scholarly journals On the Newton polytope of a Jacobian pair

2021 ◽  
Vol 85 (3) ◽  
Author(s):  
Leonid Grigor'evich Makar-Limanov
Keyword(s):  
Newton Polytope

scholarly journals The number of maximal torsion cosets in subvarieties of tori

2019 ◽  
Vol 2019 (755) ◽  
pp. 103-126
Author(s):  
César Martínez
Keyword(s):  
Newton Polytope ◽  
Torsion Point ◽  
Torsion Points ◽  
Sharp Bounds ◽  
Maximal Torsion ◽  
Algebraic Torus

AbstractWe present sharp bounds on the number of maximal torsion cosets in a subvariety of the complex algebraic torus {\mathbb{G}_{\mathrm{m}}^{n}}. Our first main result gives a bound in terms of the degree of the defining polynomials. We also give a bound for the number of isolated torsion point, that is maximal torsion cosets of dimension 0, in terms of the volume of the Newton polytope of the defining polynomials. This result proves the conjectures of Ruppert and of Aliev and Smyth on the number of isolated torsion points of a hypersurface. These conjectures bound this number in terms of the multidegree and the volume of the Newton polytope of a polynomial defining the hypersurface, respectively.

Amoebas, Monge-Amp�re measures, and triangulations of the Newton polytope

2004 ◽  
Vol 121 (3) ◽  
pp. 481-507 ◽  
Author(s):  
Hans Rullg�rd ◽  
Mikael Passare
Keyword(s):  
Newton Polytope

scholarly journals The Newton Polytope of the Implicit Equation

2007 ◽  
Vol 7 (2) ◽  
pp. 327-346 ◽  
Author(s):  
B. Sturmfels ◽  
J. Tevelev ◽  
J. Yu
Keyword(s):  
Newton Polytope ◽  
Implicit Equation

scholarly journals Newton Polytopes of Cluster Variables of Type $A_n$

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Adam Kalman
Keyword(s):  
Type A ◽  
Laurent Expansion ◽  
Cluster Algebras ◽  
Newton Polytope ◽  
Newton Polytopes ◽  
Affine Hull ◽  
Cluster Variable ◽  
Nous Montrons ◽  
The Face ◽  
International Audience

International audience We study Newton polytopes of cluster variables in type $A_n$ cluster algebras, whose cluster and coefficient variables are indexed by the diagonals and boundary segments of a polygon. Our main results include an explicit description of the affine hull and facets of the Newton polytope of the Laurent expansion of any cluster variable, with respect to any cluster. In particular, we show that every Laurent monomial in a Laurent expansion of a type $A$ cluster variable corresponds to a vertex of the Newton polytope. We also describe the face lattice of each Newton polytope via an isomorphism with the lattice of elementary subgraphs of the associated snake graph. Nous étudions polytopes de Newton des variables amassées dans les algèbres amassées de type A, dont les variables sont indexés par les diagonales et les côtés d’un polygone. Nos principaux résultats comprennent une description explicite de l’enveloppe affine et facettes du polytope de Newton du développement de Laurent de toutes variables amassées. En particulier, nous montrons que tout monôme Laurent dans un développement de Laurent de variable amassée de type A correspond à un sommet du polytope de Newton. Nous décrivons aussi le treillis des facesde chaque polytope de Newton via un isomorphisme avec le treillis des sous-graphes élémentaires du “snake graph” qui est associé.

scholarly journals Bezout-type theorems for differential fields

2017 ◽  
Vol 153 (4) ◽  
pp. 867-888 ◽  
Author(s):  
Gal Binyamini
Keyword(s):  
Elliptic Curves ◽  
Lattice Points ◽  
Newton Polytope ◽  
Algebraic Varieties ◽  
Number Of Solutions ◽  
Asymptotic Growth ◽  
Closed Fields ◽  
Exponential Bounds ◽  
Differential Fields ◽  
Derivatives Of

We prove analogs of the Bezout and the Bernstein–Kushnirenko–Khovanskii theorems for systems of algebraic differential conditions over differentially closed fields. Namely, given a system of algebraic conditions on the first $l$ derivatives of an $n$-tuple of functions, which admits finitely many solutions, we show that the number of solutions is bounded by an appropriate constant (depending singly-exponentially on $n$ and $l$) times the volume of the Newton polytope of the set of conditions. This improves a doubly-exponential estimate due to Hrushovski and Pillay. We illustrate the application of our estimates in two diophantine contexts: to counting transcendental lattice points on algebraic subvarieties of semi-abelian varieties, following Hrushovski and Pillay; and to counting the number of intersections between isogeny classes of elliptic curves and algebraic varieties, following Freitag and Scanlon. In both cases we obtain bounds which are singly-exponential (improving the known doubly-exponential bounds) and which exhibit the natural asymptotic growth with respect to the degrees of the equations involved.

scholarly journals Combinatorics of Nahm sums, quiver resultants and the K-theoretic condition

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Dmitry Noshchenko
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Adjacency Matrix ◽  
Knot Theory ◽  
Newton Polytope ◽  
Polynomial Equations ◽  
Quiver Representation ◽  
Conformal Field ◽  
Combinatorial Pattern ◽  
Initial Forms

