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Finitely generated extension field of positive transcendence degree
function field (often abbreviated as function field) of n {\displaystyle n} variables over a field k {\displaystyle k} is a finitely generated field extension
Algebraic_function_field
Topics referred to by the same term
Function field may refer to: Function field of an algebraic variety Function field (scheme theory) Algebraic function field Function field sieve Function
Function_field
Ratio of polynomial functions
rational functions over a field K is a field, the field of fractions of the ring of the polynomial functions over K. A function f {\displaystyle f} is called
Rational_function
Distance from a point to the boundary of a set
In mathematics and its applications, the signed distance function or signed distance field (SDF) is the orthogonal distance of a given point x to the
Signed_distance_function
Topics referred to by the same term
Party or function, a social event Function Drinks, an American beverage company Function Health, an American health technology company Function field (disambiguation)
Function
Algebraic structure with addition, multiplication, and division
such as fields of rational functions, algebraic function fields, algebraic number fields, finite fields, and p-adic fields are commonly used and studied
Field_(mathematics)
The sheaf of rational functions KX of a scheme X is the generalization to scheme theory of the notion of function field of an algebraic variety in classical
Function field (scheme theory)
Function_field_(scheme_theory)
Mathematical concept in algebraic geometry
algebraic geometry, the function field of an algebraic variety V consists of objects that are interpreted as rational functions on V. In classical algebraic
Function field of an algebraic variety
Function_field_of_an_algebraic_variety
Assignment of numbers to points in space
In mathematics and physics, a scalar field is a function associating a single[dubious – discuss] number to each point in a region of space – possibly
Scalar_field
Expectation value of time-ordered quantum operators
In quantum field theory, correlation functions, often referred to as correlators or Green's functions, are vacuum expectation values of time-ordered products
Correlation function (quantum field theory)
Correlation_function_(quantum_field_theory)
Function that encodes the dependence of a coupling parameter on the energy scale
In theoretical physics, specifically quantum field theory, a beta function or Gell-Mann–Low function, β(g), encodes the dependence of a coupling parameter
Beta_function_(physics)
Mathematical function
algebraic function. Basic examples of algebraic functions are polynomial functions, rational functions, the nth root function, and functions obtained from
Algebraic_function
Mathematical concept
kinds of global fields: Algebraic number field: A finite extension of Q {\displaystyle \mathbb {Q} } Global function field: The function field of an irreducible
Global_field
function of a number field Duursma zeta function of error-correcting codes Epstein zeta function of a quadratic form Goss zeta function of a function
List_of_zeta_functions
Association of one output to each input
mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the
Function_(mathematics)
Algorithm to solve the discrete logarithm problem
mathematics, the Function Field Sieve is one of the most efficient algorithms to solve the Discrete Logarithm Problem (DLP) in a finite field. It has heuristic
Function_field_sieve
Class of mathematical function
mathematical field of complex analysis, a meromorphic function on an open subset D {\displaystyle D} of the complex plane is a function that is holomorphic
Meromorphic_function
Generating function for quantum correlation functions
In quantum field theory, partition functions are generating functionals for correlation functions, making them key objects of study in the path integral
Partition function (quantum field theory)
Partition_function_(quantum_field_theory)
Curve defined as zeros of polynomials
over a field F are categorically equivalent to algebraic function fields in one variable over F. Such an algebraic function field is a field extension
Algebraic_curve
over the field of algebraic numbers is a global height function with local contributions coming from Fubini–Study metrics on the Archimedean fields and the
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
Construction of a larger algebraic field by "adding elements" to a smaller field
function defined on M. More generally, given an algebraic variety V over some field K, the function field K(V), consisting of the rational functions defined
Field_extension
Algebraic variety
means such a function field has a single transcendental function as generator: for example the j-function generates the function field of X(1) = PSL(2
Modular_curve
Algebraic structure with addition and multiplication
is associated its function field. The points of an algebraic variety correspond to valuation rings contained in the function field and containing the
Ring_(mathematics)
Extension of the factorial function
and other formulas in the fields of probability, statistics, analytic number theory, and combinatorics. The gamma function can be seen as a solution to
