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Algebraic function

An algebraic function is a function f(x) that satisfies a polynomial equation p(x, f(x)) = 0, where p(x, y) is a polynomial in two variables with integer coefficients.^[1] These functions encompass a broad class built from rational functions through finite compositions of algebraic operations, including addition, subtraction, multiplication, division, and root extractions.^[1] Key examples include polynomials, such as f(x) = x^2 + 3x - 1, which satisfy the equation y - (x^2 + 3x - 1) = 0; rational functions, like f(x) = \frac{x^2 + 1}{x - 2}, satisfying (x - 2)y - (x^2 + 1) = 0; and radical functions, such as f(x) = \sqrt{x}, which solves y^2 - x = 0. More generally, algebraic functions arise as solutions to implicit polynomial equations, such as the unit circle defined by x^2 + y^2 - 1 = 0, where explicit branches include y = \pm \sqrt{1 - x^2}.^[2] Unlike transcendental functions—such as the exponential function e^x or the natural logarithm \ln x, which cannot be expressed algebraically—algebraic functions are closed under algebraic operations and form the basis for much of classical algebra and analysis.^[1] They play a central role in solving polynomial equations, algebraic geometry, and the study of Riemann surfaces, where their multi-valued nature in the complex plane leads to concepts like branching and monodromy.^[3]

Definition and Fundamentals

Definition

In mathematics, an algebraic function over a field K (such as the field of rational numbers \mathbb{Q} or complex numbers \mathbb{C}) is defined as a function f: U \to \overline{K}, where U \subseteq K is an open set and \overline{K} is an algebraic closure of K, such that f(x) satisfies a polynomial equation with coefficients in the field K(x) of rational functions in one variable over K.^[4] More formally, there exists an irreducible polynomial P(t) \in K(x) such that P(f(x)) = 0 for all x \in U, where the degree of P is the degree of the algebraic function.^[5] This definition arises in the context of field extensions: the function f generates a finite algebraic extension K(x, f) of K(x), and P(t) serves as the minimal polynomial of f over K(x), which is the monic irreducible polynomial of least degree annihilating f.^[4] An element \alpha \in L, where L is a field extension of K(x), is algebraic over K(x) if it has a minimal polynomial over K(x); otherwise, it is transcendental. The algebraic closure of K(x) is the smallest extension containing all roots of polynomials over K(x), and algebraic functions are precisely the non-constant elements algebraic over K(x) in such extensions.^[6] Algebraic functions must be distinguished from algebraic numbers: the latter are constant elements algebraic over K (i.e., roots of polynomials in K), whereas algebraic functions vary with the indeterminate x and satisfy polynomials in K(x).^[5] This distinction underscores that algebraic functions inhabit extensions of rational function fields, enabling the study of their properties through algebraic geometry and analysis.

Examples and Classifications

Algebraic functions encompass a variety of concrete forms that satisfy polynomial equations. A primary example is the rational function, defined as the quotient of two polynomials, such as f(x) = \frac{x^2 + 1}{x - 2}, where the denominator is nonzero; these functions are algebraic because they can be expressed through polynomial division and satisfy a linear equation in the function value.^[7] Another basic case is the radical function, like f(x) = \sqrt{x}, which implicitly satisfies the polynomial equation y^2 - x = 0 and represents an algebraic function involving roots.^[7] A illustrative example of a higher-degree algebraic function arises from solving the quadratic equation y^2 - x y + 1 = 0 for y, yielding the explicit form y = \frac{x \pm \sqrt{x^2 - 4}}{2}; this demonstrates two distinct branches corresponding to the choice of sign, highlighting the multi-valued nature typical of irrational algebraic functions.^[8] Algebraic functions are classified based on their defining polynomial's degree in the dependent variable y. Rational functions are those of degree 1 in y, expressible directly as ratios of polynomials without roots. Irrational algebraic functions have degree greater than 1, involving roots or other non-rational expressions, such as the square root example above. They may also be presented in explicit form, where y is solved algebraically in terms of x, or implicit form, defined solely by the polynomial equation like y^2 - x y + 1 = 0.^[9] Near singularities, the local behavior of algebraic functions is captured by Puiseux series expansions, which generalize power series to include fractional exponents and describe the parametrization of branches around singular points. For instance, these series provide asymptotic approximations for the solutions of polynomial equations at branch points.^[10] A key structural feature is that every algebraic function defined by an irreducible polynomial of degree n in y possesses at most n branches, reflecting the limited number of solution paths in the complex plane.^[8]

