Root Separation for Trinomials
Abstract
We give a separation bound for the complex roots of a trinomial $f \in \mathbb{Z}[X]$. 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 $\log (°f)$. It is known that no such bound is possible for 4nomials (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 4nomials.
 Publication:

arXiv eprints
 Pub Date:
 September 2017
 DOI:
 10.48550/arXiv.1709.03294
 arXiv:
 arXiv:1709.03294
 Bibcode:
 2017arXiv170903294K
 Keywords:

 Computer Science  Symbolic Computation;
 Computer Science  Computational Complexity;
 Mathematics  Number Theory