Gamma_function
Abstract algebra concept
] {\displaystyle k[t]} is the rational function field k ( t ) {\displaystyle k(t)} . For any field k, the field of fractions of the formal power series
Field_of_fractions
Concept in mathematics
In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called
Morphism of algebraic varieties
Morphism_of_algebraic_varieties
Finite extension of the rationals
function fields, the local fields are completions of the local rings at all points of the curve for function fields. Many results valid for function fields
Algebraic_number_field
Assignment of a vector to each point in a subset of Euclidean space
fields are one kind of tensor field. Given a subset S of Rn, a vector field is represented by a vector-valued function V: S → Rn in standard Cartesian
Vector_field
Theoretical object in mathematics
fields starts with a curve C over a finite field k, which comes equipped with a function field F, which is a field extension of k. Each such function
Field_with_one_element
number field or a global function field). It is used to encode ramification data for abelian extensions of a global field. Let K be a global field with
Modulus (algebraic number theory)
Modulus_(algebraic_number_theory)
Concept in number theory
curves. Let K {\displaystyle K} be a global field, meaning either a number field or a global function field. Let v {\displaystyle v} run over the places
Adele_ring
Method of solution to differential equations
In mathematics, a Green's function (or Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with
Green's_function
Branch of number theory
algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring
Algebraic_number_theory
Concept in abstract algebra
\mathbb {R} [x],\,g(0)\neq 0\},} considered as a subring of the field of rational functions R ( x ) {\displaystyle \mathbb {R} (x)} . R {\displaystyle R}
Discrete_valuation_ring
S-shaped curve
networks. There are various generalizations, depending on the field. The logistic function was introduced in a series of three papers by Pierre François
Logistic_function
Generalization of the Riemann zeta function for algebraic number fields
Dedekind zeta function of an algebraic number field K, usually denoted ζ K ( s ) {\displaystyle \zeta _{K}(s)} , is an analytic function that represents
Dedekind_zeta_function
Algebraic curve
characteristic of the ground field is not 2, one can take h(x) = 0). A hyperelliptic function is an element of the function field of such a curve, or of the
Hyperelliptic_curve
Type of mathematical function
elementary function is a function of a single variable (real or complex) that is typically encountered by beginners. The basic elementary functions are polynomial
Elementary_function
Mathematical function having a characteristic S-shaped curve or sigmoid curve
sigmoid function is the logistic function. Other sigmoid functions are given in the Examples section. In some fields, most notably in the context of artificial
Sigmoid_function
Mathematical conjecture about zeros of L-functions
the algebraic function field case (not the number field case). Global L-functions can be associated to elliptic curves, number fields (in which case
Generalized Riemann hypothesis
Generalized_Riemann_hypothesis
Abelian group
{\displaystyle A|_{k(t)}} (the pullback of A {\displaystyle A} to the function field k ( t ) = k ( P 1 ) {\displaystyle k(t)=k(\mathbb {P} ^{1})} ) by a
Mordell–Weil_group
Type of function in mathematics
an analytic function is a function that is locally represented by a convergent power series. More precisely, a real or complex function is analytic at
Analytic_function
Conjecture on zeros of the zeta function
for curves over finite fields, which was proved by André Weil. The Riemann zeta function ζ {\displaystyle \zeta } is a function whose argument may be any
Riemann_hypothesis
Branch of mathematics
such as the field of rational numbers, number fields, finite fields, function fields, and p-adic fields. A large part of singularity theory is devoted
Algebraic_geometry
Family of solutions to related differential equations
Bessel functions are a class of special functions that commonly appear in problems involving wave motion, heat conduction, and other physical phenomena
Bessel_function
Type of energy
close to the solid to be influenced by ambient electric fields in the vacuum. The work function is not a characteristic of a bulk material, but rather
Work_function
One-dimensional complex manifold
meromorphic function on T {\displaystyle T} . This function and its derivative ℘ τ ′ ( z ) {\displaystyle \wp '_{\tau }(z)} generate the function field of T
Riemann_surface
Concept in algebraic geometry
nonsingular model for the function field of a variety X, in other words a complete non-singular variety X′ with the same function field. In practice it is more
Resolution_of_singularities
Algebraic variety
over a given field K, which is birationally equivalent to a projective space of some dimension over K. This means that its function field is isomorphic
Rational_variety
Class of periodic mathematical functions
In the mathematical field of complex analysis, elliptic functions are special kinds of meromorphic functions, that satisfy two periodicity conditions.
