A $$\tau $$ τ -Conjecture for Newton Polygons

Pascal Koiran; Natacha Portier; Sébastien Tavenas; Stéphan Thomassé

doi:10.1007/s10208-014-9216-x

A $$\tau $$ τ -Conjecture for Newton Polygons

Foundations of Computational Mathematics ◽

10.1007/s10208-014-9216-x ◽

2014 ◽

Vol 15 (1) ◽

pp. 185-197 ◽

Cited By ~ 9

Author(s):

Pascal Koiran ◽

Natacha Portier ◽

Sébastien Tavenas ◽

Stéphan Thomassé

Keyword(s):

Newton Polygons

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Newton polygons of L-functions associated to Deligne polynomials

Finite Fields and Their Applications ◽

10.1016/j.ffa.2021.101880 ◽

2021 ◽

Vol 75 ◽

pp. 101880

Author(s):

Jiyou Li

Keyword(s):

Newton Polygons

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Solution of tetrahedron equation and cluster algebras

Journal of High Energy Physics ◽

10.1007/jhep05(2021)103 ◽

2021 ◽

Vol 2021 (5) ◽

Author(s):

P. Gavrylenko ◽

M. Semenyakin ◽

Y. Zenkevich

Keyword(s):

Integrable System ◽

Cluster Algebra ◽

Newton Polygon ◽

Cluster Algebras ◽

Weyl Groups ◽

Newton Polygons ◽

Tetrahedron Equation ◽

Affine Weyl Groups ◽

New Formalism ◽

Bruhat Cell

Abstract We notice a remarkable connection between the Bazhanov-Sergeev solution of Zamolodchikov tetrahedron equation and certain well-known cluster algebra expression. The tetrahedron transformation is then identified with a sequence of four mutations. As an application of the new formalism, we show how to construct an integrable system with the spectral curve with arbitrary symmetric Newton polygon. Finally, we embed this integrable system into the double Bruhat cell of a Poisson-Lie group, show how triangular decomposition can be used to extend our approach to the general non-symmetric Newton polygons, and prove the Lemma which classifies conjugacy classes in double affine Weyl groups of A-type by decorated Newton polygons.

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