Properties and Characteristics

Algebraic Properties

Algebraic functions exhibit closure under fundamental algebraic operations, reflecting their role in generating extensions of the rational function field. Specifically, if f and g are algebraic functions over a base field K, then their sum f + g, product f \cdot g, and composition f \circ g are also algebraic functions. This closure arises because adjoining algebraic elements to a field preserves algebraicity, as established in the theory of field extensions.^[11] From the perspective of field extensions, an algebraic function f over K generates a finite extension K(x, f) of the rational function field K(x), where x is the independent variable. The degree of f, denoted [K(x, f) : K(x)], is the degree of the minimal polynomial of f over K(x), which is the monic irreducible polynomial of least degree satisfied by f. This degree measures the algebraic complexity of f.^[6] The set of all algebraic functions over K, equipped with pointwise addition and multiplication, forms a field. This structure follows from the fact that the algebraic elements over any base field, including the rational function field K(x), constitute an algebraically closed field under the field's operations.^[12] A key quantitative aspect is the bound on degrees for sums: if f and g are algebraic functions of degrees m and n over K(x), respectively, then f + g is algebraic over K(x) with degree at most mn. This follows from the tower law in field extensions, where [K(x)(f, g) : K(x)] \leq mn, ensuring f + g lies in a finite extension of bounded degree.

Analytic Properties

Algebraic functions, defined as roots of polynomial equations with rational coefficients, exhibit holomorphic properties in the complex plane except at branch points. Specifically, away from these singularities, an algebraic function satisfies the Cauchy-Riemann equations and is infinitely differentiable, allowing representation by a convergent power series in local neighborhoods.^[13] This holomorphicity stems from the implicit function theorem applied to the defining polynomial, ensuring local analyticity where the derivative does not vanish.^[14] In the broader complex plane, algebraic functions are meromorphic, meaning they are holomorphic except at isolated poles arising from the rational components of their expression. For instance, a function like f(z) = \frac{\sqrt{z}}{z-1} is meromorphic, with a pole at z=1 and a branch point at z=0. However, the presence of roots introduces algebraic singularities, primarily branch points, which are not poles but points of ramification where the function becomes multi-valued. These branch points, such as at z=0 for \sqrt{z}, mark locations where the function fails to be single-valued and holomorphic in a punctured neighborhood.^[13]^[15] The multi-valued nature of algebraic functions limits the direct application of the residue theorem. While residues can be computed along contours avoiding branch cuts, encircling a branch point alters the function's branch, preventing straightforward integration over closed paths that enclose such points without accounting for the monodromy. This restriction contrasts with single-valued meromorphic functions, where the theorem yields precise sums of residues.^[16]^[14] Algebraic functions are analytic on their domains excluding branch cuts, which are chosen to connect branch points and render the function single-valued in the cut plane. Near a regular point z_0 (neither a pole nor branch point), the function admits a Taylor series expansion:

f(z) = \sum_{k=0}^{\infty} a_k (z - z_0)^k,

which converges in a disk around z_0 up to the nearest singularity. This local analyticity underscores the function's conformity and open mapping properties in suitable regions.^[13]^[14]

Algebraic Functions in One Variable

Overview and Construction

Algebraic functions in one variable arise as the non-rational solutions to polynomial equations over the field of rational functions in that variable. Specifically, given a field K (typically the complex numbers or rationals), an algebraic function y over K(x) is defined such that there exists an irreducible polynomial P(x, y) \in K(x) of degree n \geq 2 for which P(x, y) = 0. This polynomial is monic in y, ensuring uniqueness up to scalar multiples in the coefficients. The general form of such an equation is

