Root Separation for Trinomials

09/11/2017

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by   Pascal Koiran, et al.

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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 ( 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.