Elliptic_function
Vector function in optics
leaving a four-dimensional function variously termed the photic field, the 4D light field or lumigraph. Formally, the field is defined as radiance along
Light_field
Class of mathematical functions
functions that are doubly periodic. A ℘-function together with its derivative can be used to parameterize elliptic curves and they generate the field
Weierstrass_elliptic_function
Completes the Langlands program for general linear groups over algebraic function fields
completes the Langlands program for general linear groups over algebraic function fields, by giving a correspondence between automorphic forms on these groups
Lafforgue's_theorem
Field extension that is not algebraic
transcendence degree of its function field. Also, global function fields are transcendental extensions of degree one of a finite field, and play in number theory
Transcendental_extension
Analytic function that does not satisfy a polynomial equation
mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable
Transcendental_function
Mathematical description of quantum state
In quantum mechanics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common
Wave_function
mathematics, the local zeta function Z(V, s) (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as Z ( V , s ) =
Local_zeta_function
Unsolved problem in mathematics
conjecture comes from Srinivasa Ramanujan, who proposed it for Ramanujan tau function, and Hans Petersson, who generalized it for coefficients of modular forms
Ramanujan–Petersson conjecture
Ramanujan–Petersson_conjecture
Mathematics award
geometric interpretations for the higher derivatives of L-functions in the function field case." Wei Zhang – "For deep work on the global Gan-Gross-Prasad
Breakthrough Prize in Mathematics
Breakthrough_Prize_in_Mathematics
Linear map or polynomial function of degree one
the term linear function refers to two distinct but related notions: In calculus and related areas, a linear function is a function whose graph is a
Linear_function
Function in quantum field theory showing probability amplitudes of moving particles
In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one
Propagator
Sigmoid shape special function
mathematics, the error function (also called the Gauss error function), often denoted by e r f {\displaystyle \mathbf {erf} } , is the function erf ( z ) = 2
Error_function
Definite integral of a scalar or vector field along a path
plane. The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points
Line_integral
Mathematical theory
correspondence obtained by replacing the number fields appearing in the original number theoretic version by function fields and applying techniques from algebraic
Geometric Langlands correspondence
Geometric_Langlands_correspondence
Symmetric holomorphic function
fractional linear action of the congruence group Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the modular
Modular_lambda_function
Concept in mathematics
over a ring of functions on a curve over a finite field, generalizing the Carlitz module. Loosely speaking, they provide a function field analogue of complex
Drinfeld_module
Correlation as a function of distance
stochastic processes they are applied to. In quantum field theory there are correlation functions over quantum distributions. For possibly distinct random
Correlation_function
Mathematical function, used to describe magnetization
Langevin function is derived using statistical mechanics and describes how magnetic dipoles are aligned by an applied field. The Brillouin function was developed
Brillouin and Langevin functions
Brillouin_and_Langevin_functions
Study of classical optics using Fourier transforms
Mathematically, a real-valued component of a vector field describing a wave is represented by a scalar wave function u that depends on both space and time: u =
Fourier_optics
Generalized function whose value is zero everywhere except at zero
Dirac delta function (or δ {\displaystyle {\boldsymbol {\delta }}} distribution), also known as the unit impulse, is a generalized function on the real
Dirac_delta_function
mathematics, Tsen's theorem states that a function field K of an algebraic curve over an algebraically closed field is quasi-algebraically closed (i.e., C1)
Tsen's_theorem
Conjectures connecting number theory and geometry