\sum_{k=0}^n a_k(x) y^k = 0,

where a_n(x) = 1 and the a_k(x) for k = 0, \dots, n-1 are rational functions in x with coefficients in K, and the polynomial is irreducible over K(x).^[17]^[18] The construction of these functions proceeds by successive field extensions of K(x), adjoining roots of irreducible polynomials step by step to build the full extension field F = K(x, y_1, \dots, y_m) where the y_i satisfy the defining relations. For instance, starting from K(x), one adjoins a root of a quadratic polynomial (such as \sqrt{x} for the equation y^2 - x = 0), forming a quadratic extension, and then continues by adjoining roots of further polynomials over this new field until the minimal polynomial of degree n is considered. This process yields a finite extension of K(x) of degree n, namely the function field K(x, y) where y satisfies the irreducible polynomial of degree n. The associated Galois closure has degree dividing n! over K(x), corresponding to the algebraic function field generated by the roots.^[17]^[19] Each such algebraic function in one variable corresponds to an algebraic curve defined by the equation P(x, y) = 0. For a smooth irreducible plane curve of degree n, the smooth projective model over K has genus g = \frac{(n-1)(n-2)}{2}. In general, for singular curves, the genus is lower, given by g = \frac{(n-1)(n-2)}{2} - \sum \delta_s where \delta_s are the delta-invariants of the singularities. This genus formula holds when resolved to smoothness, and g \geq 0 characterizes all possible genera for these curves, with g = 0 for rational functions (though excluded here by n \geq 2) and higher values for more complex algebraic functions. For example, the square root function corresponds to a genus 0 curve, while smooth cubic equations yield genus 1.^[18]^[17]

Branch Points and Riemann Surfaces

Branch points are isolated points in the complex plane where an algebraic function fails to be single-valued, meaning that analytic continuation around such a point results in a different branch of the function.^[20] These points arise because the function satisfies a polynomial equation, leading to multiple solutions that permute upon encircling the singularity.^[21] Branch points are classified into algebraic and logarithmic types based on the nature of the singularity. Algebraic branch points, also known as branch points of finite order, occur when a finite number of sheets of the function meet, and the function can be parametrized locally by a Puiseux series expansion. At such a point z_0, the expansion includes a leading term of the form c (z - z_0)^{1/m} where m \geq 2 is the ramification index, indicating the order of the branching.^[21] In contrast, logarithmic branch points involve an infinite number of sheets, typically arising from compositions with the complex logarithm, though pure algebraic functions exhibit only algebraic branch points.^[20] To resolve the multi-valuedness at branch points, algebraic functions are defined on Riemann surfaces, which are multi-sheeted coverings of the complex plane (or the Riemann sphere after compactification). The construction involves taking n copies of the complex plane, where n is the degree of the defining polynomial, and gluing them along branch cuts connecting the branch points; crossing a cut transitions between sheets.^[14] This yields a surface where the function becomes single-valued and holomorphic everywhere except possibly at the branch points themselves, which are resolved as regular points on the surface. For a degree-n algebraic function, the resulting Riemann surface is a compact orientable surface of genus g, determined by the Riemann-Hurwitz formula: $2g - 2 = -2n + \sum_p (e_p - 1), where the sum is over all branch points p and e_p is the ramification index at p.^[22] A classic example is the square root function w = \sqrt{z}, defined by w^2 = z, which has algebraic branch points at z = 0 and z = \infty, each with ramification index 2. The associated Riemann surface consists of two sheets glued along a branch cut from 0 to \infty (typically the negative real axis), forming a two-sheeted covering where traversing the cut switches sheets, rendering \sqrt{z} single-valued and holomorphic on the entire surface, which is topologically a sphere (genus 0).^[23]

