Perron-Frobenius theory of some algebraic equations associated with context free grammars

Abstract

We define the infinite matrices associated with algebraic equations in the form X = a 0(x)X k + a k-1(x)X + a k(x) (a j(x) ∈ x R +[x]) . For each of the associated matrices we prove the following property: Let D( x) be the discriminant of the algebraic equation. Let D ̃ (x)= D(x) (a k-2 0 k k) . Then there exists a zero r of D̃( x) such that: (1) r is positive, and r ⩽¦ y¦ for any zero y of D̃( x); (2) if d is the period of the infinite matrix associated with the algebraic equation, then there are precisely d distinct zeros y of D̃( x) with ¦y¦ = r, namely, r exp( i2π j/ d), j = 0,1,…, d - 1; (3) each of these is a simple zero of D̃( x).