Matrices of polynomials, positivity, and finite equivalence of Markov chains

Abstract

We consider the ring R = ]R[xt, ... ,xt] of Laurent polynomials with real coefficients in the variables x k and its positive cone R+ = ]R+[xt, ... , xt]. Letting]R++ denote the positive reals, we also have the set R++ = ]R++[xt ' ... ,xt] of polynomials with positive coefficients, so that R+ = R++ U {O}. We say that pER is numerically positive if p gives a positive value whenever we substitute positive numbers for XI' ... , xk ' that is, if P(XI' ... , xk) > 0 for XI ' ... , xk > O. Every element of R++ is numerically positive; however, there are numerically positive polynomials that are not elements of R++ . This distinction lies at the heart of the paper; we will discuss it in greater detail in §2. Let B be a square matrix over R+ . Whenever we substitute positive numbers for xk ' we get a nonnegative real-valued matrix. We assume that the matrix resulting from one, hence every, such evaluation is irreducible and, for XI' ... , xk > 0, let P(x l , ••• , xk) > 0 be the maximum eigenvalue of B(xI' ... ,xk) furnished by the Perron-Frobenius theorem. The function P = P B: (]R++)k --+]R++ is called the p-function of B; it satisfies the characteristic polynomial XB of B, a monic polynomial whose coefficients lie in R. Thus, P is algebraic over R. We assume P has degree one; that is, we consider the case pER. This is a significant case. For instance, for any P E R++ , we obtain numerous examples of matrices with P B = P by simply requiring that B have its row (or column) sums equal to p. There are also many examples of B with PB E R\R+ (see [D]). Since PER, the entries of the adjoint Adj(PI -B) belong to R. Moreover, by Perron-Frobenius theory [Se], the entries are numerically positive and any column r of Adj(PI B) satisfies Br = pr. So, as an immediate consequence of Perron-Frobenius theory, we find an eigenvector r whose entries are numerically positive polynomials. One of the main purposes of this paper is to show that B has an eigenvector whose entries lie in R++ .