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Diskr. Mat., 2017, Volume 29, Issue 4, Pages 106–120 (Mi dm1435)  

On the number of integer points in a multidimensional domain

A. S. Rybakov


Abstract: We provide a new upper estimate for the modulus of the difference $|\Lambda\cap {\cal S}|-vol _n({\cal S})/det  \Lambda$, where ${\cal S}\subset \mathbb R^n$ is a set of volume $vol _n({\cal S})$ and $\Lambda\subset \mathbb R^n$ is a complete lattice with determinant $det  \Lambda$. This result has an important practical application, for example, in estimating the number of integer solutions of an arbitrary system of linear and nonlinear inequalities.

Keywords: integer lattice, number of integer points, Gaussian volume heuristic.

DOI: https://doi.org/10.4213/dm1435

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English version:
Discrete Mathematics and Applications, 2018, 28:6, 385–395

Bibliographic databases:

UDC: 511.9
Received: 22.05.2017

Citation: A. S. Rybakov, “On the number of integer points in a multidimensional domain”, Diskr. Mat., 29:4 (2017), 106–120; Discrete Math. Appl., 28:6 (2018), 385–395

Citation in format AMSBIB
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\paper On the number of integer points in a multidimensional domain
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\vol 29
\issue 4
\pages 106--120
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\jour Discrete Math. Appl.
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\vol 28
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\pages 385--395
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