fields (with subcases corresponding to number fields or function fields). Analogues for finite fields. More general fields, such as function fields over
Langlands_program
Theoretical framework in physics
classical field is a function of spatial and time coordinates. Examples include the gravitational field g(x, t) in Newtonian gravity and the electric field E(x
Quantum_field_theory
Analytic function in mathematics
The Riemann zeta function or Euler–Riemann zeta function, denoted by the lowercase Greek letter ζ (zeta), is a mathematical function of a complex variable
Riemann_zeta_function
Kind of partial function between algebraic varieties
equivalent function fields. That is, every rational function f : X → P 1 {\displaystyle f:X\to \mathbb {P} ^{1}} can be restricted to a rational function U →
Rational_mapping
Type of function in linear algebra
below, that also goes by the name "sublinear function." Let X {\displaystyle X} be a vector space over a field K , {\displaystyle \mathbb {K} ,} where K
Sublinear_function
Sum of elements on the main diagonal
linear vector fields. Given a matrix A, define a vector field F on Rn by F(x) = Ax. The components of this vector field are linear functions (given by the
Trace_(linear_algebra)
Programming construct
usually with the same syntax (a function parameter that can also be a function). In some languages, particularly C++, function objects are often called functors
Function_object
Function valued in a vector space; typically a real or complex one
A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional
Vector-valued_function
Function with a repeating pattern
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which are used to describe waves
Periodic_function
(Mathematical) ring with a unique maximal ideal
"local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of algebraic number fields examined at a particular place, or
Local_ring
Mathematics award
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical
Fields_Medal
Field theory is the branch of algebra that studies fields
is a function field of n variables over a finite field of characteristic p > 0, then its imperfect degree is pn. Algebraically closed field A field F is
Glossary_of_field_theory
Conjecture in algebraic geometry
projective curve over a finite field. Suppose that the generic fiber F of f, which is a curve over the function field k(C), is smooth over k(C). Then
Tate_conjecture
Functions in mathematics
the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R {\displaystyle f:U\to \mathbb {R} }
Harmonic_function
Function with a multiplicative scaling behaviour
to functions whose domain and codomain are vector spaces over a field F: a function f : V → W {\displaystyle f:V\to W} between two F-vector spaces is
Homogeneous_function
Best results achieved to date
variant of the medium-sized base field function field sieve, for binary fields, to compute a discrete logarithm in a field of 21971 elements. In order to
Discrete_logarithm_records
Order-preserving mathematical function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept
Monotonic_function
Relation between genus, degree, and dimension of function spaces over surfaces
transferred to function fields with finite base field. Actually, his proof of the Riemann–Roch theorem works for arbitrary perfect base fields, not necessarily
Riemann–Roch_theorem
Used to count, measure, and label
elements of an algebraic function field over a finite field and algebraic numbers have many similar properties (see Function field analogy). Therefore, they
Number
Curves of genus > 1 over the rationals have only finitely many rational points
conjectures have been put forth by Paul Vojta. The Mordell conjecture for function fields was proved by Yuri Ivanovich Manin and by Hans Grauert. In 1990, Robert
Faltings'_theorem
Mapping arbitrary data to fixed-size values
A hash function is any function that can be used to map data of arbitrary size to fixed-size values, though there are some hash functions that support
Hash_function
plane for function fields, introduced by Drinfeld (1976). It is defined to be the set difference P1(C) \ P1(F∞), where F is a function field of a curve
Drinfeld_upper_half_plane
Continuous function whose value increases to infinity
In constrained optimization, a field of mathematics, a barrier function is a continuous function whose value increases to infinity as its argument approaches
