The NIEP is solvable by reality and finitely many polynomial inequalities

Abstract: The nonnegative inverse eigenvalue problem (NIEP) is shown to be solvable by the reality condition, spectrum equal to its conjugate, as well as by a finite union and intersection of polynomial inequalities. It is also shown that the symmetric NIEP and real NIEP form semi-algebraic sets and can therefore be solved just by a finite union and intersection of polynomial inequalities. An overview of ideas are given in how tools from real algebraic geometry may be applied to the NIEP and related sub-problems.

Keywords: Algebraic Geometry; Polynomial; Nonnegative matrix; Nonnegative inverse eigenvalue problem;