Askold Khovanskii - Topological Galois Theory: Solvability and Unsolvability of Equations in Finite Terms

Topological Galois Theory: Solvability and Unsolvability of Equations in Finite Terms — read online for free the complete book (whole text) full work

Below is the text of the book, divided by pages. System saving the place of the last page read, allows you to conveniently read the book "Topological Galois Theory: Solvability and Unsolvability of Equations in Finite Terms" online for free, without having to search again every time where you left off. Put a bookmark, and you can go to the page where you finished reading at any time.

Light

Font size:

↓

↑

Georgia Tahoma Arial Verdana

Reset

Interval:

↓

↑

Bookmark:

Make

1234567...159

Springer-Verlag Berlin Heidelberg 2014

Askold Khovanskii Topological Galois Theory Springer Monographs in Mathematics 10.1007/978-3-642-38871-2_1

1. Construction of Liouvillian Classes of Functions and Liouvilles Theory

Askold Khovanskii 1

(1)

Department of Mathematics, University of Toronto, Toronto, Ontario, Canada

Some algebraic and differential equations are explicitly solvable. What does this mean? If an explicit solution is presented, the question answers itself. However, in most cases, every attempt to solve an equation explicitly is doomed to failure. We are then tempted to prove that certain equations have no explicit solutions. It is now necessary to define exactly what we mean by explicit solutions (otherwise, it is unclear what we are trying to prove).

From the modern viewpoint, the classical works on the subject lack rigorous definitions and statements of theorems. Nonetheless, it is clear that Liouville understood exactly what he was proving. He not only stated the problems on solvability of equations by elementary functions and by quadratures, but he also algebraized them. His work made it possible to define all such notions over an arbitrary differential field. But the standards of mathematical rigor were different in the time of Liouville. Indeed, according to Kolchin [], even Picard failed to give accurate, unambiguous definitions. Kolchins work satisfies modern standards, but his definitions are given for abstract differential fields from the very beginning.

However, the indefinite integral of an elementary function and the solution of a linear differential equation are functions rather than elements of an abstract differential field. In function spaces, for example, apart from differentiation and algebraic operations, an absolutely nonalgebraic operation is defined, namely composition. Anyhow, function spaces provide greater means for writing explicit formulas than abstract differential fields. Moreover, we should take into account that functions can be multivalued, can have singularities, and so on.

In function spaces, it is not hard to formalize the problem of unsolvability of equations in explicit form, and in this book, we are interested in this particular problem. One can proceed as follows: fix a class of functions and say that an equation is solvable explicitly if its solution belongs to this class. Different classes of functions correspond to different notions of solvability.

1.1 Defining Classes of Functions by Lists of Basic Functions and Admissible Operations

A class of functions can be introduced by specifying a list of basic functions and a list of admissible operations . Given these two lists, the class of functions is defined as the set of all functions that can be obtained from the basic functions by repeated application of admissible operations. In Sect., we define Liouvillian classes of functions in exactly this way.

Liouvillian classes of functions, which appear in problems of solvability in finite terms, contain multivalued functions. Thus the basic terminology should be made clear. In this section, we work with multivalued functions globally, which leads to a more general understanding of classes of functions defined by lists of basic functions and admissible operations. In this global version, a multivalued function is regarded as a single entity, and we can define operations on multivalued functions .

The result of such an operation is a set of multivalued functions; every element of this set is referred to as a function obtained from the given functions by the given operation. The class of functions is defined as the set of all (multivalued) functions that can be obtained from the basic functions by repeated application of admissible operations.

Let us define, for example, the sum of two multivalued functions of one variable.

Definition 1.1

Take an arbitrary point a on the complex line, a germ f a of an analytic function f at the point a , and a germ g a of an analytic function g at the same point a . We say that the multivalued function generated by the germ is representable as the sum of the functions f and g .

For example, it is easy to see that exactly two functions are representable in the form , namely, and f 20. Other operations on multivalued functions are defined in exactly the same way. For a class of multivalued functions, being stable under addition means that together with any pair of its functions, this class contains all functions representable as their sum. The same applies to all other operations on multivalued functions understood in the same sense asabove.

In the definition given above, it is not only the operation of addition that plays a key role but also the operation of analytic continuation hidden in the notion of multivalued function. Indeed, consider the following example. Let f 1 be an analytic function defined on an open subset U of the complex line and admitting no analytic continuation outside of U , and let f 2 be an analytic function on U given by the formula . According to our definition, the zero function is representable in the form f 1 + f 2 on the entire complex line . From the commonly accepted viewpoint, the equality holds inside the region U but not outside.

In working with multivalued functions globally, we do not insist on the existence of a common region where all necessary operations would be performed on single-valued branches of multivalued functions. A first operation can be performed in a first region, then a second operation can be performed in a second, different, region on analytic continuations of functions obtained in the first step. In essence, this more general understanding of operations is equivalent to including analytic continuation in the list of admissible operations on analytic germs. For functions of a single variable, it is possible to obtain topological obstructions even with this more general understanding of operations on multivalued analytic functions.

In the sequel, in considering topological obstructions to the membership of an analytic function of a single variable in a certain class, we will always mean this global definition of the function class via lists of basic functions and admissible operations.

For functions of several variables, things do not work in this general setting, and we are forced to adopt a more restrictive formulation (see Sect., in which we deal with multivariable functions.

1.2 Liouvillian Classes of Functions of a Single Variable

In this section, we define Liouvillian classes of functions of a single variable (for the multivariable case, the corresponding definitions are given in Chap.). We will describe these classes by lists of basic functions and admissible operations.

Next page

1234567...159

Light

Font size:

↓

↑

Georgia Tahoma Arial Verdana

Reset

Interval:

↓

↑

Bookmark:

Make

Similar books «Topological Galois Theory: Solvability and Unsolvability of Equations in Finite Terms»

Look at similar books to Topological Galois Theory: Solvability and Unsolvability of Equations in Finite Terms. We have selected literature similar in name and meaning in the hope of providing readers with more options to find new, interesting, not yet read works.

Harendra Singh (editor)

Methods of Mathematical Modelling: Fractional Differential Equations (Mathematics and its Applications)

Joël Bellaïche

The Eigenbook: Eigenvarieties, families of Galois representations, p-adic L-functions

N. U. Ahmed

Optimal Control of Dynamic Systems Driven by Vector Measures: Theory and Applications

T. A. Burton

Stability by Fixed Point Theory for Functional Differential Equations

Galois Evariste

The equation that couldnt be solved: how mathematical genius discovered the language of symmetry

Alexey L. Gorodentsev

Algebra II: Textbook for Students of Mathematics

coll.

Equations and Inequalities : Elementary Problems and Theorems in Algebra and Number Theory

Frédéric Butin

Algebra: Polynomials, Galois Theory and Applications

Zemanian

Distribution Theory and Transform Analysis : An Introduction to Generalized Functions, with Applications

Constantin Corduneanu

Functional Differential Equations: Advances and Applications

Avner Friedman

Generalized Functions and Partial Differential Equations

Mak Trifkovic

Algebraic Theory of Quadratic Numbers

Discussion, reviews of the book Topological Galois Theory: Solvability and Unsolvability of Equations in Finite Terms and just readers' own opinions. Leave your comments, write what you think about the work, its meaning or the main characters. Specify what exactly you liked and what you didn't like, and why you think so.