Monodromy and Analytic Continuation

Algebraic functions in one variable, being multivalued, admit analytic continuation along paths in the complex plane punctured at branch points, where the continuation around a closed loop induces a permutation of the function's branches on the associated Riemann surface. The monodromy theorem asserts that this permutation action is well-defined up to homotopy of the loop and depends only on the conjugacy class in the fundamental group of the punctured plane, thereby defining a monodromy group that acts on the set of branches. This group captures the global structure of the function's multivaluedness, ensuring that continuations along homotopic paths yield the same branch permutation.^[24] The monodromy group is computed by considering generators corresponding to loops encircling individual branch points, each of which permutes the n sheets of the Riemann surface according to the local ramification structure at that point. For a generic algebraic function of degree n, these local permutations generate a transitive subgroup of the symmetric group S_n, reflecting the irreducibility of the defining polynomial. The full monodromy action is encoded by a homomorphism \rho: \pi_1(\mathbb{C} \setminus \{b_1, \dots, b_m\}) \to S_n, where \{b_1, \dots, b_m\} are the branch points, mapping each generator (a small loop around b_i) to the corresponding sheet permutation.^[25] In the specific case of hyperelliptic functions, defined by equations of the form y^2 = f(x) where f is a polynomial of degree $2g+1 or $2g+2, the Riemann surface consists of two sheets, and the monodromy group is generated by transpositions corresponding to the simple branch points of the covering. Each such transposition swaps the two sheets around a Weierstrass point, and the overall group action reflects the hyperelliptic involution while ensuring the covering's connectivity.^[26] Monodromy plays a crucial role in the Frobenius method for solving the linear differential equations satisfied by algebraic functions, particularly around regular singular points that coincide with branch points. The method constructs formal power series solutions (Frobenius solutions) whose analytic continuations around these points exhibit monodromy permutations, allowing determination of the solution space's dimension and the equation's global properties via the representation on the fundamental group. This connection facilitates the study of algebraic invariants through differential equations, as the monodromy group encodes information about the function's ramification and solvability.^[27]

Algebraic Functions in Several Variables

Definition and Implicit Representations

An algebraic function in several variables over a field K is a function f: U \to K, where U \subseteq K^m is a Zariski-open subset, such that there exists a non-constant polynomial P \in K[x_1, \dots, x_m, y] satisfying P(x_1, \dots, x_m, f(x_1, \dots, x_m)) = 0 for all (x_1, \dots, x_m) \in U.^[5] This polynomial equation provides an implicit representation of f, treating it as an algebraic dependence relation among the variables.^[5] In the multivariable setting, such functions are primarily defined implicitly through ideals in the polynomial ring K[x_1, \dots, x_m, y], rather than parametric expressions that explicitly solve for f in terms of the inputs, which become impractical beyond low dimensions due to the complexity of solving multivariate systems.^[5] For a single defining equation, the ideal is principal, generated by P; more generally, f may satisfy a system of polynomials, corresponding to a prime or primary ideal capturing the relation.^[5] The polynomial P is assumed to be irreducible in K[x_1, \dots, x_m, y], ensuring the relation defines a hypersurface without redundant factors, and the equation holds generically where the projection to the x_1, \dots, x_m-space is dominant.^[5] In m variables, algebraic functions of this form relate to algebraic varieties of dimension m, as the solution set projects onto an m-dimensional ambient space while incorporating the algebraic constraint.^[5] The algebraic structure is further illuminated by dimension theory in function fields: adjoining f to the rational function field K(x_1, \dots, x_m) yields a finite field extension K(x_1, \dots, x_m, f) of transcendence degree m over K, with the degree of the extension equal to the degree of P in the variable y, provided P is irreducible and monic in y.^[28] This generalizes the one-variable case, where the extension degree similarly measures the branching of the function.^[28]

