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

Searching for Hyperbolic Polynomials with Span Less than 4

“…The present article follows the research line that started with the pioneering work [8] by Robinson and was continued by several other authors-see [1,2,3,6]; our main object of study are therefore monic, irreducible polynomials in one variable having integer coefficients and all real roots (i.e., hyperbolic), whose span (the smallest interval containing the roots) is required to be smaller than 4, excluding the case where the roots lie in an interval of length 4 with integer end-points. The importance of the threshold 4 for the span will be recalled in the next lines.…”

“…In [3] it was proved that the list is exhaustive up to degree 15. Finally, in [2], with the aid of linear programing the authors seemed to suggest that the list of polynomials of degree 16 and 17 found in previous papers is indeed complete, though no proof of this is yet available. Moreover, as we remarked, the authors exhibited three polynomials of degree 18; in spite of similar computations being conducted, no such polynomials of degree 19 or 20 were found that were not of cosine type.…”

“…The novelty in our approach is that the construction of such polynomials avails of purely algebraic operations (linear combinations and products of suitable polynomials, with the unity 1 included) which can be performed inside the whole algebra of polynomials in one variable. As an emblematic example, let us consider one of the polynomials in N with degree 18 (the largest known degree) that were remarkably found in [2] using integer linear programing-see f 156 in Appendix A. In their article, the authors assumed some inequality constraints on the coefficients so as to reduce the computational complexity; in particular, they reasoned on the typical distribution of the roots of the already known polynomials.…”

“…Other regularities were observed such as alternating signs and a weak monotonicity, in absolute value. For example, x 8 À 3x 7 À 5x 6 þ 18x 5 þ 7x 4 À 33x 3 À 3x 2 þ 18x þ 1 has coordinates ½7, À 6, 6, À 6, 5, À 3, 3, À 3, 1, and one of the three polynomials of degree 18 found in [2] it is exactly of this form: ½15, À 15, 15, À 14, 14, À 13, 12, À 11, 10, À 9, 8, À 7, 6, À 5, 4, À 3, 2, À 2, 1:…”

Chebyshev coordinates and Salem numbers

“…In [6] it was proved that the list is exhaustive up to degree 15. Finally, in [7], using ideas from linear programming, the authors seem to suggest that the list of polynomials of degree 16 and 17 found in previous papers is indeed complete, though no proof of this is yet available. Moreover, the four authors were able to exhibit three polynomials of the desired type of degree 18 and, in spite of similar computations being conducted, no such polynomials of degree 19 and 20 were found, that were not of cosine type.…”