Root separation for trinomials. (November 2019)

• Record Type: Journal Article
• Authors: Koiran, Pascal
• Is Part Of: Journal of symbolic computation. Volume 95(2019)
• Page Start: 151
• Page End: 161
• Publication Date: 2019-11
• Subjects: Root separation -- Sparse polynomials -- Linear forms in logarithms; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2019.02.004
• Languages: English
• ISSNs: 0747-7171

Abstract

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.