This article is cited in 1 paper

The distribution of solutions of a determinantal equation

A. V. Ustinov

Institute for Applied Mathematics, Khabarovsk Division, Far-Eastern Branch of the Russian Academy of Sciences, Dzerzhinsky st., 54, Khabarovsk, 680000, Russia

Abstract: In 1964, Linnik and Skubenko established the equidistribution of the integral points on the determinantal surface $\det X=P$, where $X$ is a $(3\times 3)$ matrix with independent entries and $P$ is an increasing parameter. Their method involved reducing the problem by one dimension (that is, to the determinantal equations with a $(2\times 2)$ matrix). In this paper a more precise version of the Linnik-Skubenko reduction is proposed. It can be applied to a wider range of problems arising in the geometry of numbers and in the theory of three-dimensional Voronoi-Minkowski continued fractions.
Bibliography: 24 titles.

Keywords: lattices, Kloosterman sums.

UDC: 511.336+514.174.6

MSC: Primary 55R80; Secondary 11N45

Received: 20.05.2014 and 27.01.2015

DOI: 10.4213/sm8387

 English version: Sbornik: Mathematics, 2015, 206:7, 988–1019

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