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**Puiseux series expansions for the eigenvalues of transfer matrices and partition functions from the Newton polygon method for nanotubes and ribbons.**
*(English)*
Zbl 1334.82089

The Newton polygon algorithm offers at least three advantages for the study of nanotubes and ribbons. The first one – series expansions for all eigenvalues of the transfer matrix can be found by symbolic-numerical methods as functions of some quantity such as fugacity or a Boltzmann weight. Both very large and very low fugacity regimes, and the cases of ferromagnetic or anti-ferromagnetic systems, are equally approachable. The second one – the technique is not an approximation, i.e., each iteration gives the next order in the series expansion exactly. Besides, if a root truncates to a finite sum of terms, the Newton polygon process itself simply halts, no new segments are produced once the last term in the series is reached. The third one – the degeneracy of a root presents none of the problems normally encountered by numerical root extraction methods in this case. The level of degeneracy of a root has tell-tale signs in the calculation. It is noted that the technique proposed is not limited to a single expansion variable. There is the example to use an extended Hensel construction based upon the Newton polygon algorithm to factor multivariate polynomials.

Reviewer: Yulianna Perepelkina (Moskva)