The lower bound for the number of local minima of integral lattices (Q950788)

A lower bound for the maximal number of local minima of integral lattices \(\mathfrak M(\Gamma)\) with preassigned determinant has been derived. This bound coincides with the upper one up to a constant. The desired lower bound \[ T_s(n)=\sup_{\Gamma}\#\mathfrak M(\Gamma)\geq \frac 1{s^2}\left(\frac 1{2(s+1)}\right)^s \log^s_2N+O_s(\log^{s-1}_2N) \] is an immediate consequence of the author's main result. Theorem. For any positive integer \(N\) and \(s\geq 2\), \[ \sum_{a_1,\dots,a_s=1}^N \#\mathfrak M(\Gamma_N(a))\geq \frac 1{s^2}\left(\frac 1{2(s+1)}\right)^s N^s\log^s_2N+O_s(N^s\log^{s-1}_2N). \]