The lower bound for the number of local minima of integral lattices
type: Item
instance of: scholarly article
title: The lower bound for the number of local minima of integral lattices
zbMATH Open document ID: 1204.11101
author: M. O. Avdeeva
published in: Journal of Mathematical Sciences (New York)
publication date: 2008-10-28T00:00:00Z
review text: 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 \(s\geq 2\), \[ \sum_{a_1,\dots,a_s=1}^N \#\mathfrak M(\Gamma_N(a))\geq \frac 1{s^2}\left(\frac 1{2(s+1)}\right)^s N^s\log^s_2N+O_s(N^s\log^{s-1}_2N). \]
reviewed by: Olaf Ninnemann
Mathematics Subject Classification ID: 11H50
zbMATH DE Number: 5358111
MaRDI profile type: Publication
full work available at URL: s10958-007-0377-x
OpenAlex ID: W2023095590
cites work: Q5460324
cites work: On the error of number-theoretic quadrature formulas
cites work: Q3418759
DOI: 10.1007/S10958-007-0377-X
description: scientific article; zbMATH DE number 5358111