Abstract Algebraic Nahm equations, considered in the paper, are polynomial equations, governing the q → 1 limit of the q-hypergeometric Nahm sums. They make an appearance in various fields: hyperbolic geometry, knot theory, quiver representation theory, topological strings and conformal field theory. In this paper we focus primarily on Nahm sums and Nahm equations that arise in relation with symmetric quivers. For a large class of them, we prove that quiver A-polynomials — specialized resultants of the Nahm equations, are tempered (the so-called K-theoretic condition). This implies that they are quantizable. Moreover, we find that their face polynomials obey a remarkable combinatorial pattern. We use the machinery of initial forms and mixed polyhedral decompositions to investigate the edges of the Newton polytope. We show that this condition holds for the diagonal quivers with adjacency matrix C = diag(α, α, . . . , α), α ≥ 2, and provide several checks for non-diagonal quivers. Our conjecture is that the K-theoretic condition holds for all symmetric quivers.

scholarly journals Tropical secant graphs of monomial curves

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
María Angélica Cueto ◽  
Shaowei Lin
Keyword(s):  
Tropical Geometry ◽  
Newton Polytope ◽  
Projective Curve ◽  
Secant Variety ◽  
Monomial Curves ◽  
Surface Graph ◽  
International Audience ◽  
Monomial Curve

International audience We construct and study an embedded weighted balanced graph in $\mathbb{R}^{n+1}$ parametrized by a strictly increasing sequence of $n$ coprime numbers $\{ i_1, \ldots, i_n\}$, called the $\textit{tropical secant surface graph}$. We identify it with the tropicalization of a surface in $\mathbb{C}^{n+1}$ parametrized by binomials. Using this graph, we construct the tropicalization of the first secant variety of a monomial projective curve with exponent vector $(0, i_1, \ldots, i_n)$, which can be described by a balanced graph called the $\textit{tropical secant graph}$. The combinatorics involved in computing the degree of this classical secant variety is non-trivial. One earlier approach to this is due to K. Ranestad. Using techniques from tropical geometry, we give algorithms to effectively compute this degree (as well as its multidegree) and the Newton polytope of the first secant variety of any given monomial curve in $\mathbb{P}^4$. On construit et on étude un graphe plongé dans $\mathbb{R}^{n+1}$ paramétrisé par une suite strictement croissante de $n$ nombres entiers $\{ i_1, \ldots, i_n\}$, premiers entre eux. Ce graphe s'appelle $\textit{graphe tropical surface sécante}$. On montre que ce graphe est la tropicalisation d'une surface dans $\mathbb{C}^{n+1}$ paramétrisé par des binômes. On utilise ce graphe pour construire la tropicalisation de la première sécante d'une courbe monomiale ayant comme vecteur d'exponents $(0, i_1, \ldots, i_n)$. On représente cette variété tropicale pour un graphe balancé (le $\textit{graphe tropical sécante}$). La combinatoire qu'on utilise pour le calcul du degré de ces variétés sécantes classiques n'est pas triviale, et a été developpée par K. Ranestad. En utilisant des techniques de la géométrie tropicale, on donne des algorithmes qui calculent le degré (même le multidegré) et le polytope de Newton de la première sécante d'une courbe monomiale de $\mathbb{P}^4$.

scholarly journals Euclidean Distance Degree and Mixed Volume

2021 ◽  
Author(s):  
P. Breiding ◽  
F. Sottile ◽  
J. Woodcock
Keyword(s):  
Computational Complexity ◽  
Lagrange Multiplier ◽  
Euclidean Distance ◽  
Mixed Volume ◽  
Rectangular Parallelepiped ◽  
Newton Polytope ◽  
Newton Polytopes ◽  
Sparse Polynomials

AbstractWe initiate a study of the Euclidean distance degree in the context of sparse polynomials. Specifically, we consider a hypersurface $$f=0$$ f = 0 defined by a polynomial f that is general given its support, such that the support contains the origin. We show that the Euclidean distance degree of $$f=0$$ f = 0 equals the mixed volume of the Newton polytopes of the associated Lagrange multiplier equations. We discuss the implication of our result for computational complexity and give a formula for the Euclidean distance degree when the Newton polytope is a rectangular parallelepiped.

scholarly journals AN ORACLE-BASED, OUTPUT-SENSITIVE ALGORITHM FOR PROJECTIONS OF RESULTANT POLYTOPES

2013 ◽  
Vol 23 (04n05) ◽  
pp. 397-423 ◽  
Author(s):  
IOANNIS Z. EMIRIS ◽  
VISSARION FISIKOPOULOS ◽  
CHRISTOS KONAXIS ◽  
LUIS PEÑARANDA
Keyword(s):  
Orthogonal Projection ◽  
Geometric Modeling ◽  
Tropical Geometry ◽  
Newton Polytope ◽  
Newton Polytopes ◽  
Outer Approximations

We design an algorithm to compute the Newton polytope of the resultant, known as resultant polytope, or its orthogonal projection along a given direction. The resultant is fundamental in algebraic elimination, optimization, and geometric modeling. Our algorithm exactly computes vertex- and halfspace-representations of the polytope using an oracle producing resultant vertices in a given direction, thus avoiding walking on the polytope whose dimension is α - n - 1, where the input consists of α points in ℤn. Our approach is output-sensitive as it makes one oracle call per vertex and per facet. It extends to any polytope whose oracle-based definition is advantageous, such as the secondary and discriminant polytopes. Our publicly available implementation uses the experimental CGAL package triangulation. Our method computes 5-, 6- and 7- dimensional polytopes with 35K, 23K and 500 vertices, respectively, within 2hrs, and the Newton polytopes of many important surface equations encountered in geometric modeling in < 1sec, whereas the corresponding secondary polytopes are intractable. It is faster than tropical geometry software up to dimension 5 or 6. Hashing determinantal predicates accelerates execution up to 100 times. One variant computes inner and outer approximations with, respectively, 90% and 105% of the true volume, up to 25 times faster.

Sign in / Sign up

Export Citation Format

Share Document