Algebraic function

A function $ y = f ( x _ {1}, \dots, x _ {n} ) $ of the variables $ x _ {1}, \dots, x _ {n} $ that satisfies an equation

$$ \tag{1 } F ( y , x _ {1}, \dots, x _ {n} ) = 0 , $$

where $ F $ is an irreducible polynomial in $ y, x _ {1}, \dots, x _ {n} $ with coefficients in some field $ K $, known as the field of constants. The algebraic function is said to be defined over this field, and is called an algebraic function over the field $ K $. The polynomial $ F ( y , x _ {1} \dots x _ {n} ) $ is often written in powers of the variable $ y $, so that equation (1) assumes the form

$$ P _ {k} ( x _ {1}, \dots, x _ {n} ) y ^ {k} + P _ {k - 1 } ( x _ {1}, \dots, x _ {n} ) y ^ {k - 1 } + \dots + P _ {0} ( x _ {1}, \dots, x _ {n} ) = 0, $$

where $ P _ {k} ( x _ {1}, \dots, x _ {n} ), \dots, P _ {0} ( x _ {1}, \dots, x _ {n} ) $ are polynomials in $ x _ {1}, \dots, x _ {n} $, and with $ P _ {k} ( x _ {1}, \dots, x _ {n} ) \not\equiv 0 $. The number $ k $ is the degree of $ F $ with respect to $ y $, and is called the degree of the algebraic function. If $ k = 1 $, an algebraic function may be represented as a quotient

$$ y = - \frac{P _ {0} ( x _ {1}, \dots, x _ {n} ) }{P _ {1} ( x _ {1}, \dots, x _ {n} ) } $$

of polynomials, and is called a rational function of $ x _ {1}, \dots, x _ {n} $. For $ k = 2, 3, 4 $, an algebraic function can be expressed as square and cube roots of rational functions in the variables $ x _ {1}, \dots, x _ {n} $; if $ k > 4 $, this is impossible in general.

The theory of algebraic functions was studied in the past from three different points of view: the function-theoretical point of view taken, in particular, by N.H. Abel, K. Weierstrass and B. Riemann; the arithmetic-algebraic point of view taken by R. Dedekind, H. Weber and K. Hensel; and the algebraic-geometrical point of view, which originated with the studies of A. Clebsch, M. Noether and others (cf. Algebraic geometry). The first direction of the theory of algebraic functions of a single variable is connected with the study of algebraic functions over the field of complex numbers, in which they are regarded as meromorphic functions on Riemann surfaces and complex manifolds; the most important methods applied are the geometrical and topological methods of the theory of analytic functions. The arithmetic-algebraic approach involves the study of algebraic functions over arbitrary fields. The methods employed are purely algebraic. The theory of valuations and extensions of fields are especially important. In the algebraic-geometrical approach algebraic functions are considered to be rational functions on an algebraic variety, and are studied by methods of algebraic geometry (cf. Rational function). These three points of view originally differed not only in their methods and their ways of reasoning, but also in their terminology. This differentiation has by now become largely arbitrary, since function-theoretical studies involve the extensive use of algebraic methods, while many results obtained at first using function-theoretical and topological methods can be successfully applied to more general fields using algebraic analogues of these methods.

Algebraic functions of one variable

Over the field $ \mathbf C $ of complex numbers, an algebraic function of one variable $ y = f(x) $ (or $ y(x) $ for short) is a $ k $-valued analytic function. If $ D(x) $ is the discriminant of the polynomial

$$ \tag{2 } F ( x , y ) = P _ {k} ( x ) y ^ {k} + \cdots + P _ {1} ( x ) y + P _ {0} ( x ) , $$

$$ P _ {k} ( x ) \not\equiv 0 , $$

(i.e. of the polynomial for which $ F ( x, f (x ) ) = 0 $), which is obtained by eliminating $ y $ from the equations

$$ F ( x , y ) = 0 ,\ \frac{\partial F ( x , y ) }{\partial y } = 0 $$

to yield the equation

$$ P _ {k} ( x ) D ( x ) = 0 , $$

then the roots $ x _ {1}, \dots, x _ {m} $ of this last equation are known as the critical values of $ y = f(x) $. The complementary set $ G = \mathbf C \setminus \{ x _ {1}, \dots, x _ {m} \} $ is known as the non-critical set. For any point $ x _ {0} \in G $ equation (2) has $ k $ different roots $ y _ {0} ^ {1}, \dots, y _ {0} ^ {k} $ and the condition

