SHORT EIGENVECTORS AND MULTIDIMENSIONAL THETA-FUNCTIONS

A certain family of symmetric matrices, with entries +/- 1, is known t o determine all the quartic relations that hold between multidimension al theta constants. Attention is drawn here to combinatorial propertie s of the shortest possible quartic relations, corresponding to vectors with minimal support in a certain eigenspace of such a matrix. A lowe r bound for the size of the support is established, exhibiting a ''pha se transition'' at dimension four. The multiplicity-free eigenvectors with minimal support form an interesting combinatorial design, with a rich group of symmetries. (C) Elsevier Science Inc., 1997.