Demjanenko matrix and recursion formula for relative class number over function fields. (Q1869782)

Let \(M=P^n\), where \(P\in\mathbb F_q[T]\) is irreducible, let \(K_M\) be the \(M\)th cyclotomic function field and \(K_M^+\) its maximal real subfield. Let \(h^-(\mathcal O_{K_M})=h(\mathcal O_{K_M})/h(\mathcal O_{K_M^+})\) denote its relative ideal class number. \textit{S. Bae} and \textit{P. Kang} [Acta Arith. 102, No. 3, 251--259 (2002; Zbl 0989.11064)] have defined analogues of the classical Maillet and Demyanenko matrices. The present authors give a new analogue of the latter, and show that its determinant equals \(h^-(\mathcal O_{K_M})\). Further, they give a recursion formula which enables one to compute the value of this determinant. There are results about the highest power of a prime divisor of \(q-1\) dividing \(h^-(\mathcal O_{K_M})\). There are also several numerical examples.