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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 $$y_n=\min(y_{n-k_1}-y_{n-m_1},y_{n-k_2}-y_{n-m_2},\dots, y_{n-k_L}-y_{n-m_L})\tag{1}$$ with delay matrix $$K=\left(\matrix k_1&m_1\\ k_2&m_2,\\ \vdots&\vdots\\ k_L&m_L\endmatrix\right),\tag{2}$$ where $k_i,\,m_i\ge1$ for $1\le i \le L$. 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 $$P(x)=\cases x^k+x^{k-m}-1,&\text{if } k>m,\\ x^{m}-x^{m-k}+1,&\text{if } m>k.\endcases$$ For the polynomial $P(x)$ the authors obtain the following result: Theorem 4. (i) Suppose $j\ge1.$ The characteristic polynomial $P$ has a zero which is a primitive $6j$-th root of unity if and only if $$(k,\,m)\in\{(j,\,2j),(5j,\,4j)\}.$$ (ii) Any primitive root of unity $\rho$ which satisfies $P(\rho)=0$ must be a primitive $6j$-th root of unity for some $j\ge1.$ In addition each root of $P$ of modulus one has multiplicity $1$. In the case $L=2$, i.e for the equation $$ y_n=\min(y_{n-k_1}-y_{n-m_1},\,y_{n-k_2}-y_{n-m_2}).\tag{3}$$ 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 $k_2\vert m_2$. 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 $k_2\nmid m_2.$ 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 $$\limsup_{n\to+\infty}\frac{\vert y_n\vert}n<+\infty.$$ 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 $$\limsup_{n\to+\infty}\frac{\vert y_n\vert}n=\frac1{(k_2,m_1+m_2)}.$$ As a corollary of Theorems 7, 9, and 12 the authors prove the following Theorem 10. Let $\gcd(k_1,\,m_1,\,k_2,\,m_2)=1,\,k_1=k_2+m_1$. $k_2\vert m_2$. Then all integer solutions to equation ({3}) have period $p=m_1+m_2$ if and only if $k_2\vert m_2$. 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)
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