On the Newton Polygon of a Jacobian Mate

2014 ◽  
pp. 469-476 ◽  
Author(s):  
Leonid Makar-Limanov
Keyword(s):  
Newton Polygon

scholarly journals Solution of tetrahedron equation and cluster algebras

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.

The Newton polygon

2001 ◽  
pp. 197-203
Author(s):  
Gerd Fischer
Keyword(s):  
Newton Polygon

scholarly journals Wiener-Hopf equation and Fredholm property of the Goursat problem in Gevrey space

1994 ◽  
Vol 135 ◽  
pp. 165-196 ◽  
Author(s):  
Masatake Miyake ◽  
Masafumi Yoshino
Keyword(s):  
Differential Equations ◽  
Differential Operators ◽  
Newton Polygon ◽  
Index Theorem ◽  
Fredholm Property ◽  
Goursat Problem ◽  
Point Of View ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Gevrey Spaces

In the study of ordinary differential equations, Malgrange ([Ma]) and Ramis ([R1], [R2]) established index theorem in (formal) Gevrey spaces, and the notion of irregularity was nicely defined for the study of irregular points. In their studies, a Newton polygon has a great advantage to describe and understand the results in visual form. From this point of view, Miyake ([M2], [M3], [MH]) studied linear partial differential operators on (formal) Gevrey spaces and proved analogous results, and showed the validity of Newton polygon in the study of partial differential equations (see also [Yn]).

scholarly journals The Alexander polynomial of a rational link

2019 ◽  
Vol 28 (03) ◽  
pp. 1950004
Author(s):  
Mark E. Kidwell ◽  
Kerry M. Luse
Keyword(s):  
Standard Form ◽  
Newton Polygon ◽  
Laurent Polynomial ◽  
Alexander Polynomial ◽  
Total Degree ◽  
Polynomial Formula ◽  
Positive Coefficients

We relate some terms on the boundary of the Newton polygon of the Alexander polynomial [Formula: see text] of a rational link to the number and length of monochromatic twist sites in a particular diagram that we call the standard form. Normalize [Formula: see text] to be a true polynomial (as opposed to a Laurent polynomial), in such a way that terms of even total degree have positive coefficients and terms of odd total degree have negative coefficients. If the rational link has a reduced alternating diagram with no self-crossings, then [Formula: see text]. If the standard form of the rational link has [Formula: see text] monochromatic twist sites, and the [Formula: see text]th monochromatic twist site has [Formula: see text] crossings, then [Formula: see text]. Our proof employs Kauffman’s clock moves and a lattice for the terms of [Formula: see text] in which the [Formula: see text]-power cannot decrease.

scholarly journals The Newton Polygon and Elliptic Problems with Parameter

1998 ◽  
Vol 192 (1) ◽  
pp. 125-157 ◽  
Author(s):  
Robert Denk ◽  
Reinhard Mennicken ◽  
Leonid Volevich
Keyword(s):  
Elliptic Problems ◽  
Newton Polygon

INVARIANTS OF DEFECTLESS IRREDUCIBLE POLYNOMIALS

2010 ◽  
Vol 09 (04) ◽  
pp. 603-631 ◽  
Author(s):  
RON BROWN ◽  
JONATHAN L. MERZEL
Keyword(s):  
Irreducible Polynomial ◽  
Newton Polygon ◽  
Extension Field ◽  
Irreducible Polynomials ◽  
Technical Result ◽  
Valued Field ◽  
Minimal Pairs ◽  
Simple Characterization ◽  
Polynomial Extensions

Defectless irreducible polynomials over a Henselian valued field (F, v) have been studied by means of strict systems of polynomial extensions and complete (also called "saturated") distinguished chains. Strong connections are developed here between these two approaches and applications made to both. In the tame case in which a root α of an irreducible polynomial f generates a tamely ramified extension of (F, v), simple formulas are given for the Krasner constant, the Brink separant and the diameter of f. In this case a (best possible) result is given showing that a sufficiently good approximation in an extension field K of F to a root of a defectless polynomial f over F guarantees the existence of an exact root of f in K. Also in the tame case a (best possible) result is given describing when a polynomial is sufficiently close to a defectless polynomial so as to guarantee that the roots of the two polynomials generate the same extension fields. Another application in the tame case gives a simple characterization of the minimal pairs (in the sense of N. Popescu et al.). A key technical result is a computation in the tame case of the Newton polygon of f(x+α). Invariants of defectless polynomials are discussed and the existence of defectless polynomials with given invariants is proven. Khanduja's characterization of the tame polynomials whose Krasner constants equal their diameters is generalized to arbitrary defectless polynomials. Much of the work described here will be seen not to require the hypothesis that (F, v) is Henselian.

scholarly journals Fourier transforms of powers of well-behaved 2D real analytic functions

2018 ◽  
Vol 30 (3) ◽  
pp. 723-732
Author(s):  
Michael Greenblatt
Keyword(s):  
Analytic Functions ◽  
Fourier Transforms ◽  
Newton Polygon ◽  
Companion Paper ◽  
Two Dimensional ◽  
Compactly Supported ◽  
Sharp Estimates ◽  
Real Analytic Functions ◽  
Real Analytic ◽  
Compactly Supported Functions

AbstractThis paper is a companion paper to [6], where sharp estimates are proven for Fourier transforms of compactly supported functions built out of two-dimensional real-analytic functions. The theorems of [6] are stated in a rather general form. In this paper, we expand on the results of [6] and show that there is a class of “well-behaved” functions that contains a number of relevant examples for which such estimates can be explicitly described in terms of the Newton polygon of the function. We will further see that for a subclass of these functions, one can prove noticeably more precise estimates, again in an explicitly describable way.

scholarly journals Factorization of the polar curve and the Newton polygon

2003 ◽  
Vol 26 (3) ◽  
pp. 288-303 ◽  
Author(s):  
Andrzej Lenarcik ◽  
Mateusz Masternak ◽  
Arkadiusz Płoski
Keyword(s):  
Newton Polygon ◽  
Polar Curve

The Newton polygon of a product of power series

1994 ◽  
Vol 85 (1) ◽  
pp. 109-118 ◽  
Author(s):  
Detlev W. Hoffmann
Keyword(s):  
Power Series ◽  
Newton Polygon
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