| FazBrowse Wikipedia Viewer | Main Page | | |
| Viewing: https://en.wikipedia.org/wiki/Algebraic_function |
In mathematics, an algebraic function is a function that satisfies a polynomial equation. Thus an equation of the following form holds: f(x)=1−x2 (for x2+y2−1=0 illustrates this, since it includes both the top semicircle y satisfying a polynomial equation y and complex coefficients. The example of y=−1−x2, with branch points where the two branches come together, at x is defined algebraically as an element algebraic over the rational function field an(x)yn+an−1(x)yn−1+⋯+a0(x)=0,
where the coefficients K. If the irreducible defining polynomial has degree n.
An algebraic function in K(x1,…,xm). Equivalently, it satisfies a polynomial equation
with poles at the zeros of yn−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 P(x,y)=0, then interchanging the roles of P(x,y)=0 is an algebraic curve; away from exceptional points, its local branches may be represented as graphs over the p(x,y) is a nonconstant polynomial of degree p(x,y)=0 has y3−xy+1=0.
Using the cubic formula, we get
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 x0 ∈ C is such that the polynomial p(x0, y) of y has n distinct zeros. We shall show that the algebraic function is analytic in a neighborhood of x0. Choose a system of n non-overlapping discs Δi containing each of these zeros. Then by the argument principle
since the fi 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 ℂ(x), equivalently the Galois closure of the extension generated by one branch, then the Galois group 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.
Algebraic functions are closed under addition, subtraction, multiplication, division, and composition, wherever the operations are defined. Algebraically, this follows from the fact that if K(x), then the field u and f is algebraic over K(t), then, under suitable interpretation of branches, Q(t,x)=0 define two such correspondences, then their composite is contained in the algebraic relation obtained by eliminating P(x,y)=0 implicitly gives
| View original on Wikipedia | Back | FazBrowse Home |