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երեն bahasa indonesia italiano 日本語 қазақша 한국어 кыргызча मगही नेपाली nederlands norsk nynorsk norsk bokmål polski português română русский தமிழ் українська اردو tiếng việt 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikimedia commons wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia mathematical function in mathematics an algebraic function is a function that satisfies a polynomial equation thus an equation of the following form holds a n x f x n a n 1 x f x n 1 a 1 x f x a 0 x 0 displaystyle a_ n x f x n a_ n 1 x f x n 1 cdots a_ 1 x f x a_ 0 x 0 where the a k x displaystyle a_ k x are polynomials not all zero basic examples of algebraic functions are polynomial functions rational functions the nth root function and functions obtained from these by composition and algebraic operations addition multiplication subtraction and division thus an example of an algebraic function is the function f x 1 x 2 displaystyle f x sqrt 1 x 2 for 1 x 1 displaystyle 1 x 1 whose graph is the top half of the standard unit circle this function satisfies x 2 f x 2 1 0 displaystyle x 2 f x 2 1 0 algebraic functions are contrasted with transcendental functions such as the exponential function logarithm and the trigonometric functions algebraic functions are usually treated more generally as multivalued functions the example of x 2 y 2 1 0 displaystyle x 2 y 2 1 0 illustrates this since it includes both the top semicircle y 1 x 2 displaystyle y sqrt 1 x 2 and bottom semicircle y 1 x 2 displaystyle y sqrt 1 x 2 in the same package algebraic functions are most often studied over the complex numbers formally an algebraic function over the complex numbers is defined to be a multivalued function y displaystyle y satisfying a polynomial equation p x y 0 displaystyle p x y 0 where p x y displaystyle p x y is an irreducible polynomial of two variables having positive degree in y displaystyle y and complex coefficients 1 the example of x 2 y 2 1 0 displaystyle x 2 y 2 1 0 can be expressed as having the two single valued branches y 1 x 2 displaystyle y sqrt 1 x 2 and y 1 x 2 displaystyle y sqrt 1 x 2 with branch points where the two branches come together at x 1 displaystyle x pm 1 this particular function can be written using finitely many algebraic operations and extraction of nth roots but this is not generally the case such as with the bring radical over the complex numbers algebraic functions have local holomorphic branches away from finitely many branch points and poles and are naturally studied as meromorphic functions on compact riemann surfaces 2 more generally over a field k displaystyle k an algebraic function in one variable x displaystyle x is defined algebraically as an element algebraic over the rational function field k x displaystyle k x equivalently it satisfies a polynomial equation of positive degree in y displaystyle y a n x y n a n 1 x y n 1 a 0 x 0 displaystyle a_ n x y n a_ n 1 x y n 1 cdots a_ 0 x 0 where the coefficients a i x displaystyle a_ i x are polynomials in x displaystyle x with coefficients in k displaystyle k if the irreducible defining polynomial has degree n displaystyle n in y displaystyle y the algebraic function is said to have degree n displaystyle n an algebraic function in m displaystyle m variables over k displaystyle k is an element algebraic over the field of rational functions k x 1 x m displaystyle k x_ 1 ldots x_ m equivalently it satisfies a polynomial equation p y x 1 x 2 x m 0 displaystyle p y x_ 1 x_ 2 dots x_ m 0 in one variable algebraic functions are closely related to algebraic curves and their function fields in the separable case they may also be studied via finite or ramified covers of the projective line 3 algebraic functions in one variable edit basic examples edit polynomial and rational functions are algebraic a polynomial function y p x displaystyle y p x satisfies y p x 0 displaystyle y p x 0 a rational function y p x q x displaystyle y p x q x satisfies q x y p x 0 displaystyle q x y p x 0 with poles at the zeros of q displaystyle q more generally the n displaystyle n th root of a polynomial or rational function is algebraic since it satisfies an equation such as y n p x 0 displaystyle y n p x 0 many elementary algebraic functions can be obtained from rational functions by algebraic operations and extraction of roots however algebraic functions are more general than functions expressible by radicals by galois theory roots of a general polynomial equation of degree five or higher cannot be expressed by radicals where a local inverse branch of an algebraic function exists it is again algebraic more generally if x displaystyle x and y displaystyle y satisfy a polynomial relation p x y 0 displaystyle p x y 0 then interchanging the roles of x displaystyle x and y displaystyle y gives an algebraic correspondence whose branches include