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added a comment on $m<n.$
Igor Rivin
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Well, the probability that the kernel of $M$ is nontrivial is bounded above by around $(3/4)^n,$ by Tao-Vu (J. Amer. Math. Soc. 20 (2007), 603-628), so there is that. If $m<n,$ the probability is certainly smaller than that...

Igor Rivin
  • 96.4k
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  • 366