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Hilbert's thirteenth problem

December 15, 2023

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 7th-degree equations using algebraic (variant: continuous) functions of two arguments. It was first presented in the context of nomography, 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 edit

Using the methods pioneered by Tschirnhaus (1683), Bring (1786), and Jerrard (1834), William Rowan Hamilton showed in 1836 that every seventh-degree equation can be reduced via radicals to the form  .

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 of a finite number of two-variable functions.

History edit

Hilbert originally posed his problem for algebraic functions (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 of 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 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 edit

  • Shreeram S. 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 G. 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.
  • D. Hilbert, "Über die Gleichung neunten Grades", Math. Ann. 97 (1927), 243–250, available online
  • G. G. Lorentz, Approximation of Functions (1966), Ch. 11
  • A. G. Vitushkin, "On Hilbert's thirteenth problem and related questions", Uspekhi Mat. Nauk 59:1 (2004), 11 24. (English Translation in Russian Math. Surveys 59 (2004), no. 1, 11–25) DOI: 10.1070/RM2004v059n01ABEH000698.
  • Farb, Benson; Wolfson, Jesse (2020). "Resolvent degree, Hilbert's 13th Problem and geometry". L'Enseignement Mathématique. 65 (3): 303–376. arXiv:1803.04063. doi:10.4171/LEM/65-3/4-2. ISSN 0013-8584. S2CID 14000951.

See also edit

External links edit

  • Ornes, Stephen (14 January 2021). "Mathematicians Resurrect Hilbert's 13th Problem". Quanta Magazine.

December 15, 2023
hilbert, thirteenth, problem, this, article, includes, list, references, related, reading, external, links, sources, remain, unclear, because, lacks, inline, citations, please, help, improve, this, article, introducing, more, precise, citations, january, 2021,. This article includes a list of references related reading or external links but its sources remain unclear because it lacks inline citations Please help to improve this article by introducing more precise citations January 2021 Learn how and when to remove this template message 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 7th degree equations using algebraic variant continuous functions of two arguments It was first presented in the context of nomography 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 Contents 1 Introduction 2 History 3 References 4 See also 5 External linksIntroduction editUsing the methods pioneered by Tschirnhaus 1683 Bring 1786 and Jerrard 1834 William Rowan Hamilton showed in 1836 that every seventh degree equation can be reduced via radicals to the form x 7 a x 3 b x 2 c x 1 0 displaystyle x 7 ax 3 bx 2 cx 1 0 nbsp 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 of a finite number of two variable functions History editHilbert originally posed his problem for algebraic functions 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 of 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 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 editShreeram S Abhyankar Hilbert s Thirteenth Problem Algebre non commutative groupes quantiques et invariants Reims 1995 1 11 Semin Congr 2 Soc Math France Paris 1997 V I Arnold and G 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 D Hilbert Uber die Gleichung neunten Grades Math Ann 97 1927 243 250 available online G G Lorentz Approximation of Functions 1966 Ch 11 A G Vitushkin On Hilbert s thirteenth problem and related questions Uspekhi Mat Nauk 59 1 2004 11 24 English Translation in Russian Math Surveys 59 2004 no 1 11 25 DOI 10 1070 RM2004v059n01ABEH000698 Farb Benson Wolfson Jesse 2020 Resolvent degree Hilbert s 13th Problem and geometry L Enseignement Mathematique 65 3 303 376 arXiv 1803 04063 doi 10 4171 LEM 65 3 4 2 ISSN 0013 8584 S2CID 14000951 See also editSeptic equationExternal links editOrnes Stephen 14 January 2021 Mathematicians Resurrect Hilbert s 13th Problem Quanta Magazine Retrieved from https en wikipedia org w index php title Hilbert 27s thirteenth problem amp oldid 1162836501, wikipedia, wiki, book, books, library,

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