Geodesic
On the number of local minima of integer lattices
Dalʹnevostočnyj matematičeskij žurnal, Tome 11 (2011) no. 2, pp. 149-154

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Let $E_s(N)$ be the average number of local minima of $s$-dimensional integer lattices with determinant equals $N$. We prove the following estimates $$ \frac{2^{-1}}{(s-1)!}+O_s\left(\frac{1}{\ln N}\right)\le\frac{E_s(N)}{\ln^{s-1}N}\le\frac{2^s}{(s-1)!}+O_s\left(\frac{1}{\ln N}\right) $$ for any prime $N$. Using this result we have a new lower bound for maximum number of local minima of integer lattices. 
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     author = {A. A. Illarionov and Y. A. Soyka},
     title = {On the number of local minima of integer lattices},
     journal = {Dalʹnevosto\v{c}nyj matemati\v{c}eskij \v{z}urnal},
     pages = {149--154},
     publisher = {mathdoc},
     volume = {11},
     number = {2},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DVMG_2011_11_2_a2/}
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A. A. Illarionov; Y. A. Soyka. On the number of local minima of integer lattices. Dalʹnevostočnyj matematičeskij žurnal, Tome 11 (2011) no. 2, pp. 149-154. http://geodesic.mathdoc.fr/item/DVMG_2011_11_2_a2/