generally on a residual set, with the dual concept being a nowhere dense setNowhere dense setIn mathematics, a nowhere dense set in a topological space is a set whose closure has empty interior. The order of operations is important. For example, the set of rational numbers, as a subset of R has the property that the closure of the interior is empty, but it is not nowhere dense; in fact it...
, or more generally a meagre setMeagre setIn the mathematical fields of general topology and descriptive set theory, a meagre set is a set that, considered as a subset of a topological space, is in a precise sense small or negligible...
.
Definitions: measure theory
In measure theory, a generic property is one that holds almost everywhereAlmost everywhere
In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
, meaning "with probability 1", with the dual concept being null set
Null set
In mathematics, a null set is a set that is negligible in some sense. For different applications, the meaning of "negligible" varies. In measure theory, any set of measure 0 is called a null set...
, meaning "with probability 0".
Probability
In probability, one speaks of trials instead of a space, so one instead says that a property holds almost surelyAlmost surely
In probability theory, one says that an event happens almost surely if it happens with probability one. The concept is analogous to the concept of "almost everywhere" in measure theory...
for "with probability 1". For example, the Law of large numbers
Law of large numbers
In probability theory, the law of large numbers is a theorem that describes the result of performing the same experiment a large number of times...
states that the sample mean converges almost surely to the population mean.
Discrete mathematics
In discrete mathematicsDiscrete mathematics
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not...
, one uses the term almost all
Almost all
In mathematics, the phrase "almost all" has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set" ; see almost....
to mean cofinite
Cofinite
In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X...
(all but finitely many), cocountable
Cocountable
In mathematics, a cocountable subset of a set X is a subset Y whose complement in X is a countable set. In other words, Y contains all but countably many elements of X. For example, the irrational numbers are a cocountable subset of the reals...
(all but countably many), for sufficiently large numbers, or, sometimes, asymptotically almost surely. The concept is particularly important in the study of random graph
Random graph
In mathematics, a random graph is a graph that is generated by some random process. The theory of random graphs lies at the intersection between graph theory and probability theory, and studies the properties of typical random graphs.-Random graph models:...
s.
Definitions: topology
In topologyTopology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
and algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, a generic property is one that holds on a dense
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
open set, or more generally on a residual set (a countable intersection of dense open sets), with the dual concept being a closed nowhere dense set
Nowhere dense set
In mathematics, a nowhere dense set in a topological space is a set whose closure has empty interior. The order of operations is important. For example, the set of rational numbers, as a subset of R has the property that the closure of the interior is empty, but it is not nowhere dense; in fact it...
, or more generally a meagre set
Meagre set
In the mathematical fields of general topology and descriptive set theory, a meagre set is a set that, considered as a subset of a topological space, is in a precise sense small or negligible...
(a countable union of nowhere dense closed sets).
However, density alone is not sufficient to characterize a generic property. This can be seen even in the real numbers, where both the rational numbers and their complement, the irrational numbers, are dense. Since it does not make sense to say that both a set and its complement exhibit typical behavior, both the rationals and irrationals cannot be examples of sets large enough to be typical. Consequently we rely on the stronger definition above which implies that the irrationals are typical and the rationals are not.
For applications, if a property holds on a residual set, it may not hold for every point, but perturbing it slightly will generally land one inside the residual set (by nowhere density of the components of the meagre set), and these are thus the most important case to address in theorems and algorithms.
Function spaces
A property is generic in CrFunction space
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...
if the set holding this property contains a residual subset in the Cr topology
Whitney topologies
In mathematics, and especially differential topology, functional analysis and singularity theory, the Whitney topologies are a countably infinite family of topologies defined on the set of smooth mappings between two smooth manifolds. They are named after the American mathematician Hassler...
. Here Cr is the function space
Function space
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...
whose members are continuous functions with r continuous derivatives from a manifold M to a manifold N.
The space Cr(M, N), of Cr mappings between M and N, is a Baire space
Baire space
In mathematics, a Baire space is a topological space which, intuitively speaking, is very large and has "enough" points for certain limit processes. It is named in honor of René-Louis Baire who introduced the concept.- Motivation :...
, hence any residual set is dense
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
. This property of the function space is what makes generic properties typical.
Algebraic varieties
A property of an algebraic varietyAlgebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
X is said to be true generically if it holds except on a proper Zariski-closed
Zariski topology
In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
subset of X, in other words, if it holds on a non-empty Zariski-open subset. By the definition of algebraic varieties, they are irreducible spaces, hence any non-empty open set is dense.
For example, by the Jacobian criterion for regularity, a generic point of a variety over a field of characteristic zero is smooth. (This statement is known as generic smoothness.) This is true because the Jacobian criterion can be used to find equations for the points which are not smooth: They are exactly the points where the Jacobian matrix of a point of X does not have full rank. In characteristic zero, these equations are non-trivial, so they cannot be true for every point in the variety. Consequently, the set of all non-regular points of X is a proper Zariski-closed subset of X.
Here is another example. Let f : X → Y be a regular map between two algebraic varieties. For every point y of Y, consider the dimension of the fiber of f over y, that is, dim f−1(y). Generically, this number is constant. It is not necessarily constant everywhere. If, say, X is the blowup of Y at a point and f is the natural projection, then the relative dimension of f is zero except at the point which is blown up, where it is dim Y - 1.
Some properties are said to hold very generically. Frequently this means that the ground field
Ground field
In mathematics, a ground field is a field K fixed at the beginning of the discussion. It is used in various areas of algebra: for example in linear algebra where the concept of a vector space may be developed over any field; and in algebraic geometry, where in the foundational developments of André...
is uncountable and that the property is true except on a countable union of proper Zariski-closed subsets (i.e., the property holds on a dense Gδ set). For instance, this notion of very generic occurs when considering rational connectedness. However, other definitions of very generic can and do occur in other contexts.
Generic point
In scheme theoryScheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
, one formalizes the notion of a generic property by adding additional points for each subvariety, called the "generic point
Generic point
In mathematics, in the fields general topology and particularly of algebraic geometry, a generic point P of a topological space X is an algebraic way of capturing the notion of a generic property: a generic property is a property of the generic point.- Definition and motivation :A generic point of...
" of the subvariety. Then a generic property is a property of the generic point. For any reasonable property, it turns out that the property is true generically on the subvariety (in the sense of being true on an open dense subset) if and only if the property is true at the generic point. Such results are frequently proved using the methods of limit
Limit (category theory)
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse limits....
s of affine schemes developed in EGA
Éléments de géométrie algébrique
The Éléments de géométrie algébrique by Alexander Grothendieck , or EGA for short, is a rigorous treatise, in French, on algebraic geometry that was published from 1960 through 1967 by the Institut des Hautes Études Scientifiques...
IV 8.
General position
A related concept in algebraic geometry is general positionGeneral position
In algebraic geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the general case situation, as opposed to some more special or coincidental cases that are possible...
– points are in general position if they satisfy no more equations than are necessary. For instance, no 3 points are collinear
Line (geometry)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
. This is a notion of a generic property of a configuration space
Configuration space
- Configuration space in physics :In classical mechanics, the configuration space is the space of possible positions that a physical system may attain, possibly subject to external constraints...
.
Genericity results
- Sard's theorem: If
is a smooth function between smooth manifolds, then a generic point of N is not a critical value of f – critical values of f are a null setNull setIn mathematics, a null set is a set that is negligible in some sense. For different applications, the meaning of "negligible" varies. In measure theory, any set of measure 0 is called a null set...
in N. - Jacobian criterion / generic smoothness: A generic point of a variety over a field of characteristic zero is smooth.