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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


(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).

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