Root separation for trinomials. (November 2019) - Koiran, Pascal
Abstract: We give a separation bound for the complex roots of a trinomial f ∈ Z [ X ] . The logarithm of the inverse of our separation bound is polynomial in the size of the sparse encoding of f ; in particular, it is polynomial in log ⁡ ( deg ⁡ f ) . It is known that no such bound is possible for 4-nomials (polynomials with 4 monomials). For trinomials, the classical results (which are based on the degree of f rather than the number of monomials) give separation bounds that are exponentially worse. As an algorithmic application, we show that the number of real roots of a trinomial f can be computed in time polynomial in the size of the sparse encoding of f . The same problem is open for 4-nomials.