We say that a vector $\mathbf{x}$ in $\mathbb{Z}^j$ is non-negative if it is of the form \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_j \\ \end{bmatrix}

where $x_{i} \geq 0$.

Suppose that $A$ is a $j \times j$ integer valued matrix with entries in the interval $[-n,n]$. What is the smallest possible value of $\delta_{j} \geq 0$ so that if $\varepsilon > 0$ then there exists an integer valued vector $\mathbf{x}_{A} \neq \mathbf{0}$ such that $A\cdot\mathbf{x}_{A}$ is non-negative, and $|\mathbf{x}_{A}| \leq C_{\epsilon}n^{\delta_{j} + \varepsilon}$ (here $C_{\epsilon}$ is a positive constant dependent only on $\varepsilon$)?

Note that we can find candidate values $\delta_{j} < \infty$, but I have not proved that here. I mainly seek a “non-trivial” upper bound for $\delta_{j}$.

We say that a vector $\mathbf{x}$ in $\mathbb{Z}^j$ is non-negative if it is of the form \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_j \\ \end{bmatrix}

where $x_{i} \geq 0$ for all $i=1,\dots,j$.

Suppose that $A$ is a $j \times j$ integer valued matrix with entries in the interval $[-n,n]$. What is the smallest possible value of $\delta_{j} \geq 0$ so that if $\varepsilon > 0$ then there exists an integer valued vector $\mathbf{x}_{A} \neq \mathbf{0}$ such that $A\cdot\mathbf{x}_{A}$ is non-negative, and $|\mathbf{x}_{A}| \leq C_{\epsilon}n^{\delta_{j} + \varepsilon}$ (here $C_{\epsilon}$ is a positive constant dependent only on $\varepsilon$)?

Note that we can find candidate values $\delta_{j} < \infty$, but I have not proved that here. I mainly seek a “non-trivial” upper bound for $\delta_{j}$.

We say that a vector $\mathbf{x}$ in $\mathbb{Z}^j$ is non-negative if it is of the form \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_j \\ \end{bmatrix}

where $x_{j} \geq 0$.

Suppose that $A$ is a $j \times j$ integer valued matrix with entries in the interval $[-n,n]$. What is the smallest possible value of $\delta_{j} \geq 0$ so that if $\varepsilon > 0$ then there exists an integer valued vector $\mathbf{x}_{A} \neq \mathbf{0}$ such that $A\cdot\mathbf{x}_{A}$ is non-negative, and $|\mathbf{x}_{A}| \leq C_{\epsilon}n^{\delta_{j} + \varepsilon}$ (here $C_{\epsilon}$ is a positive constant dependent only on $\varepsilon$)?

Note that we can find candidate values $\delta_{j} < \infty$, but I have not proved that here. I mainly seek a “non-trivial” upper bound for $\delta_{j}$.

We say that a vector $\mathbf{x}$ in $\mathbb{Z}^j$ is non-negative if it is of the form \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_j \\ \end{bmatrix}

where $x_{i} \geq 0$.

Suppose that $A$ is a $j \times j$ integer valued matrix with entries in the interval $[-n,n]$. What is the smallest possible value of $\delta_{j} \geq 0$ so that if $\varepsilon > 0$ then there exists an integer valued vector $\mathbf{x}_{A} \neq \mathbf{0}$ such that $A\cdot\mathbf{x}_{A}$ is non-negative, and $|\mathbf{x}_{A}| \leq C_{\epsilon}n^{\delta_{j} + \varepsilon}$ (here $C_{\epsilon}$ is a positive constant dependent only on $\varepsilon$)?

Note that we can find candidate values $\delta_{j} < \infty$, but I have not proved that here. I mainly seek a “non-trivial” upper bound for $\delta_{j}$.

We say that a vector $\mathbf{x}$ in $\mathbb{Z}^j$ is non-negative if it is of the form \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_j \\ \end{bmatrix}

where $x_{j} \geq 0$.

Suppose that $A$ is a $j \times j$ integer valued matrix with entries in the interval $[-n,n]$. What is the smallest possible value of $\delta_{j} \geq 0$ so that if $\varepsilon > 0$ then there exists an integer valued vector $\mathbf{x}_{A} \neq \mathbf{0}$ such that $A\cdot\mathbf{x}_{A}$ is non-negative, and $|\mathbf{x}_{A}| \leq C_{\epsilon}n^{\delta_{j} + \varepsilon}$ (here $C_{\epsilon}$ is a positive constant dependent only on $\varepsilon$)?

Note that we can find candidate values $\delta_{j} < \infty$, but I have not proved that here. I mainly seek a ``non-trivial'' upper bound for $\delta_{j}$.

We say that a vector $\mathbf{x}$ in $\mathbb{Z}^j$ is non-negative if it is of the form \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_j \\ \end{bmatrix}

where $x_{j} \geq 0$.

Suppose that $A$ is a $j \times j$ integer valued matrix with entries in the interval $[-n,n]$. What is the smallest possible value of $\delta_{j} \geq 0$ so that if $\varepsilon > 0$ then there exists an integer valued vector $\mathbf{x}_{A} \neq \mathbf{0}$ such that $A\cdot\mathbf{x}_{A}$ is non-negative, and $|\mathbf{x}_{A}| \leq C_{\epsilon}n^{\delta_{j} + \varepsilon}$ (here $C_{\epsilon}$ is a positive constant dependent only on $\varepsilon$)?

Note that we can find candidate values $\delta_{j} < \infty$, but I have not proved that here. I mainly seek a “non-trivial” upper bound for $\delta_{j}$.