Skolem problem

{{Short description|Unsolved problem in mathematics}}

{{Unsolved|mathematics|Is there an algorithm to test whether a constant-recursive sequence has a zero?}}

In mathematics, the Skolem problem is the problem of determining whether the values of a constant-recursive sequence include the number zero. The problem can be formulated for recurrences over different types of numbers, including integers, rational numbers, and algebraic numbers. It is not known whether there exists an algorithm that can solve this problem.{{citation

| last1 = Ouaknine | first1 = Joël

| last2 = Worrell | first2 = James

| contribution = Decision problems for linear recurrence sequences

| doi = 10.1007/978-3-642-33512-9_3

| mr = 3040104

| pages = 21–28

| publisher = Springer-Verlag | location = Heidelberg

| series = Lecture Notes in Computer Science

| title = Reachability Problems: 6th International Workshop, RP 2012, Bordeaux, France, September 17–19, 2012, Proceedings

| volume = 7550

| year = 2012| isbn = 978-3-642-33511-2

}}.

A linear recurrence relation expresses the values of a sequence of numbers as a linear combination of earlier values; for instance, the Fibonacci numbers may be defined from the recurrence relation

:{{math|1=F(n) = F(n − 1) + F(n − 2)}}

together with the initial values {{math|1=F(0) = 0}} and {{math|1=F(1) = 1}}.

The Skolem problem is named after Thoralf Skolem, because of his 1933 paper proving the Skolem–Mahler–Lech theorem on the zeros of a sequence satisfying a linear recurrence with constant coefficients.{{citation|last=Skolem|first=Th.|title=Einige Sätze über gewisse Reihenentwicklungen und exponentiale Beziehungen mit Anwendung auf diophantische Gleichungen|journal=Oslo Vid. Akad. Skrifter|volume=I|number=6|year=1933}}. {{harvtxt|Ouaknine|Worrell|2012}} instead cite a 1934 paper of Skolem for this result. This theorem states that, if such a sequence has zeros, then with finitely many exceptions the positions of the zeros repeat regularly. Skolem proved this for recurrences over the rational numbers, and Mahler and Lech extended it to other systems of numbers. However, the proofs of the theorem do not show how to test whether there exist any zeros.

There does exist an algorithm to test whether a constant-recursive sequence has infinitely many zeros, and if so to construct a decomposition of the positions of those zeros into periodic subsequences, based on the algebraic properties of the roots of the characteristic polynomial of the given recurrence.{{citation

| last1 = Berstel | first1 = Jean

| last2 = Mignotte | first2 = Maurice

| issue = 2

| journal = Bulletin de la Société Mathématique de France

| language = French

| mr = 0414475

| pages = 175–184

| title = Deux propriétés décidables des suites récurrentes linéaires

| url = http://www.numdam.org/item?id=BSMF_1976__104__175_0

| volume = 104

| year = 1976| doi = 10.24033/bsmf.1823

| doi-access = free

| url-access = subscription

}}. The remaining difficult part of the Skolem problem is determining whether the finite set of non-repeating zeros is empty or not.

Partial solutions to the Skolem problem are known, covering the special case of the problem for recurrences of degree at most four. However, these solutions do not apply to recurrences of degree five or more.{{citation

| last1 = Mignotte | first1 = M.

| last2 = Shorey | first2 = T. N. | author2-link = Tarlok Nath Shorey

| last3 = Tijdeman | first3 = R. | author3-link = Robert Tijdeman

| journal = Journal für die Reine und Angewandte Mathematik

| mr = 743965

| pages = 63–76

| title = The distance between terms of an algebraic recurrence sequence

| volume = 349

| year = 1984}}.{{citation

| last = Vereshchagin | first = N. K.

| issue = 2

| journal = Matematicheskie Zametki

| language = Russian

| mr = 808885

| pages = 177–189, 347

| title = The problem of the appearance of a zero in a linear recursive sequence

| volume = 38

| year = 1985}}.

For integer recurrences, the Skolem problem is known to be NP-hard.{{citation

| last1 = Blondel | first1 = Vincent D.

| last2 = Portier | first2 = Natacha

| doi = 10.1016/S0024-3795(01)00466-9

| journal = Linear Algebra and Its Applications

| mr = 1917474

| pages = 91–98

| title = The presence of a zero in an integer linear recurrent sequence is NP-hard to decide

| volume = 351/352

| year = 2002}}.