1-inverses for polynomial matrices of non-constant rank
This paper is, in a sense, a sequel to the authors' work [Syst. Control Lett. 4, 33-39 (1984; Zbl 0538.15005)] in which explicit constructions were provided for the 1-inverses of constant rank polynomial matrices (a more general situation is, actually, considered in the above quoted paper). Here one deals with a generalization of the problem which arises from control theory. Let F be an \((r+q,r)\) matrix with polynomial entries, and suppose that the common zeroes of all the \(r\times r\) minors of F are contained in an open set U such that \(C^ nU\) is the closure of a complete Reinhardt domain. Then we can explicitly construct a left inverse Z of F, such that the entries of Z are rational functions whose singularities are contained in U. The same theorem gives an explicit construction of the 1-inverse Z of F if, in the hypotheses, the \(r\times r\) minors are replaced by the \(m\times m\) minors \((m<r).\) The proof is in two steps: first one employs some explicit formulas due to \textit{M. Andersson} and \textit{B. Berndtsson} [Ann. Inst. Fourier 32, No.3, 91-110 (1982; Zbl 0466.32001)] to explicitly solve the corresponding scalar problem: given polynomials \(p_ 1,...,p_ t\) with common zeroes in U, find \(q_ 1,...,q_ t\) rational (with singularities in U), such that \(p_ 1q_ 1+...+p_ tq_ t=1\); then the result is achieved via the use of some known determinantal identities. An interesting related problem is to investigate to which domains U the above result can be extended.
scientific article; zbMATH DE number 3931184