Geometric Interpretations

In algebraic geometry, an algebraic function f: \mathbb{A}^m \to \mathbb{A}^n in several variables is geometrically interpreted through its association with algebraic varieties, where the graph of f, denoted \Gamma_f = \{(x, f(x)) \mid x \in \mathbb{A}^m\}, forms a subvariety of codimension n in the product space \mathbb{A}^m \times \mathbb{A}^n, which is an affine variety of dimension m defined by the vanishing of polynomials encoding the components of f.^[18]^[29] This graph is closed and isomorphic to the domain \mathbb{A}^m via the natural projection, providing a spatial representation that embeds the function's behavior within a higher-dimensional ambient space.^[18] For a more general algebraic function f: V \to \mathbb{A}^m, where V \subset \mathbb{A}^k is an affine variety defined by the vanishing of a system of polynomials P_1 = \cdots = P_r = 0, the graph \Gamma_f similarly constitutes a subvariety of codimension equal to the dimension of the codomain in \mathbb{A}^k \times \mathbb{A}^m, capturing the function's values over the variety V.^[29] The geometric structure of such functions is further illuminated by viewing f as a projection morphism from its graph variety to the target space \mathbb{A}^n, where the fibers over points in the image correspond to preimages under f, often forming subvarieties themselves.^[18]^[29] For instance, if V is irreducible, the projection \pi: \Gamma_f \to \mathbb{A}^m is dominant, with generic fibers being finite sets determined by the degree of the defining equations, reflecting the multivalued nature of algebraic functions in higher dimensions.^[29] This projection perspective highlights how algebraic functions induce fibrations over their images, with the total space \Gamma_f serving as a resolution or compactification tool in the ambient affine space. Singularities in the graph hypersurface or the domain variety can be resolved geometrically through birational maps, such as blowing up along smooth subvarieties, which replaces singular points with exceptional divisors to yield a smooth model birational to the original.^[30]^[18] For example, in the case of a singular hypersurface defined by P = 0, successive blow-ups at permissible centers—smooth subvarieties contained in the singular locus—produce a resolution \tilde{V} \to V where the strict transform is nonsingular and the exceptional locus has simple normal crossings, preserving the birational equivalence while simplifying the geometry.^[30] This process, effective in characteristic zero for varieties of any dimension, facilitates the study of invariants like the Euler characteristic or cohomology by transferring computations to the smooth birational model.^[30] In applications within algebraic geometry, these geometric interpretations enable the implicit parametrization of curves and surfaces, where an algebraic function f: V \to \mathbb{A}^m provides coordinates for points on a variety defined implicitly by polynomials, as in the case of plane curves V(F) \subset \mathbb{A}^2 parametrized via rational maps from the graph.^[18]^[31] For higher-dimensional surfaces, such as cubic hypersurfaces in \mathbb{P}^3, the projection from the Fano variety of lines on the surface yields implicit representations that encode intersection properties and rationality questions, linking the function's graph to moduli spaces of subvarieties.^[31] This approach, building on implicit polynomial representations, underscores the role of algebraic functions in bridging explicit parametrizations with geometric invariants like Picard groups or monodromy actions.^[29]^[31]

Historical Development

Early Foundations

The origins of algebraic functions can be traced to ancient civilizations, where early solutions to quadratic equations laid implicit groundwork for algebraic methods. Around 1800 BCE, Babylonian mathematicians solved problems equivalent to quadratic equations through geometric procedures, such as finding the dimensions of rectangles given their area and the sum or difference of sides, using techniques that anticipated algebraic manipulation without explicit symbolic notation.^[32] In ancient India, during the 7th century CE, Brahmagupta systematized these approaches in his Brāhmasphuṭasiddhānta, providing rules for solving quadratic equations of various types, including those involving negative quantities and zero, which marked a significant advancement in handling indeterminate forms.^[33] The Renaissance brought explicit algebraic solutions for higher-degree equations, introducing radicals as a key tool. In 1545, Girolamo Cardano published Ars Magna, revealing the general formula for cubic equations—derived from the independent discoveries of Scipione del Ferro and Niccolò Tartaglia—which expressed roots using cube roots and square roots, thereby extending the radical-based solvability beyond quadratics.^[34] This work highlighted the power of algebraic operations in resolving polynomial roots, influencing subsequent developments in function theory. In the 17th and 18th centuries, figures like Isaac Newton, Leonhard Euler, and Joseph-Louis Lagrange refined the conceptual framework for algebraic equations and functions. During the 1670s, Newton classified plane curves as algebraic if they could be defined by polynomial equations of finite degree, distinguishing them from transcendental curves based on their geometric and analytic properties; he enumerated 72 types of cubic curves using this degree-based criterion.^[35] Euler, in his Introductio in analysin infinitorum (1748), formalized the distinction between algebraic functions—those expressible through finite rational operations, roots, and powers—and transcendental functions, which require infinite series or non-algebraic processes like exponentials or logarithms, thereby providing a rigorous basis for analyzing polynomial solvability.^[36] Lagrange built on this in works like Théorie des fonctions analytiques (1797), emphasizing power series expansions to represent functions analytically and exploring methods for solving higher-degree equations without relying solely on radicals.^[37] A pivotal limitation emerged in the early 19th century with the Abel-Ruffini theorem, which delineated the boundaries of algebraic solvability by radicals. In 1824 (with full publication in 1826), Niels Henrik Abel proved that the general quintic equation cannot be solved using radicals alone, confirming and rigorizing Paolo Ruffini's earlier inconclusive attempt from 1799, and thus establishing that algebraic functions for degrees five and higher generally transcend radical expressions.^[38] Building on this, Évariste Galois, in the early 1830s, developed a group-theoretic framework to determine the solvability of polynomial equations by radicals. His theory introduced the Galois group of a polynomial, showing that solvability corresponds to the group being solvable, providing a complete criterion for radical expressibility and laying the foundations for abstract group theory in algebra. Galois's work, published posthumously after his death in 1832, profoundly influenced the study of algebraic functions and their symmetries.^[39]

