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In this answer, Alex Ravsky proves the following lemma:

Lemma. Let $K$ and $N$ be positive integers, $V=\{v_1,\dots, v_k\}\subset [0,K]^N$ be a linearly dependent over $\mathbb R$ system of vectors with integer entries. Then there exist integers $f_1,\dots, f_k$ which are not all zeroes such that $|f_i|\le (kK)^{k-1}$ for each $i$ and $f_1v_1+\dots+f_kv_k=0$.

In other words: if some linear combination of the vectors in $V$ equals $0$, then there exists an integer linear combination of the vectors in $V$ that equals $0$, and the size of the integer coefficients is upper-bounded by $(kK)^{k-1}$.

MY QUESTION: Is the upper bound of $(kK)^{k-1}$ on the size of coefficients asymptotically tight? I.e.: is there a set $V$ as in the lemma for which, in every zero linear combination, some coefficients are that large? Alternatively, can we prove a tighter upper bound - can there be an upper bound that is polynomial in $k$?

Alex Ravsky
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Erel Segal-Halevi
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1 Answers1

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We can improve the upper bound as follows.

Recall that given a set $X$, the space $\ell_2(X)$ consists of elements $(t_\alpha)_{\alpha\in X}\in\mathbb R^X$ such that $\sum_{\alpha\in X} t_\alpha^2<\infty$ (in particular, the set $\{\alpha\in X:t_\alpha\ne 0\}$ is countable. If $X$ is finite, then $\ell_2(X)=\mathbb R^X$.

Lemma 2. Let $X$ be a nonempty set. For each nonempty subset $Y$ of $X$ let $\pi_Y:\ell_2(X)\to \mathbb R^Y$ be the natural projection. Let $V\subset \ell_2(X)$ be a nonempty finite linearly dependent set of nonzero vectors with integer entries. Let $k=|V|-1$ and $$K=\sup\{\|\pi_Y(v)\|:v\in V,\, Y\subset X,\, |Y|=k\}.$$ Then there exist integers $(\Delta_v)_{v\in V}$ which are not all zeroes such that $|\Delta_v|\le K^k$ for each $v\in V$ and $\sum_{v\in V} \Delta_vv=0$.

Proof. Let $V'$ be a maximal linearly independent subset of $V$. It is easy to show by induction on $|V'|$ that there exists a subset $Y$ of $X$ with $|Y|\le |V'|$ such that the set $\pi(V')$ is linearly independent. Pick any vector $u\in V\setminus V'$. Since the set $\pi(V')\cup\{\pi(u)\}$ is linearly dependent, the system $\sum_{v\in V'} x_v\pi(v)=\pi(u)$ has a unique solution. The solution can be calculated by Cramer's rule, which provides the integer determinants $(\Delta_v)_{v\in V'\cup\{u\}}$ such that $|\Delta_v|\le K^k$, by Hadamard's inequality for each $v\in V'\cup\{u\}$ and $x_v=\Delta_v/\Delta_u$ for each $v\in V'$. Since the set $V'\cup\{u\}$ is linearly dependent and the system $\sum_{v\in V'} x_v\pi(v)=\pi(u)$ has a unique solution, the system $\sum_{v\in V'} x_vv=u$ has (the same) unique solution. It remains to put $\Delta_v=0$ for each $v\in V\setminus (V'\cup\{u\})$. $\square$

Alex Ravsky
  • 106,166