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Sample $n^2$ integers $a_{11},\dots,a_{nn}$ in $[-d,d]\not\subseteq[0,0]$ uniformly.

What is the probability that the resulting matrix $[a_{ij}]$ has rank $r$?

Is there a nice parametrization of such matrices that helps us generate such a rank $r$ matrix quickly deterministically?

In general what is a good strategy?

Sample $n^2$ integers $a_{11},\dots,a_{nn}$ in $\{-d,\dots,-1,0,1\dots,d\}$ uniformly.

What is the probability that the resulting matrix $[a_{ij}]$ has rank $r$?

Is there a nice parametrization of such matrices that helps us generate such a rank $r$ matrix quickly deterministically?

In general what is a good strategy?

Sample $n^2$ integers $a_{11},\dots,a_{nn}$ in $0\subsetneq[-d,d]$ uniformly.

What is the probability that the resulting matrix $[a_{ij}]$ has rank $r$?

Is there a nice parametrization of such matrices that helps us generate such a rank $r$ matrix quickly deterministically?

In general what is a good strategy?

Sample $n^2$ integers $a_{11},\dots,a_{nn}$ in $[-d,d]\not\subseteq[0,0]$ uniformly.

What is the probability that the resulting matrix $[a_{ij}]$ has rank $r$?

Is there a nice parametrization of such matrices that helps us generate such a rank $r$ matrix quickly deterministically?

In general what is a good strategy?

Sample $n^2$ integers $a_{11},\dots,a_{nn}$ in $0\subsetneq[-d,d]$ uniformly.

What is the probability that the resulting matrix $[a_{ij}]$ has rank $r$?

What is a good strategy to generate such a rank $r$ matrix quickly deterministically?

Is there a nice parametrization of such matrices?

Sample $n^2$ integers $a_{11},\dots,a_{nn}$ in $0\subsetneq[-d,d]$ uniformly.

What is the probability that the resulting matrix $[a_{ij}]$ has rank $r$?

Is there a nice parametrization of such matrices that helps us generate such a rank $r$ matrix quickly deterministically?

In general what is a good strategy?