the local inverse branches the solution set p x y 0 displaystyle p x y 0 is an algebraic curve away from exceptional points its local branches may be represented as graphs over the x displaystyle x line the role of complex numbers edit from an algebraic perspective complex numbers enter quite naturally into the study of algebraic functions first of all by the fundamental theorem of algebra the complex numbers are an algebraically closed field hence for each value of x displaystyle x for which p x y displaystyle p x y is a nonconstant polynomial of degree n displaystyle n in y displaystyle y the equation p x y 0 displaystyle p x y 0 has n displaystyle n complex roots counted with multiplicity exceptional values of x displaystyle x such as zeros of the leading coefficient or of the discriminant are responsible for poles multiple roots and branch points a graph of three branches of the algebraic function y where y 3 xy 1 0 over the domain 3 2 2 3 x 50 furthermore even if one is ultimately interested in real algebraic functions there may be no means to express the function in terms of addition multiplication division and taking nth roots without resorting to complex numbers see casus irreducibilis for example consider the algebraic function determined by the equation y 3 x y 1 0 displaystyle y 3 xy 1 0 using the cubic formula we get y 2 x 108 12 81 12 x 3 3 108 12 81 12 x 3 3 6 displaystyle y frac 2x sqrt 3 108 12 sqrt 81 12x 3 frac sqrt 3 108 12 sqrt 81 12x 3 6 for x 3 4 3 displaystyle x leq frac 3 sqrt 3 4 the square root is real and the cubic root is thus well defined providing the unique real root on the other hand for x 3 4 3 displaystyle x frac 3 sqrt 3 4 the square root is not real and one has to choose for the square root either non real square root thus the cubic root has to be chosen among three non real numbers if the same choices are done in the two terms of the formula the three choices for the cubic root provide the three branches shown in the accompanying image it may be proven that there is no way to express this function in terms of nth roots using real numbers only even though the resulting function is real valued on the domain of the graph shown on a more significant theoretical level using complex numbers allows one to use the powerful techniques of complex analysis to discuss algebraic functions in particular the argument principle can be used to show that any algebraic function is in fact an analytic function at least in the multiple valued sense formally let p x y be a complex polynomial in the complex variables x and y suppose that x 0 c is such that the polynomial p x 0 y of y has n distinct zeros we shall show that the algebraic function is analytic in a neighborhood of x 0 choose a system of n non overlapping discs δ i containing each of these zeros then by the argument principle 1 2 π i δ i p y x 0 y p x 0 y d y 1 displaystyle frac 1 2 pi i oint _ partial delta _ i frac p_ y x_ 0 y p x_ 0 y dy 1 by continuity this also holds for all x in a neighborhood of x 0 in particular p x y has only one root in δ i given by the residue theorem f i x 1 2 π i δ i y p y x y p x y d y displaystyle f_ i x frac 1 2 pi i oint _ partial delta _ i y frac p_ y x y p x y dy which is an analytic function branch points and puiseux series edit at a critical value the local branches need not be single valued functions of x displaystyle x instead after introducing a local parameter t displaystyle t with x x 0 t e displaystyle x x_ 0 t e the branches can be represented by convergent puiseux series 1 y j j 0 a j t j j j 0 a j x x 0 j e displaystyle y sum _ j j_ 0 infty a_ j t j sum _ j j_ 0 infty a_ j x x_ 0 j e the integer e displaystyle e describes the ramification of the branch algebraic functions have no singularities other than poles and algebraic branch points monodromy edit note that the foregoing proof of analyticity derived an expression for a system of n different function elements f i x provided that x is not a critical value of the projection to the x displaystyle x line a critical value is a value of x displaystyle x for which the number of distinct zeros of p x y displaystyle p x y is smaller than the degree of p displaystyle p in y displaystyle y this occurs only where the leading coefficient in y displaystyle y or the discriminant vanishes hence there are only finitely many such values c 1 c m a close analysis of the properties of the function elements f i near the critical values can be used to show that the monodromy cover is ramified over the critical values and possibly the point at infinity thus the holomorphic extension of the f i has at worst algebraic poles and ordinary algebraic branchings over the critical values note that away from the critical values we have p x y a n x y f 1 x y f 2 x y f n x displaystyle p x y a_ n x y f_ 1 x y f_ 2 x cdots y f_ n x since the f i are by definition the distinct zeros