Modern Advancements

In the late 19th century, Richard Dedekind and Heinrich Weber advanced the abstract algebraic treatment of functions by developing an arithmetic theory analogous to that of algebraic numbers. Their 1882 paper introduced ideals and field extensions specifically for algebraic functions of one variable, laying foundational groundwork for modern algebraic geometry by treating function fields over curves in a purely algebraic manner. David Hilbert's Nullstellensatz, published in 1893, provided a pivotal link between ideals in polynomial rings over algebraically closed fields and the corresponding algebraic varieties, formalizing the geometric interpretation of radical ideals as vanishing sets. This theorem, appearing in his work on invariant theory, enabled rigorous algebraic descriptions of varieties and resolved key questions about the solvability of polynomial systems, influencing subsequent developments in multi-variable algebraic functions.^[40] The theory of Riemann surfaces, originating with Bernhard Riemann in the 1850s, matured in the early 20th century through advancements in complex analysis, notably Adolf Hurwitz's contributions to monodromy groups around the turn of the century, which clarified the analytic continuation and branching behavior of multi-valued algebraic functions. In the 1930s, Oscar Zariski's work on birational geometry, particularly in his 1935 monograph Algebraic Surfaces, established systematic methods for resolving singularities and classifying surfaces up to birational equivalence using commutative algebra, bridging classical Italian geometry with abstract approaches. Alexander Grothendieck's introduction of schemes in the 1960s, detailed in Éléments de géométrie algébrique (EGA), generalized the framework for algebraic functions in several variables by incorporating relative schemes over base rings, allowing uniform treatment of arithmetic and geometric aspects across families of varieties. This abstraction extended Hilbert's ideas to non-Noetherian settings and facilitated the study of moduli spaces and deformation theory. Algorithmic progress emerged with Bruno Buchberger's 1965 development of Gröbner bases, which provided a computational method to solve systems of polynomial equations implicitly defining algebraic varieties, enabling effective elimination and ideal membership tests in multi-variable settings. Post-2000, computational algebraic geometry advanced through software like Macaulay2, which implements efficient algorithms for Gröbner bases, free resolutions, and cohomology computations, supporting research in toric varieties, D-modules, and Calabi-Yau manifolds as demonstrated in its 2002 companion volume.^[41]

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The evolution of the function concept can be seen as a tug of war between two elements, two mental images: the geometric (expressed in the form of a curve) and.
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[PDF] Lagrange's Theory of Analytical Functions and His Ideal of Purity of ...
Aug 12, 2011 · We explain Lagrange's notions of function and algebraic quantity, and concentrate on power-series expansions, on the algorithm for derivative ...<|separator|>
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[PDF] Galois Theory - University of Oregon
In 1824, Abel published his first significant paper, ”Memoir on algebraic equations, ... ARNOLD'S PROOF OF THE ABEL-RUFFINI THEOREM. 19. Figure 1.1: Restrict ...Missing: unsolvability | Show results with:unsolvability
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https://mathshistory.st-andrews.ac.uk/Projects/Brunk/chapter-1/
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[PDF] Computations in algebraic geometry with Macaulay 2
Macaulay 2 is a computer algebra system devoted to supporting research in algebraic geometry, commutative algebra, and their applications. The reader of this ...