View source for Hilbert's thirteenth problem

{{Short description|On solutions of 7th-degree equations}}
{{No footnotes|date=January 2021}}
'''Hilbert's thirteenth  problem''' is one of the 23 [[Hilbert problems]] set out in a celebrated list compiled in 1900 by [[David Hilbert]].  It entails proving whether a solution exists for all [[Septic equation|7th-degree equations]] using [[algebraic function|algebraic]] (variant: [[continuous function|continuous]]) [[Function (mathematics)|functions]] of two [[argument of a function|arguments]]. It was first presented in the context of [[nomograph]]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]] (1683), [[Erland Samuel Bring]] (1786), and [[George Jerrard]] (1834), [[William Rowan Hamilton]] showed in 1836 that every [[Septic equation|seventh-degree]] equation can be reduced via radicals to the form <math>x^7 + ax^3 + bx^2 + cx + 1 = 0</math>. 

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 [[function composition|composition]] of a finite number of two-variable functions.

==History==
Hilbert originally posed his problem for [[algebraic function]]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 (set theory)|class]] of [[continuous function|continuous functions]]. 

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 [[function composition|composition]] 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]]. 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]], "[http://www.emis.de/journals/SC/1997/2/pdf/smf_sem-cong_2_1-11.pdf  Hilbert's Thirteenth Problem]", ''Algèbre non commutative, groupes quantiques et invariants'' (Reims, 1995), 1–11, ''Sémin. Congr.'', 2, Soc. Math. France, Paris, 1997. {{MR|1601178}}
* [[Vladimir Arnold|V. I. Arnold]] and [[Goro Shimura]], ''Superposition of algebraic functions'' (1976), in ''Mathematical Developments Arising From Hilbert Problems'', Volume 1, [[American Mathematical Society|Proceedings of Symposia in Pure Mathematics]] 28 (1976), pp. 45-46.
* {{cite journal|first=David |last=Hilbert|author-link=David Hilbert|title=Über die Gleichung neunten Grades|journal= [[Mathematische Annalen]] |volume=97|issue=1|year=1927| pages=243–250|url=https://eudml.org/doc/182637| jfm=52.0103.02|doi=10.1007/BF01447867|mr=1512361}}
* {{cite book|first=George G. |last=Lorentz|author-link= George G. Lorentz|title=Approximation of Functions| year=1966|publisher=[[Holt McDougal|Holt, Rinehart and Winston]]|location=New York&nbsp;Chicago&nbsp;Toronto|at=Chapter 11|mr=0213785}}
* {{cite journal|first=Anatoli Georgievich|last= Vitushkin|author-link=Anatoli Vitushkin|url=http://www.mathnet.ru/links/dd8ef2eead217b7f2f2b8c3b8e0b0b5b/rm698.pdf|title= 13-я проблема Гильберта и смежные вопросы|journal=[[Russian Mathematical Surveys|Uspekhi Matematicheskikh Nauk]]|volume=59|issue=1|year=2004| pages=11–24|doi=10.4213/rm698}} English translation in: {{cite journal| url=https://iopscience.iop.org/article/10.1070/RM2004v059n01ABEH000698|title=On Hilbert's thirteenth problem and related questions |journal=[[Russian Mathematical Surveys]]|volume=59 |year=2004|issue= 1|pages=11–25|doi= 10.1070/RM2004v059n01ABEH000698|mr=2068840 |last1=Vitushkin |first1=A. G. |bibcode=2004RuMaS..59...11V |url-access=subscription}}
*{{Cite journal|last1=Farb|first1=Benson|author1-link=Benson Farb|last2=Wolfson|first2=Jesse|date=2020|title=Resolvent degree, Hilbert's 13th Problem and geometry|url=https://www.ems-ph.org/doi/10.4171/LEM/65-3/4-2|journal=[[L'Enseignement mathématique]]|language=en|volume=65|issue=3|pages=303–376|arxiv=1803.04063|doi=10.4171/LEM/65-3/4-2|s2cid=14000951|issn=0013-8584|mr=4113045}}

== See also ==
* [[Septic equation]]
* [[Thomae's formula]]

== External links ==
* {{Cite web|first=Stephen|last=Ornes|date=14 January 2021|title=Mathematicians Resurrect Hilbert's 13th Problem
|url=https://www.quantamagazine.org/mathematicians-resurrect-hilberts-13th-problem-20210114/|website=[[Quanta Magazine]]}}
{{Hilbert's problems}}

[[Category:Polynomials]]
[[Category:Hilbert's problems|#13]]
[[Category:Disproved conjectures]]