of p analytic continuation of the local branches around loops avoiding the critical values permutes the branches these permutations form the monodromy group of the algebraic function the monodromy action on the universal covering space is related but different notion in the theory of riemann surfaces algebraically if l displaystyle l is the splitting field of p x y displaystyle p x y over c x displaystyle mathbb c x equivalently the galois closure of the extension generated by one branch then the galois group gal l c x displaystyle operatorname gal l mathbb c x acts by permuting the roots f 1 f n displaystyle f_ 1 ldots f_ n under the correspondence between finite branched covers of the riemann sphere and finite extensions of c x displaystyle mathbb c x this galois group is identified with the monodromy group of the covering thus the monodromy action realizes the galois group of the splitting field as a permutation group on the branches closure properties edit algebraic functions are closed under addition subtraction multiplication division and composition wherever the operations are defined algebraically this follows from the fact that if u displaystyle u and v displaystyle v are algebraic over k x displaystyle k x then the field k x u v displaystyle k x u v is a finite algebraic extension of k x displaystyle k x hence any rational expression in u displaystyle u and v displaystyle v is again algebraic over k x displaystyle k x similarly if f displaystyle f is algebraic over k x displaystyle k x and g displaystyle g is algebraic over k t displaystyle k t then under suitable interpretation of branches f g t displaystyle f g t is again algebraic equivalently if algebraic functions are regarded as algebraic correspondences on the projective line the composite correspondence is again algebraic if p x y 0 displaystyle p x y 0 and q t x 0 displaystyle q t x 0 define two such correspondences then their composite is contained in the algebraic relation obtained by eliminating x displaystyle x for instance by the resultant res x q t x p x y 0 displaystyle operatorname res _ x q t x p x y 0 4 on a nonsingular branch the derivative of an algebraic function is also algebraic differentiating p x y 0 displaystyle p x y 0 implicitly gives y p x x y p y x y displaystyle y p_ x x y over p_ y x y where this expression is valid away from points at which p y 0 displaystyle p_ y 0 by contrast an antiderivative of an algebraic function need not be algebraic integrals of algebraic functions lead more generally to abelian integrals such as elliptic integrals 5 history edit the ideas surrounding algebraic functions go back at least as far as rené descartes the first discussion of algebraic functions appears to have been in edward waring s 1794 an essay on the principles of human knowledge in which he writes let a quantity denoting the ordinate be an algebraic function of the abscissa x by the common methods of division and extraction of roots reduce it into an infinite series ascending or descending according to the dimensions of x and then find the integral of each of the resulting terms see also edit algebraic expression analytic function complex analysis elementary function function mathematics generalized function list of eponyms of special functions list of types of functions polynomial rational function special functions transcendental function references edit 1 2 bliss gilbert ames 2004 1933 algebraic functions dover phoenix editions dover publications isbn 978 0 486 49568 2 chapter ii forster otto 1981 lectures on riemann surfaces graduate texts in mathematics vol 81 springer isbn 978 0 387 90617 1 section i 8 fulton william 2008 algebraic curves an introduction to algebraic geometry pdf kozen dexter landau susan zippel richard 1994 decomposition of algebraic functions algorithmic number theory lecture notes in computer science vol 877 springer pp 99 112 doi 10 1007 3 540 58691 1_46 lang serge 1982 introduction to algebraic and abelian functions graduate texts in mathematics vol 89 2nd ed springer isbn 978 0 387 90710 9 ahlfors lars 1979 complex analysis mcgraw hill van der waerden b l 1931 modern algebra volume ii springer external links edit wikimedia commons has media related to algebraic functions definition of algebraic function in the encyclopedia of math weisstein eric w algebraic function mathworld algebraic function at planetmath definition of algebraic function archived 2020 10 26 at the wayback machine in david j darling s internet encyclopedia of science v t e function history list of specific functions types by domain codomain x 𝔹 𝔹 x 𝔹ⁿ 𝔹 x ℤ ℤ x x ℝ ℝ x ℝⁿ x x ℂ ℂ x ℂⁿ x classes properties constant identity linear polynomial rational algebraic analytic smooth continuous measurable injective surjective bijective constructions restriction composition λ inverse generalizations relation binary relation set valued multivalued partia...