The NIEP is solvable by reality and finitely many polynomial inequalities

TL;DR

This paper demonstrates that the nonnegative inverse eigenvalue problem (NIEP) can be characterized and solved using polynomial inequalities and real algebraic geometry, establishing its semi-algebraic nature.

Contribution

It introduces a novel approach to solving the NIEP through polynomial inequalities and shows that related problems are semi-algebraic sets, expanding the mathematical tools available.

Findings

NIEP solvable via reality and conjugate spectrum conditions

Symmetric and real NIEP form semi-algebraic sets

Application of real algebraic geometry tools to NIEP

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.