Existence of matrices with prescribed entries (Q1068161)

This paper solves the following problem. Let F be a field and \(a_ 1,...,a_ n\) elements of F. Let \((i_ 1,j_ 1),...,(i_ n,j_ n)\) be n distinct positions on an \(n\times n\) matrix and \(f(\lambda)=\lambda^ n-c_ 1\lambda^{n-1}-...-c_ n\) a polynomial with coefficients in F. Find a necessary and sufficient condition for the existence of an \(n\times n\) matrix \(A=[a_{ij}]\) over F with characteristic polynomial f(\(\lambda)\), and \(a_{i_ t,j_ t}=a_ t.\) The author proves that for \(n>4\) there always exists a nonderogatory matrix satisfying the required conditions except in the following cases: (i) All the n prescribed entries are the principal ones and their sum is not \(c_ 1\). (ii) There exists a row (or column) all of whose non principal entries are prescribed to be zero, and the principal one is prescribed to be something that is not a root of f(\(\lambda)\). (iii) There is a row (or column) all of whose non principal entries are prescribed to be zero and f(\(\lambda)\) has no root in F. The author also considers and solves the case \(n\leq 4\) but there are more exceptions. The resolution of the problem is very intricate (it needs 53 pages) and is an interesting contribution to a series of problems that have been investigated by several authors.