zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Periodicity and boundedness for the integer solutions to a minimum-delay difference equation. (English) Zbl 1202.39012

The authors consider the difference equations

y n =min(y n-k 1 -y n-m 1 ,y n-k 2 -y n-m 2 ,,y n-k L -y n-m L )(1)

with delay matrix

K=k 1 m 1 k 2 m 2 ,k L m L ,(2)

where k i ,m i 1 for 1iL. The authors put k=k 1 ,m=m 1 in the case L=1, and note that the characteristic polynomial P(x) for the equation (1) is

P(x)=x k +x k-m -1,ifk>m,x m -x m-k +1,ifm>k·

For the polynomial P(x) the authors obtain the following result:

Theorem 4. (i) Suppose j1· The characteristic polynomial P has a zero which is a primitive 6j-th root of unity if and only if

(k,m){(j,2j),(5j,4j)}·

(ii) Any primitive root of unity ρ which satisfies P(ρ)=0 must be a primitive 6j-th root of unity for some j1· In addition each root of P of modulus one has multiplicity 1.

In the case L=2, i.e for the equation

y n =min(y n-k 1 -y n-m 1 ,y n-k 2 -y n-m 2 )·(3)

the authors obtain the following results.

Theorem 7. Let gcd(k 1 ,m 1 ,k 2 ,m 2 )=1,k 1 =k 2 +m 1 and k 2 |m 2 . Then all integer solutions to equation (3) must have period p=m 1 +m 2 .

Theorem 8. Let gcd(k 1 ,m 1 ,k 2 ,m 2 )=1 and k 1 =k 2 +m 1 . Then there exists a nontrivial integer solutions to equation (3) which has period p=m 1 +m 2 . The authors give an effective construction of such a solution.

Theorem 9. Let gcd(k 1 ,m 1 ,k 2 ,m 2 )=1,k 1 =k 2 +m 1 and k 2 m 2 · Then there exists a periodic integer solution to equation (3) with minimal period k 2 .

Theorem 11. Let gcd(k 1 ,m 1 ,k 2 ,m 2 )=1 and k 1 =k 2 +m 1 · Then any integer solutions to the equation (3) must satisfy the inequality

lim sup n+ |y n | n<+·

Theorem 12. Let gcd(k 1 ,m 1 ,k 2 ,m 2 )=1,k 1 =k 2 +m 1 and d=gcd(k 2 ,m 1 +m 2 )>1. Then there exists an integer solution to the equation (3) such that

lim sup n+ |y n | n=1 (k 2 ,m 1 +m 2 )·

As a corollary of Theorems 7, 9, and 12 the authors prove the following

Theorem 10. Let gcd(k 1 ,m 1 ,k 2 ,m 2 )=1,k 1 =k 2 +m 1 . k 2 |m 2 . Then all integer solutions to equation (3) have period p=m 1 +m 2 if and only if k 2 |m 2 . All constructions are effective.

Unexpected results, detailed proofs, supplementary material (interesting observations, a Conjecture, some Open Questions) – all this left a very good impression (some stylistic bugs and misprints, which do not deserve to be mentioned, cannot destroy it).

MSC:
39A23Periodic solutions (difference equations)
39A10Additive difference equations
39A22Growth, boundedness, comparison of solutions (difference equations)