A rank theorem for Vandermonde matrices
Let \(n\), \(r\) and \(d\) be three positive integers and \(A\) be an \(r\times (d+1)\) matrix with entries in an algebraically closed field \(K\subset {\mathbb C}\). Let \(x=(x_1, \ldots, x_n)\) be a vector of \(n\) distinct elements of \({\mathbb C}\setminus K\). Let \(V(x,A)\) be an \((n+r)\times (d+1)\) matrix such that the first \(n\) rows of \(V(x,A)\) form the \(n\times (d+1)\) Vandermonde matrix on the vector \(x=(x_1,\ldots, x_n)\) and the last \(r\) rows of \(V(x,A)\) form the matrix \(A\). If \(A\) has rank \(n\) then the matrix \(V(x,A)\) has rank \(n+r\) as soon as \(d+1\geq n(r+1)\). The authors obtain two different proofs of the above statements, the first one is based on dimension arguments and alternative proof is based on the methods from the theory of linear recurrences. The authors discuss also some extensions of the above results and its applications in the construction of the limit theory for generic polynomials.
scientific article; zbMATH DE number 2081749