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Nonlinear objectives, nonlinear constraints, non-convex objective, non-convex feasible region.
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Is the matrix $\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}$ nonsingular?
This is an instance of Holte's Amazing matrix. Consider addition of binary digits. Start with a carry of $c \in \{0,1,\ldots,2(m-1)\}$. Choose $2m-1$ bits uniformly at random, and add their sum to $c$ …