Hilbert's thirteenth problem
Encyclopedia
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 or not a solution exists for all 7th-degree equations
using 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 actual question is more easily posed however in terms of continuous function
s. Hilbert considered the general seventh-degree equation
and asked whether its solution, x, a function of the three variables a, b and c, can be expressed using a finite number of two-variable functions.
A more general question is: 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.
Arnold later returned to the question, jointly with Goro Shimura
(V. I. Arnold and G. Shimura, Superposition of algebraic functions (1976), in Mathematical Developments Arising From Hilbert's Problems).
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
. It entails proving whether or not a solution exists for all 7th-degree equations
Septic equation
In mathematics, a septic equation, heptic equation or septimic equation is a polynomial equation of degree seven. It is of the form:ax^7+bx^6+cx^5+dx^4+ex^3+fx^2+gx+h=0,\,...
using 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 actual question is more easily posed however in terms of continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
s. Hilbert considered the general seventh-degree equation
and asked whether its solution, x, a function of the three variables a, b and c, can be expressed using a finite number of two-variable functions.
A more general question is: can every continuous function of three variables be expressed as a composition
Function composition
In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...
of finitely many continuous functions of two variables? The affirmative answer to this general question was given in 1957 by Vladimir Arnold
Vladimir Arnold
Vladimir Igorevich Arnold was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable Hamiltonian systems, he made important contributions in several areas including dynamical systems theory, catastrophe theory,...
, then only nineteen years old and a student of Andrey Kolmogorov
Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov was a Soviet mathematician, preeminent in the 20th century, who advanced various scientific fields, among them probability theory, topology, intuitionistic logic, turbulence, classical mechanics and computational complexity.-Early life:Kolmogorov was born at Tambov...
. 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.
Arnold later returned to the question, jointly with Goro Shimura
Goro Shimura
is a Japanese mathematician, and currently a professor emeritus of mathematics at Princeton University.Shimura was a colleague and a friend of Yutaka Taniyama...
(V. I. Arnold and G. Shimura, Superposition of algebraic functions (1976), in Mathematical Developments Arising From Hilbert's Problems).