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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).

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