Let \({\mathcal T}_ c^ n\) and \({\mathcal T}_ r^ n\) be the linear spaces of complex symmetric and real symmetric Toeplitz matrices, respectively. The inverse eigenvalue problem for symmetric complex-valued Toeplitz matrices (IEPSCTM) consists of finding \(T\in {\mathcal T}_ c^ n\) with prescribed \(n\) complex eigenvalues \(\{\sigma_ 1,\dots,\sigma_ n\}\). The inverse eigenvalue problem for symmetric real-valued Toeplitz matrices (IEPSRTM) consists of finding \(T\in{\mathcal T}_ r^ n\) with prescribed \(n\) real eigenvalues \(\{\sigma_ 1,\dots,\sigma_ n\}\).
The author outlines approaches to the IEPSCTM and IEPSRTM using methods of complex and real algebraic geometry. In particular on the base of Bézout’s theorem he shows that for \(n\leq 4\) the IEPSCTM is always solvable and the number of distinct Toeplitz matrices with a prescribed spectrum is \(n!\) counted with multiplicities. For the IEPSRTM he develops the following method. Let \(\sigma_ 1=\sigma_ 1(T)\leq\dots\leq \sigma_ n=\sigma_ n(T)\), \(\sigma=\sigma(T)=(\sigma_ 1,\dots,\sigma_ n)\) be the eigenvalues of \(T\in {\mathcal T}_ r^ n\) arranged in an increasing order. Let \({\mathcal T}_{r,0}^ m\subset {\mathcal T}_ r^ n\) be the subset of all matrices with pairwise distinct eigenvalues and \(K_ i^ n\), \(i=1,\dots,\kappa_ n\) be the connected components of \({\mathcal T}_{r,0}^ n\). Then he defines the map \(\sigma: K_ i^ n \to \Lambda_ 0^ n\), where \(\Lambda_ 0^ n=\{x\), \(x=(x_ 1,\dots,x_ n)\), \(x_ 1<x_ 2<\dots<x_ n\}\) and the topological degree \(d_ i^ n\) of this map is well defined.
The author proves that the IEPSRTM is solvable if \(d_ i^ n\neq 0\) for some \(i\) and shows that this is the case for \(n=2,3,4\) [P. Delsarte and Y. Genin, Lect. Notes Control Inf., Sci. 58, 194-213 (1984; Zbl 0559.15017)]. Moreover he considers the IEPSRTM for odd Toeplitz matrices and proves some results concerning the full map \(\sigma:{\mathcal T}_ r^ 4\to \Lambda^ 4\) which describes all 10 components of \({\mathcal T}_{r,0}^ 4\) and those components on which the degree of the map is equal to zero.