# 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 $\log (\deg f)$. 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.
Keywords :
Type de document :
Pré-publication, Document de travail
2017
Domaine :

https://hal-ens-lyon.archives-ouvertes.fr/ensl-01585049
Contributeur : Pascal Koiran <>
Soumis le : lundi 11 septembre 2017 - 10:11:56
Dernière modification le : mardi 12 septembre 2017 - 01:08:28

### Fichiers

trinomial.pdf
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### Identifiants

• HAL Id : ensl-01585049, version 1
• ARXIV : 1709.03294

### Citation

Pascal Koiran. Root Separation for Trinomials. 2017. <ensl-01585049>

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## 49

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