Barrier_function
Special function in the physical sciences
mathematics, the Airy function (or Airy function of the first kind) A i ( x ) {\displaystyle \mathbf {Ai({\boldsymbol {x}})} } is a special function named after
Airy_function
larger class of L-functions (Artin-Hecke L-functions); and to the global function field case. Here the inclusion of Artin L-functions, in particular, implicates
Weil's_criterion
Multivalued function in mathematics
In mathematics, the Lambert W function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse
Lambert_W_function
Set of functions between two fixed sets
equipped with possibly some extra structure. Let F be a field and let X be any set. The functions X → F can be given the structure of a vector space over
Function_space
In the field of mathematics, the Goss zeta function, named after David Goss, is an analogue of the Riemann zeta function for function fields. Sheats (1998)
Goss_zeta_function
Mathematical function, denoted exp(x) or e^x
In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. It is denoted
Exponential_function
FUNCTION FIELD
FUNCTION FIELD
Boy/Male
Australian, British, English
A Field
Surname or Lastname
English
English : nickname from the animal, Middle English catte ‘cat’. The word is found in similar forms in most European languages from very early times (e.g. Gaelic cath, Slavic kotu). Domestic cats were unknown in Europe in classical times, when weasels fulfilled many of their functions, for example in hunting rodents. They seem to have come from Egypt, where they were regarded as sacred animals.English : from a medieval female personal name, a short form of Catherine.Variant spelling of German and Dutch Katt.
Surname or Lastname
English
English : habitational name from any of various places, such as Merryfield in Devon and Cornwall or Mirfield in West Yorkshire, all named with the Old English elements myrige ‘pleasant’ + feld ‘pasture’, ‘open country’ (see Field).
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Girl/Female
Hindu, Indian
Fraction of the Cosmos
Surname or Lastname
South German
South German : occupational name for an official in charge of the legal auction of property confiscated in default of a fine; such a sale was known in Middle High German as a gant (from Italian incanto, a derivative of Late Latin inquantare ‘to auction’, from the phrase In quantum? ‘To how much (is the price raised)?’).German : metonymic occupational name for a cooper, from Middle High German ganter, kanter ‘barrel rack’.German : variant of Gander 3.English : occupational name for a glover, from Old French gantier, an agent derivative of gant ‘glove’ (see Gant).
Girl/Female
Afghan, Arabic, Australian, Indian, Muslim
Fiction; Romance; Story
Girl/Female
Tamil
Ankshika | அஂகà¯à®·à¯€à®•à®¾
It’s derived from the root word - anksh that means a fraction. Ankshika means the fraction of the cosmos
Ankshika | அஂகà¯à®·à¯€à®•à®¾
Biblical
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Surname or Lastname
English (chiefly West Midlands and northern England)
English (chiefly West Midlands and northern England) : topographic name for someone who lived in a house (Middle English hous) in open pasture land (see Field). Reaney draws attention to the form de Felhouse (Staffordshire 1332), and suggests that this may have become Fellows.
Boy/Male
American, British, English
Lives in the Field
Surname or Lastname
English
English : variant of Field, from the dative plural of Old English feld ‘open country’.
Surname or Lastname
English
English : topographic name for someone who lived by a watercourse or road junction, Old English gelǣt, or a habitational name from Leat in Devon, or The Leete in Essex, named with this element.
Surname or Lastname
English (chiefly Gloucestershire and Worcestershire)
English (chiefly Gloucestershire and Worcestershire) : variant of Millward.French (northern) : from a Germanic personal name composed of the elements mil ‘good’, ‘gracious’ + hard ‘hardy’, ‘brave’, ‘strong’.Southern French : from a variant spelling of Occitan milhar ‘millet field’ (from mil ‘millet’).
Boy/Male
Indian
Friction
Girl/Female
Bengali, Indian
Fraction of Time
Girl/Female
Indian
It’s derived from the root word - anksh that means a fraction. Ankshika means the fraction of the cosmos
Boy/Male
English
In the field.
