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Suppose that $\Lambda \subseteq \mathbb{R}^n$ is a lattice and $H \subseteq \mathbb{R}^n$ is a hyperplane such that $H \cap \Lambda$ has rank $n - 1$. I would like to know an upper bound on the shortest vector of $\Lambda$ not in $H$, ideally something in terms of the discriminant of $\Lambda$ and the discriminant of $H \cap \Lambda$.

For the situation I have in mind, the discriminant of $H \cap \Lambda$ will be very large compared to the discriminant of $\Lambda$, so intuitively I would expect there to be a short element of $\Lambda$ not on $H$, but I do not know a good way to show this.

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    $\begingroup$ One can readily derive lower bounds, see for example Prop. 1.2.9 or theorem 2.1.3 of Martinet's Perfect Lattices in Euclidean Space. This is of course not what you want, but at least theorem 2.1.3 has an equality condition which may be useful for constructing a potential counterexample to what you want. $\endgroup$
    – Mark Schultz-Wu
    Commented Sep 17, 2021 at 2:42

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