Inertially arbitrary sign patterns with no nilpotent realization

An n by n sign pattern S is inertially arbitrary if each ordered triple (n1, n2, n3) of nonnegative integers with n1 + n2 + n3 = n is the inertia of some real matrix in Q(S), the sign pattern class of S. If every real, monic polynomial of degree n having a positive coefficient of xn−2 is the characteristic polynomial of some matrix in Q(S), then it is shown that S is inertially arbitrary. A new family of irreducible sign patterns G2k+1(k⩾2) is presented and proved to be inertially arbitrary, but not potentially nilpotent (and thus not spectrally arbitrary). The well-known Nilpotent-Jacobian method cannot be used to prove that G2k+1 is inertially arbitrary, since G2k+1 has no nilpotent realization. In order to prove that Q(G2k+1) allows each inertia with n3 ⩾ 1, a realization of G2k+1 with only zero eigenvalues except for a conjugate pair of pure imaginary eigenvalues is identified and used with the Implicit Function Theorem. Matrices in Q(G2k+1) with inertias having n3 = 0 are constructed by a recursive procedure from those of lower order. Some properties of the coefficients of the characteristic polynomial of an arbitrary matrix having certain fixed inertias are derived, and are used to show that G5 and G7 are minimal inertially arbitrary sign patterns.