Surname or Lastname
English
English : topographic name for someone who lived by the gates of a medieval walled town. The Middle English singular gate is from the Old English plural, gatu, of geat ‘gate’ (see Yates). Since medieval gates were normally arranged in pairs, fastened in the center, the Old English plural came to function as a singular, and a new Middle English plural ending in -s was formed. In some cases the name may refer specifically to the Sussex place Eastergate (i.e. ‘eastern gate’), known also as Gates in the 13th and 14th centuries, when surnames were being acquired.Americanized spelling of German Götz (see Goetz).Translated form of French Barrière (see Barriere).In New England, Gates was the preferred English version of the name of an extensive French family, called Barrière dit Langevin.
Boy/Male
French Greek
Cyrano de Bergerac was a seventeenth-century soldier and science-fiction writer.
FUNCTION FIELD
FUNCTION FIELD
Boy/Male
Arabic, Hindu, Indian, Muslim
Safeguard
Female
Italian
Feminine form of Italian Eulalio, EULALIA means "well-spoken."Â
Girl/Female
Greek American
Goddess of discord.
Surname or Lastname
English
English : habitational name from any of various minor places named Duncombe, probably from Old English dūn ‘hill’ + Old English cumb ‘valley’.English : probably a variant of Duncan.
Boy/Male
Muslim
Civilization
Boy/Male
Tamil
Ramadeep | ராமாதீப
Lord Rama
Surname or Lastname
English
English : occupational name for a man responsible for the maintenance and provision of saddle-horses (see Palfrey).
Girl/Female
Hindu, Indian
Prayer of Sai Baba
Girl/Female
Hindu
Prayer, Request, Humility
Boy/Male
Arabic
River; Pilgrim to Makkah
FUNCTION FIELD
FUNCTION FIELD
FUNCTION FIELD
FUNCTION FIELD
FUNCTION FIELD
v. t.
To supply with an organ or organs having a special function or functions.
a.
Pertaining to, or connected with, a function or duty; official.
v. t.
To sell by auction.
n.
The act of joining, or the state of being joined; union; combination; coalition; as, the junction of two armies or detachments; the junction of paths.
v. t.
To separate by means of, or to subject to, fractional distillation or crystallization; to fractionate; -- frequently used with out; as, to fraction out a certain grade of oil from pretroleum.
v. i.
Alt. of Functionate
n.
The natural or assigned action of any power or faculty, as of the soul, or of the intellect; the exertion of an energy of some determinate kind.
n.
The act of feigning, inventing, or imagining; as, by a mere fiction of the mind.
v. t.
The act of uniting, or the state of being united; junction.
n.
The office, duties, or functions of a minister, servant, or agent; ecclesiastical, executive, or ambassadorial function or profession.
a.
Pertaining to the function of an organ or part, or to the functions in general.
n.
The things sold by auction or put up to auction.
n.
The appropriate action of any special organ or part of an animal or vegetable organism; as, the function of the heart or the limbs; the function of leaves, sap, roots, etc.; life is the sum of the functions of the various organs and parts of the body.
v. t.
To give sanction to; to ratify; to confirm; to approve.
n.
The course of action which peculiarly pertains to any public officer in church or state; the activity appropriate to any business or profession.
n.
The act of anointing, or the state of being anointed; unction; specifically (Med.), the rubbing of ointments into the pores of the skin, by which medicinal agents contained in them, such as mercury, iodide of potash, etc., are absorbed.
n.
A derived function; a function obtained from a given function by a certain algebraic process.
n.
A quantity so connected with another quantity, that if any alteration be made in the latter there will be a consequent alteration in the former. Each quantity is said to be a function of the other. Thus, the circumference of a circle is a function of the diameter. If x be a symbol to which different numerical values can be assigned, such expressions as x2, 3x, Log. x, and Sin. x, are all functions of x.
n.
The place or point of union, meeting, or junction; specifically, the place where two or more lines of railway meet or cross.
n.
The act of anointing, smearing, or rubbing with an unguent, oil, or ointment, especially for medical purposes, or as a symbol of consecration; as, mercurial unction.