Determinants Whose Elements Have Equal Norm1

Hence A, B, C equal A', B', C in some order. Each of the 6 orders leads immediately to the proportionality of two rows or columns. The above theorem, in its specialization to minors of Vandermonde determinants composed of gth roots of unity in R*, was used in [l] for the proof of a theorem on power series without terms whose subscript belongs to one of 3 residue classes modulo an arbitrary integer q. In [2] I showed that the corresponding theorem for 4 residue classes is false for q = 6. This suggests that There exist vanishing i by i minors of the form \ ea>**| , eq = 1, ein R*, without proportional rows or columns. Indeed, examining the counterexample in [2] in the light of the proof in [l] we obtain the determinant