$$ \frac{\partial F (x _ {0} , y _ {0} ^ {j} ) }{\partial y } \neq 0 ,\ \ j = 1, \dots, k , $$

is satisfied. According to the implicit function theorem, in a neighbourhood of the point $ x _ {0} $ there exist $ k $ single-valued analytic functions $ f _ {0} ^ { 1 } (x), \dots, f _ {0} ^ { k } (x) $ which satisfy the conditions

$$ f _ {0} ^ { j } ( x _ {0} ) = y _ {0} ^ {j} ,\ F ( x , f _ {0} ^ { j } ( x ) ) = 0 $$

and which can be decomposed into a convergent series

$$ \tag{3 } f _ {0} ^ { j } ( x ) = y _ {0} ^ {j} + \alpha _ {1} ^ {j} ( x - x _ {0} ) + \alpha _ {2} ^ {j} ( x - x _ {0} ) ^ {2} + \cdots . $$

Thus, for each point $ x _ {0} \in G $ one can construct $ k $ elements of an analytic function, known as the function elements with centre at the point $ x _ {0} $. For any two points $ x _ {1} , x _ {2} \in G $, any elements $ f _ {1} ^ { i } (x) $ and $ f _ {2} ^ { i } (x) $ with centres at $ x _ {1} $ and $ x _ {2} $, respectively, are derived from each other by analytic continuation along some curve in $ G $; in particular, any two elements with the same centres are also connected in this way. If $ x _ {0} $ is a critical point of an algebraic function, then two cases are possible: 1) $ x _ {0} $ is a root of the discriminant, i.e. $ D (x _ {0} ) = 0 $, but $ P _ {k} (x _ {0} ) \neq 0 $; or 2) $ P _ {k} ( x _ {0} ) = 0 $.

Case 1. Let $ K _ {0} $ be a small disc with centre at $ x _ {0} $ which does not contain other critical points, and let $ f _ {1} ^ { \prime } (x), \dots, f _ {k} ^ { \prime } (x) $ be a system of regular elements with centre at $ x ^ \prime \in K _ {0} $, $ x ^ \prime \neq x _ {0} $. These functions remain bounded as $ x\rightarrow x _ {0} $. Furthermore, let $ D $ be the circle with centre $ x _ {0} $ passing through $ x ^ \prime $; it is completely contained inside $ K _ {0} $. The analytic continuation of some given element, e.g. $ f _ {1} ^ { \prime } (x) $, along $ D $ (in, say, the clockwise direction), yields an element $ f ^ { \prime } (x) $ which also belongs to the system of elements with centre $ x ^ \prime $. This system consists of $ k $ elements, and a minimum required finite number $ a _ {1} \leq k $ of such turns yields the initial element $ f _ {1} ^ { \prime } (x) $. One obtains a subsystem $ f _ {1} ^ { \prime } (x), \dots, f _ {a _ {1} } ^ { \prime } (x) $ of elements with centre $ x ^ \prime $; each one of these elements may be obtained by analytic continuation of the other by a number of turns around the point $ x _ {0} $; such a subsystem is known as a cycle. Any system $ f _ {1} (x), \dots, f _ {k} (x) $ can be decomposed into a number of non-intersecting cycles

$$ \{ f _ {1} ^ { \prime } (x), \dots, f _ {a _ {1} } ^ { \prime } (x) \} , $$

$$ \{ f _ {a _ {1} + 1 } ^ { \prime } ( x ), \dots, f _ {a _ {1} + a _ {2} } ^ { \prime } ( x ) \}, \dots $$

$$ \{ f _ {a _ {1} + \cdots + a _ {s - 1 } + 1 } ^ { \prime } (x), \dots, f _ {a _ {1} + \cdots + a _ {s} } ^ { \prime } ( x ) \} , $$

$ a _ {1} + \cdots + a _ {s} = k $. If $ a _ {1} > 1 $, then the element $ f _ {1} ^ { \prime } (x) $ is not a single-valued function of $ x $ in the disc $ K _ {0} $, but is a single-valued analytic function of the parameter $ \tau = {(x - x _ {0} ) } ^ {1/ a _ {1} } $ in a neighbourhood of $ \tau = 0 $. In a certain neighbourhood of this point the elements $ f _ {1} ^ { \prime } (x), \dots, f _ {a _ {1} } ^ { \prime } (x) $ of the first cycle can be represented as convergent series

$$ \tag{4 } f _ {1} ^ { \prime } (x) = \ \sum _ {i = 0 } ^ \infty \alpha _ {i} ^ {1} \tau ^ {i} = \sum _ {i = 0 } ^ \infty \alpha _ {i} ^ {1} ( x - x _ {0} ) ^ {i / a _ {1} } , $$

