Square-free characterization of principal nonsingularity

Establish that the Fourier matrix F_N has no zero principal minor if and only if N is square-free, for every integer N greater than or equal to 4.

Background

For a Fourier matrix F_N, principal nonsingularity means that every principal submatrix has a nonzero determinant. The paper explains that Caragea and Lee proved square-freeness is exactly the condition guaranteeing nonvanishing of all two-by-two and three-by-three principal minors, while nonsquare-free orders possess zero principal minors of every intermediate size.

These results, together with the relevance of principal Fourier minors to woven Riesz bases and related lifting results, motivate the conjectured complete characterization stated here. The paper resolves the previously unresolved cases N=70 and N=143, but the general equivalence remains the broader conjectural problem.

References

Together with the Riesz-basis formulation above, this led to the conjecture

F_N\text{ has no zero principal minor} \quad\Longleftrightarrow\quad N\text{ is square-free};

see .

— Principal nonsingularity of the Fourier matrices of orders \(70\) and \(143\)  (2608.17746 - Gu et al., 18 Aug 2026) in Introduction