Introduction and overview Many algebraic function can be formed by the basic
algebraic operations:
addition,
multiplication,
division, and taking an
nth root. However, since
not all polynomial roots can be expressed in radicals, many algebraic functions cannot be explicitly represented in that way. The basic operations can be used to form polynomials, so any
polynomial function y = p(x) is an algebraic function; it is simply the solution
y to the equation : y-p(x) = 0.\, More generally, any
rational function y=\frac{p(x)}{q(x)} is algebraic, the solution to :q(x)y-p(x)=0. Moreover, the
nth root of any polynomial y=\sqrt[n]{p(x)} is an algebraic function, solving the equation :y^n-p(x)=0. Surprisingly, the
inverse function of an algebraic function is an algebraic function. For supposing that
y is a solution to :a_n(x)y^n+\cdots+a_0(x)=0, 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, :b_m(y)x^m+b_{m-1}(y)x^{m-1}+\cdots+b_0(y)=0. 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" x = \pm\sqrt{y}. Another way to understand this is that the
set of branches of the polynomial equation defining our 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
y) 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. , we get : y=-\frac{2x}{\sqrt[3]{-108+12\sqrt{81-12x^3}}}+\frac{\sqrt[3]{-108+12\sqrt{81-12x^3}}}{6}. For x\le \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>\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
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 Δ
i containing each of these zeros. Then by the argument principle :\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
x0. In particular,
p(
x,
y) has only one root in Δ
i, given by the
residue theorem: :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.
Monodromy Note that the foregoing proof of analyticity derived an expression for a system of
n different
function elements fi(
x), provided that
x is not a
critical point of
p(
x,
y). A
critical point is a point where the number of distinct zeros is smaller than the degree of
p, and this occurs only where the highest degree term of
p or the
discriminant vanish. Hence there are only finitely many such points
c1, ...,
cm. A close analysis of the properties of the function elements
fi near the critical points can be used to show that the
monodromy cover is
ramified over the critical points (and possibly the
point at infinity). Thus the
holomorphic extension of the
fi has at worst algebraic poles and ordinary algebraic branchings over the critical points. Note that, away from the critical points, we have :p(x,y) = a_n(x)(y-f_1(x))(y-f_2(x))\cdots(y-f_n(x)) since the
fi are by definition the distinct zeros of
p. The
monodromy group acts by permuting the factors, and thus forms the
monodromy representation of the
Galois group of
p. (The
monodromy action on the
universal covering space is related but different notion in the theory of
Riemann surfaces.)
Cardinality Since the set of polynomials with rational coefficients is
countable, the set of algebraic functions is also countable. == History ==