$$ {\dots \dots \dots \dots \dots \dots \dots } $$

$$ f _ {a _ {1} } ^ { \prime } ( x ) = \sum _ {i = 0 } ^ \infty \alpha _ {i} ^ {a _ {1} } \tau ^ {i} = \sum _ {i = 0 } ^ \infty \alpha _ {i} ^ {a _ {1} } ( x - x _ {0} ) ^ {i/a _ {1} } ; $$

and similar expansions also take place for the elements of other cycles. Such expansions of elements by fractional degrees of the difference $ x - x _ {0} $, where $ x _ {0} $ is a critical point, are known as Puiseux series. The transformation $ \tau \rightarrow \tau _ {r} $, $ r = e ^ {2 \pi i / a } $, which corresponds to one turn around $ x _ {0} $, converts the Puiseux series of elements in one cycle into each other in cyclic order, i.e. there is a cyclic permutation of the series and of the corresponding elements. To turns around the critical point correspond permutations of the elements with centre at this point; these permutations consist of cycles of the orders $ a _ {1}, \dots, a _ {s} $, $ \sum {a _ {i} } = k $. The permutations defined in this way constitute the monodromy group of the algebraic function. If at least one element $ a _ {i} $ is greater than 1, the critical point $ x _ {0} $ is called an algebraic branch point of the algebraic function; the numbers $ a _ {i} $ (sometimes $ a _ {i} - 1 $) are called the branch indices (or branch orders) of the algebraic function.

Case 2. If $ P _ {k} (x) y $ is substituted for $ y $, one returns to case 1; one obtains expansions similar to (4), which may contain a finite number of terms with negative indices:

$$ \tag{5 } f ( x ) = \sum _ {i = - p } ^ \infty \alpha _ {i} \tau ^ {i} = \ \sum _ {i = - p } ^ \infty \alpha _ {i} ( x - x _ {0} ) ^ {i/a} . $$

If $ p > 0 $, the point $ x _ {0} $ is a pole of order $ p $ of the algebraic function. An algebraic function is usually considered on the Riemann sphere $ S $, i.e. on the complex plane completed by the point at infinity $ x = \infty $. The introduction of the variable $ \tau = 1/x $ reduces this case to the previous case; in a neighbourhood of the point $ \tau = 0 $ ($ x = \infty $) one has the expansion

$$ \tag{6 } y ( x ) = \sum _ {j = - r } ^ \infty \alpha _ {j} \tau ^ {j/a} = \ \sum _ {j = - r } ^ \infty \alpha _ {j} x ^ {- j / a } . $$

If $ r > 0 $, then the point $ x = \infty $ is called a pole of order $ r $.

The parameter of the expansion in the series (3), (4), (5), (6) is called the local uniformizing parameter for the algebraic function. If $ x _ {0} $ is a non-critical point of the algebraic function, then $ \tau = x - x _ {0} $ can be taken as parameter; if, on the other hand, $ x _ {0} $ is a critical point, the root $ (x - x _ {0} ) ^ {1/a} $ (where $ a $ is a positive integer) can be taken as such a parameter. The population of all elements of an algebraic function described above forms the complete algebraic function in the sense of Weierstrass. Algebraic functions have no singularities other than algebraic branch points and poles. The converse proposition is also true: A function $ y = f(x) $ which is analytic, is not more than $ s $-valued at all points of the Riemann sphere except for a finite number of points $ x _ {1}, \dots, x _ {m} $ and $ x = \infty $, and has at such points only poles or algebraic branch points, is an algebraic function of degree $ k \leq s $.

The Riemann surface of a complete algebraic function is compact and is a $ k $-sheeted covering of the Riemann sphere, branch points can be the critical points and the point $ x = \infty $. Algebraic functions are the only class of functions with a compact Riemann surface. The genus of the Riemann surface of an algebraic function is important; it is called the genus of the algebraic function. It can be calculated by the Riemann–Hurwitz formula. The genus of a rational function is zero, and its Riemann surface is the Riemann sphere. The Riemann surface of an elliptic function that satisfies a third- or fourth-degree equation is a torus; the genus of such a function is one.

The universal covering Riemann surface of an algebraic function is a simply-connected two-dimensional manifold, i.e. it has a trivial fundamental group and is conformally equivalent either to the Riemann sphere, the complex plane or the interior of the unit disc. In the first case the algebraic function is a rational, in the second case it is an elliptic, while in the third case it is a general function.

