Hilbert's thirteenth problem is one of the 23

Hilbert problems Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the pr ...

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 Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** 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 Vladimir Igorevich Arnold (or Arnol'd; , ; 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. He is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, and contributed to s ...

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 Erland Samuel Bring (19 August 1736 – 20 May 1798) was a Swedish mathematician. Bring studied at Lund University between 1750 and 1757. In 1762 he obtained a position of a reader in history and was promoted to professor in 1779. At Lund he wr ...

(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

, 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 was a Japanese mathematician and Michael Henry Strater Professor Emeritus of Mathematics at Princeton University who worked in number theory, automorphic forms, and arithmetic geometry. He was known for developing the theory of complex multip ...

(Arnold and Shimura 1976).

References

Shreeram Shankar Abhyankar Shreeram Shankar Abhyankar (22 July 1930 – 2 November 2012) was an Indian American mathematician known for his contributions to algebraic geometry. At the time of his death, he held the Marshall Distinguished Professor of Mathematics Chair ...

,
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

, ''Superposition of algebraic functions'' (1976), in ''Mathematical Developments Arising From Hilbert Problems'', Volume 1,

Proceedings of Symposia in Pure Mathematics The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

28 (1976), pp. 45-46. * * * English translation in: *

External links

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

Introduction

History

References

See also

External links