Nested sequences of principal minors and potential stability

Sufficient conditions are determined for an n-by-n zero-nonzero pattern to allow a nested sequence of nonzero principal miners. A method is then given to sign such a pattern so that it allows a nested sequence of k-by-k principal miners with sign (-1)(k) for 1 less than or equal to k less than or equal to n. It is a classical result that a sign pattern with such a sequence is potentially stable; thus classes of potentially stable sign patterns can be identified. A necessary and sufficient condition for a certain tree sign-pattern class to be potentially stable is also given. (C) Elsevier Science Inc., 1997.