The uniformization problem of algebraic functions is closely connected with the theory of Riemann surfaces of algebraic functions. The function $ y = f(x) $ can be uniformized if $ y $ and $ x $ are representable as single-valued analytic functions

$$ y = y ( t ) ,\ x = x ( t ) $$

of a parameter $ t $, which identically satisfy equation (2). The uniformization problem is locally solved by a local uniformizing parameter; however, it is the solution "in the large" that is of interest. If $ k = 1 $, i.e. if $ y(x) $ is a rational function of $ x $, this parameter may be the variable $ x - x _ {0} $; if $ k = 2 $, then uniformization is attained with the aid of a rational or a trigonometric function. For instance, if $ y(x) $ satisfies the equation

$$ y ^ {2} - x ^ {2} = 1 , $$

one can take

$$ y = \frac{t ^ {2} + 1 }{t ^ {2} - 1 } ,\ x = \frac{2t }{t ^ {2} - 1 } $$

or

$$ y = \ \mathop{\rm sec} t ,\ x = \mathop{\rm tg} t . $$

If $ k = 3, 4 $ in the case of an algebraic function of genus one, uniformization is achieved using elliptic functions. Finally, if $ k > 4 $ and the genus of the algebraic function is higher than one, uniformization is realized using automorphic functions (cf. Automorphic function).

Algebraic functions of several variables.

If $ f $ is an algebraic function in the variables $ x _ {1}, \dots, x _ {n} $, then the set of all rational functions $ R (y, x _ {1}, \dots, x _ {n} ) $ forms a field $ K _ {f} $, coinciding with the field of rational functions on the algebraic hypersurface in $ ( n + 1 ) $-dimensional space defined by the equation $ F (y, x _ {1}, \dots, x _ {n} ) = 0 $. If the field of constants $ k $ is the field of complex numbers $ \mathbf C $ and if $ n = 1 $, then $ K _ {f} $ is identical with the field of meromorphic functions on the Riemann surface of the algebraic function. The field $ K _ {f} $ is an extension of finite type of the field of constants $ k $ of transcendence degree $ n $ (cf. Extension of a field). In particular, any $ n + 1 $ elements of this field are connected by an algebraic equation, so that each of them defines an algebraic function of the remaining elements. Any extension $ K $ of finite type of a field $ k $ of transcendence degree $ n $ is known as an algebraic function field in $ n $ variables (or, sometimes, as a function field). Each such field contains a purely transcendence extension $ k( x _ {1}, \dots, x _ {n} ) $ of the field $ k $ (called the field of rational functions in $ n $ variables). Any element $ y \in K $ satisfies some algebraic equation $ \Phi (y, x _ {1}, \dots, x _ {n} ) = 0 $, and can be considered as an algebraic function in the variables $ x _ {1}, \dots, x _ {n} $. Each field $ K $ of algebraic functions in $ n $ variables is isomorphic to the field of rational functions on some algebraic variety of dimension $ n $, which is called a model of $ K $. If the field of constants $ k $ is algebraically closed and of characteristic zero, then each algebraic function field has a non-singular projective model (cf. Resolution of singularities). Let $ S $ be the set of all non-trivial valuations (cf. Valuation) of an algebraic function field $ K $ which are non-negative on the field of constants. If provided with the natural topology, it is known as the abstract Riemann surface of the field $ K $[1]. In the case of algebraic functions in one variable, the Riemann surface coincides with the set of non-singular projective models, which in this case is uniquely defined up to an isomorphism. Many concepts and results in algebraic geometry on the model of a field $ K $ can be restated in the language of the theory of valuations of fields [1], [6]. A particularly close analogy holds for algebraic functions in one variable, the theory of which is practically identical with the theory of algebraic curves.

Each algebraic function field in one variable is the field of fractions of a Dedekind ring, so that many results and concepts of the theory of divisibility in algebraic number fields can be applied to function fields [12]. Many problems and constructions in algebraic number theory motivate similar problems and constructions in fields of algebraic functions and vice versa. For instance, the application of a Puiseux expansion to the theory of algebraic numbers led to the genesis of the $ p $-adic method in number theory, due to Hensel. Class field theory, which had originally belonged to the domain of algebraic numbers, was subsequently applied to functions [2]. An especially close analogy exists between algebraic number fields and algebraic function fields over a finite field of constants. For instance, the concept of a zeta-function is defined for the latter and the analogue of the Riemann hypothesis has been demonstrated for algebraic function fields (cf. Zeta-function in algebraic geometry).

References

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