Peregrine soliton

Hermite's problem is an open problem in mathematics posed by Charles Hermite in 1848. He asked for a way of expressing real numbers as sequences of natural numbers, such that the sequence is eventually periodic precisely when the original number is a cubic irrational.

Motivation

A standard way of writing real numbers is by their decimal representation, such as:

$x=a_0.a_1{a}_2{a}_3\dots$

where a₀ is an integer, the integer part of x, and a₁, a₂, a₃… are integers between 0 and 9. Given this representation the number x is equal to

$x=\sum _{n=0}^{\mathrm{\infty }}\frac{a_n}{10^n}.$

The real number x is a rational number only if its decimal expansion is eventually periodic, that is if there are natural numbers N and p such that for every n ≥ N it is the case that a_n+p = a_n.

Another way of expressing numbers is to write them as continued fractions, as in:

$x=[a_0;a_1,a_2,a_3,\dots ],$

where a₀ is an integer and a₁, a₂, a₃… are natural numbers. From this representation we can recover x since

$x=a_0+\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\ddots }}}.$

If x is a rational number then the sequence (a_n) terminates after finitely many terms. On the other hand, Euler proved that irrational numbers require an infinite sequence to express them as continued fractions. Moreover, this sequence is eventually periodic (again, so that there are natural numbers N and p such that for every n ≥ N we have a_n+p = a_n), if and only if x is a quadratic irrational.

Hermite's question

Rational numbers are algebraic numbers that satisfy a polynomial of degree 1, while quadratic irrationals are algebraic numbers that satisfy a polynomial of degree 2. For both these sets of numbers we have a way to construct a sequence of natural numbers (a_n) with the property that each sequence gives a unique real number and such that this real number belongs to the corresponding set if and only if the sequence is eventually periodic.

In 1848 Charles Hermite wrote a letter to Carl Gustav Jacob Jacobi asking if this situation could be generalised, that is can one assign a sequence of natural numbers to each real number x such that the sequence is eventually periodic precisely when x is a cubic irrational, that is an algebraic number of degree 3? Or, more generally, for each natural number d is there a way of assigning a sequence of natural numbers to each real number x that can pick out when x is algebraic of degree d?

Approaches

Sequences that attempt to solve Hermite's problem are often called multidimensional continued fractions. Jacobi himself came up with an early example, finding a sequence corresponding to each pair of real numbers (x,y) that acted as a higher dimensional analogue of continued fractions. He hoped to show that the sequence attached to (x, y) was eventually periodic if and only if both x and y belonged to a cubic number field, but was unable to do so and whether this is the case remains unsolved.

Rather than generalising continued fractions, another approach to the problem is to generalise Minkowski's question mark function. This function ? : [0, 1] → [0, 1] also picks out quadratic irrational numbers since ?(x) is rational if and only if x is either rational or a quadratic irrational number, and moreover x is rational if and only if ?(x) is a dyadic rational, thus x is a quadratic irrational precisely when ?(x) is a non-dyadic rational number. Various generalisations of this function to either the unit square [0, 1] × [0, 1] or the two-dimensional simplex have been made, though none has yet solved Hermite's problem.

References

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