Dual Diophantine systems of linear inequalities

V. G. Zhuravlev

Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 2, Tome 479 (2019), pp. 23-51

Résumé

A modified version of the $\mathcal{L}$-algorithm is proposed. Using this algorithm anyone can build an infinite sequence of integer solutions for dual systems of linear inequalities $\mathcal{S}$ and $\mathcal{S}^*$ of $d+1$ variables, consisting respectively of $k^{\perp}$ and $k^{* \perp} $ inequalities, where $k^{\perp} + k^{* \perp} = d + 1$. Solutions are obtained by using two recurrence relations of the order $d+1$. Approximations in the systems of inequalities $\mathcal{S}$ and $ \mathcal {S}^* $ is carried out with Diophantine exponents $ \frac {d + 1-k^{\perp}} { k^{\perp}} - \varrho $ and $\frac{d + 1-k ^{*\perp}} { k^{*\perp}} - \varrho $, where the deviation $ \varrho> 0 $ can be made arbitrarily small due to a suitable choice of the recurrence relations. The $ \mathcal{L}$-algorithm is based on a method of localizing units in algebraic number fields.