Integer Linear Programming applied to determining monic hyperbolic irreducible polynomials with integer coefficients and span less than 4

In this work, we propose a new method to find monic irreducible polynomials with integer coefficients, only real roots, and span less than 4. The main idea is to reduce the search of such polynomials to the solution of Integer Linear Programming problems. In this frame, the coefficients of the polynomials we are looking for are the integer unknowns. We give inequality constraints specified by the properties that the polynomials should have, such as the typical distribution of their roots. These properties can be inferred from those of polynomials already treated in the literature on this topic.