Algebraic Functions Algebraic Functions algebraic function is informally a function that satisfies a polynomial equation whose coefficients are themselves polynomials with rational coefficients. For example, an algebraic function in one variable x is a solution y for an equation where the coefficients ai(x) are polynomial functions of x with rational coefficients. A function which is not algebraic is called a transcendental function. In more precise terms, an algebraic function may not be a function at all, at least not in the conventional sense. Consider for example the equation of a circle: However, both branches are thought of as belonging to the "function" determined by the polynomial equation. Thus an algebraic function is most naturally considered as a multiple valued function. An algebraic function in n variables is similarly defined as a function y which solves a polynomial equation in n + 1 variables:
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It is normally assumed that p should be an irreducible polynomial. The existence of an algebraic function is then guaranteed by the implicit function theorem. Formally, an algebraic function in n variables over the field K is an element of the algebraic closure of the field of rational functions K(x1,...,xn). In order to understand algebraic functions as functions, it becomes necessary to introduce ideas relating to Riemann surfaces or more generally algebraic varieties, and sheaf theory. The informal definition of an algebraic function provides a number of clues about the properties of algebraic functions. To gain an intuitive understanding, it may be helpful to regard algebraic functions as functions which can be formed by the usual algebraic operations: addition, multiplication, division, and taking an nth root. Of course, this is something of an oversimplification; because of casus irreducibilis (and more generally the fundamental theorem of Galois theory), algebraic functions need not be expressible by radicals. First, note that any polynomial is an algebraic function, since polynomials are simply the solutions for y of the equation More generally, any rational function is algebraic, being the solution of Moreover, the nth root of any polynomial is an algebraic function, solving the equation Surprisingly, the inverse function of an algebraic function is an algebraic function. For supposing that y is a solution of
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for each value of x, then x is also a solution of this equation for each value of y. Indeed, interchanging the roles of x and y and gathering terms, Writing x as a function of y gives the inverse function, also an algebraic function. However, not every function has an inverse. For example, y = x2 fails the horizontal line test: it fails to be one-to-one. The inverse is the algebraic "function" . In this sense, algebraic functions are often not true functions at all, but instead are multiple valued functions. Another way to understand this, which will become important later in the article, is that an algebraic function is the graph of an algebraic curve. The role of complex numbers :- 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 any polynomial relation p(y, x) = 0 is guaranteed to have at least one solution (and in general a number of solutions not exceeding the degree of p in x) for y at each point x, provided we allow y to assume complex as well as real values. Thus, problems to do with the domain of an algebraic function can safely be minimized.
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