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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

${y}_{n}=min\left({y}_{n-{k}_{1}}-{y}_{n-{m}_{1}},{y}_{n-{k}_{2}}-{y}_{n-{m}_{2}},\cdots ,{y}_{n-{k}_{L}}-{y}_{n-{m}_{L}}\right)\phantom{\rule{2.em}{0ex}}\left(1\right)$

with delay matrix

$K=\left(\begin{array}{cc}{k}_{1}& {m}_{1}\\ {k}_{2}& {m}_{2},\\ ⋮& ⋮\\ {k}_{L}& {m}_{L}\end{array}\right),\phantom{\rule{2.em}{0ex}}\left(2\right)$

where ${k}_{i},\phantom{\rule{0.166667em}{0ex}}{m}_{i}\ge 1$ for $1\le i\le L$. The authors put $k={k}_{1},\phantom{\rule{0.166667em}{0ex}}m={m}_{1}$ in the case $L=1,$ and note that the characteristic polynomial $P\left(x\right)$ for the equation (1) is

$P\left(x\right)=\left\{\begin{array}{cc}{x}^{k}+{x}^{k-m}-1,\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}k>m,\hfill \\ {x}^{m}-{x}^{m-k}+1,\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}m>k·\hfill \end{array}\right\$

For the polynomial $P\left(x\right)$ the authors obtain the following result:

Theorem 4. (i) Suppose $j\ge 1·$ The characteristic polynomial $P$ has a zero which is a primitive $6j$-th root of unity if and only if

$\left(k,\phantom{\rule{0.166667em}{0ex}}m\right)\in \left\{\left(j,\phantom{\rule{0.166667em}{0ex}}2j\right),\left(5j,\phantom{\rule{0.166667em}{0ex}}4j\right)\right\}·$

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

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

${y}_{n}=min\left({y}_{n-{k}_{1}}-{y}_{n-{m}_{1}},\phantom{\rule{0.166667em}{0ex}}{y}_{n-{k}_{2}}-{y}_{n-{m}_{2}}\right)·\phantom{\rule{2.em}{0ex}}\left(3\right)$

the authors obtain the following results.

Theorem 7. Let $gcd\left({k}_{1},\phantom{\rule{0.166667em}{0ex}}{m}_{1},\phantom{\rule{0.166667em}{0ex}}{k}_{2},\phantom{\rule{0.166667em}{0ex}}{m}_{2}\right)=1,\phantom{\rule{0.166667em}{0ex}}{k}_{1}={k}_{2}+{m}_{1}$ and ${k}_{2}|{m}_{2}$. Then all integer solutions to equation (3) must have period $p={m}_{1}+{m}_{2}$.

Theorem 8. Let $gcd\left({k}_{1},\phantom{\rule{0.166667em}{0ex}}{m}_{1},\phantom{\rule{0.166667em}{0ex}}{k}_{2},\phantom{\rule{0.166667em}{0ex}}{m}_{2}\right)=1$ 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 $gcd\left({k}_{1},\phantom{\rule{0.166667em}{0ex}}{m}_{1},\phantom{\rule{0.166667em}{0ex}}{k}_{2},\phantom{\rule{0.166667em}{0ex}}{m}_{2}\right)=1,\phantom{\rule{0.166667em}{0ex}}{k}_{1}={k}_{2}+{m}_{1}$ and ${k}_{2}\nmid {m}_{2}·$ Then there exists a periodic integer solution to equation (3) with minimal period ${k}_{2}$.

Theorem 11. Let $gcd\left({k}_{1},\phantom{\rule{0.166667em}{0ex}}{m}_{1},\phantom{\rule{0.166667em}{0ex}}{k}_{2},\phantom{\rule{0.166667em}{0ex}}{m}_{2}\right)=1$ and ${k}_{1}={k}_{2}+{m}_{1}·$ Then any integer solutions to the equation (3) must satisfy the inequality

$\underset{n\to +\infty }{lim sup}\frac{|{y}_{n}|}{n}<+\infty ·$

Theorem 12. Let $gcd\left({k}_{1},\phantom{\rule{0.166667em}{0ex}}{m}_{1},\phantom{\rule{0.166667em}{0ex}}{k}_{2},\phantom{\rule{0.166667em}{0ex}}{m}_{2}\right)=1,\phantom{\rule{0.166667em}{0ex}}{k}_{1}={k}_{2}+{m}_{1}$ and $d=gcd\left({k}_{2},{m}_{1}+{m}_{2}\right)>1$. Then there exists an integer solution to the equation (3) such that

$\underset{n\to +\infty }{lim sup}\frac{|{y}_{n}|}{n}=\frac{1}{\left({k}_{2},{m}_{1}+{m}_{2}\right)}·$

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

Theorem 10. Let $gcd\left({k}_{1},\phantom{\rule{0.166667em}{0ex}}{m}_{1},\phantom{\rule{0.166667em}{0ex}}{k}_{2},\phantom{\rule{0.166667em}{0ex}}{m}_{2}\right)=1,\phantom{\rule{0.166667em}{0ex}}{k}_{1}={k}_{2}+{m}_{1}$. ${k}_{2}|{m}_{2}$. 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:
 39A23 Periodic solutions (difference equations) 39A10 Additive difference equations 39A22 Growth, boundedness, comparison of solutions (difference equations)