Hilbert's thirteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by

David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental idea ...

. It entails proving whether a solution exists for all 7th-degree equations using algebraic (variant: continuous) functions of two

arguments An argument is a series of sentences, statements, or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persua ...

. It was first presented in the context of

nomograph A nomogram (), also called a nomograph, alignment chart, or abac, is a graphical calculating device, a two-dimensional diagram designed to allow the approximate graphical computation of a mathematical function. The field of nomography was inve ...

y, and in particular "nomographic construction" — a process whereby a function of several variables is constructed using functions of two variables. The variant for continuous functions was resolved affirmatively in 1957 by Vladimir Arnold when he proved the Kolmogorov–Arnold representation theorem, but the variant for algebraic functions remains unresolved.

Introduction

Using the methods pioneered by

Ehrenfried Walther von Tschirnhaus Ehrenfried Walther von Tschirnhaus or Tschirnhauß (; 10 April 1651 – 11 October 1708) was a German mathematician, physicist, physician, and philosopher. He introduced the Tschirnhaus transformation and is considered by some to have been the ...

(1683), Erland Samuel Bring (1786), and George Jerrard (1834),

William Rowan Hamilton Sir William Rowan Hamilton (4 August 1805 – 2 September 1865) was an Irish astronomer, mathematician, and physicist who made numerous major contributions to abstract algebra, classical mechanics, and optics. His theoretical works and mathema ...

showed in 1836 that every seventh-degree equation can be reduced via radicals to the form

x^7 + ax^3 + bx^2 + cx + 1 = 0

. Regarding this equation, Hilbert asked whether its solution, ''x'', considered as a function of the three variables ''a'', ''b'' and ''c'', can be expressed as the

composition Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography * Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include ...

of a finite number of two-variable functions.

History

Hilbert originally posed his problem for

algebraic function In mathematics, an algebraic function is a function that can be defined as the root of an irreducible polynomial equation. Algebraic functions are often algebraic expressions using a finite number of terms, involving only the algebraic operati ...

s (Hilbert 1927, "...Existenz von algebraischen Funktionen...", i.e., "...existence of algebraic functions..."; also see Abhyankar 1997, Vitushkin 2004). However, Hilbert also asked in a later version of this problem whether there is a solution in the

class Class, Classes, or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used d ...

continuous functions In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...

. A generalization of the second ("continuous") variant of the problem is the following question: can every continuous function of three variables be expressed as a

of finitely many continuous functions of two variables? The affirmative answer to this general question was given in 1957 by Vladimir Arnold, then only nineteen years old and a student of

Andrey Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Soviet ...

. Kolmogorov had shown in the previous year that any function of several variables can be constructed with a finite number of three-variable functions. Arnold then expanded on this work to show that only two-variable functions were in fact required, thus answering Hilbert's question when posed for the class of continuous functions. Arnold later returned to the algebraic version of the problem, jointly with Goro Shimura (Arnold and Shimura 1976).

References

* Shreeram Shankar Abhyankar,
Hilbert's Thirteenth Problem
, ''Algèbre non commutative, groupes quantiques et invariants'' (Reims, 1995), 1–11, ''Sémin. Congr.'', 2, Soc. Math. France, Paris, 1997. * V. I. Arnold and Goro Shimura, ''Superposition of algebraic functions'' (1976), in ''Mathematical Developments Arising From Hilbert Problems'', Volume 1, Proceedings of Symposia in Pure Mathematics 28 (1976), pp. 45-46. * * * English translation in: *

External links

* {{Hilbert's problems Polynomials #13 Disproved conjectures

Introduction

History

References

See also

External links