Generating Parameters for Algebraic Torus-Based Cryptosystems

2010 ◽  
pp. 156-168
Author(s):  
Tomoko Yonemura ◽  
Yoshikazu Hanatani ◽  
Taichi Isogai ◽  
Kenji Ohkuma ◽  
Hirofumi Muratani
Keyword(s):  
Algebraic Torus

scholarly journals LAURENT POLYNOMIAL LANDAU–GINZBURG MODELS FOR COMINUSCULE HOMOGENEOUS SPACES

2021 ◽  
Author(s):  
PETER SPACEK
Keyword(s):  
Homogeneous Space ◽  
Homogeneous Spaces ◽  
Laurent Polynomial ◽  
Complete Intersections ◽  
Laurent Polynomials ◽  
Young Diagrams ◽  
Similar Shape ◽  
Algebraic Torus ◽  
The One ◽  
Polynomial Potentials

AbstractIn this article we construct Laurent polynomial Landau–Ginzburg models for cominuscule homogeneous spaces. These Laurent polynomial potentials are defined on a particular algebraic torus inside the Lie-theoretic mirror model constructed for arbitrary homogeneous spaces in [Rie08]. The Laurent polynomial takes a similar shape to the one given in [Giv96] for projective complete intersections, i.e., it is the sum of the toric coordinates plus a quantum term. We also give a general enumeration method for the summands in the quantum term of the potential in terms of the quiver introduced in [CMP08], associated to the Langlands dual homogeneous space. This enumeration method generalizes the use of Young diagrams for Grassmannians and Lagrangian Grassmannians and can be defined type-independently. The obtained Laurent polynomials coincide with the results obtained so far in [PRW16] and [PR13] for quadrics and Lagrangian Grassmannians. We also obtain new Laurent polynomial Landau–Ginzburg models for orthogonal Grassmannians, the Cayley plane and the Freudenthal variety.

scholarly journals Tropical geometry and the motivic nearby fiber

2011 ◽  
Vol 148 (1) ◽  
pp. 269-294 ◽  
Author(s):  
Eric Katz ◽  
Alan Stapledon
Keyword(s):  
Euler Characteristic ◽  
Hodge Structure ◽  
Tropical Geometry ◽  
Mixed Hodge Structure ◽  
General Fiber ◽  
Algebraic Torus ◽  
Tropical Varieties

AbstractWe construct motivic invariants of a subvariety of an algebraic torus from its tropicalization and initial degenerations. More specifically, we introduce an invariant of a compactification of such a variety called the ‘tropical motivic nearby fiber’. This invariant specializes in the schön case to the Hodge–Deligne polynomial of the limit mixed Hodge structure of a corresponding degeneration. We give purely combinatorial expressions for this Hodge–Deligne polynomial in the cases of schön hypersurfaces and matroidal tropical varieties. We also deduce a formula for the Euler characteristic of a general fiber of the degeneration.

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.

scholarly journals A formula for the S-class number of an algebraic torus

2017 ◽  
Vol 181 ◽  
pp. 218-239
Author(s):  
Minh-Hoang Tran
Keyword(s):  
Class Number ◽  
Algebraic Torus

scholarly journals About almost geodesic curves

Filomat ◽  
2019 ◽  
Vol 33 (4) ◽  
pp. 1013-1018 ◽  
Author(s):  
Olga Belova ◽  
Josef Mikes ◽  
Karl Strambach
Keyword(s):  
Affine Connection ◽  
Algebraic Torus ◽  
Constant Coefficients

We determine in Rn the form of curves C for which also any image under an (n-1)-dimensional algebraic torus is an almost geodesic with respect to an affine connection ? with constant coefficients and calculate the components of ?.

scholarly journals Torus-equivariant vector bundles on projective spaces

1988 ◽  
Vol 111 ◽  
pp. 25-40 ◽  
Author(s):  
Tamafumi Kaneyama
Keyword(s):  
Vector Bundle ◽  
Toric Variety ◽  
Vector Bundles ◽  
Rational Point ◽  
Projective Spaces ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Algebraic Torus

For a free Z-module N of rank n, let T = TN be an n-dimensional algebraic torus over an algebraically closed field k defined by N. Let X = TN emb (Δ) be a smooth complete toric variety defined by a fan Δ (cf. [6]). Then T acts algebraically on X. A vector bundle E on X is said to be an equivariant vector bundle, if there exists an isomorphism ft: t*E → E for each k-rational point t in T, where t: X → X is the action of t. Equivariant vector bundles have T-linearizations (see Definition 1.2 and [2], [4]), hence we consider T-linearized vector bundles.

scholarly journals On Equivariant Vector Bundles on an Almost Homogeneous Variety

1975 ◽  
Vol 57 ◽  
pp. 65-86 ◽  
Author(s):  
Tamafumi Kaneyama
Keyword(s):  
Vector Bundles ◽  
Multiplicative Group ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Arbitrary Characteristic ◽  
Algebraic Torus ◽  
Homogeneous Variety

Let k be an algebraically closed field of arbitrary characteristic. Let T be an n-dimensional algebraic torus, i.e. T = Gm × · · · × Gm n-times), where Gm = Spec (k[t, t-1]) is the multiplicative group.

scholarly journals Gravitational Quantum Cohomology

1997 ◽  
Vol 12 (09) ◽  
pp. 1743-1782 ◽  
Author(s):  
Tohru Eguchi ◽  
Kentaro Hori ◽  
Chuan-Sheng Xiong
Keyword(s):  
Quantum Cohomology ◽  
Fano Manifold ◽  
Fano Manifolds ◽  
Hamiltonian Structures ◽  
Rational Surfaces ◽  
Algebraic Torus ◽  
Lax Operator ◽  
Dimensional Gravity ◽  
New Type ◽  
First Chern Class

We discuss how the theory of quantum cohomology may be generalized to "gravitational quantum cohomology" by studying topological σ models coupled to two-dimensional gravity. We first consider σ models defined on a general Fano manifold M (manifold with a positive first Chern class) and derive new recursion relations for its two-point functions. We then derive bi-Hamiltonian structures of the theories and show that they are completely integrable at least at the level of genus 0. We next consider the subspace of the phase space where only a marginal perturbation (with a parameter t) is turned on and construct Lax operators (superpotentials) L whose residue integrals reproduce correlation functions. In the case of M = CP N the Lax operator is given by [Formula: see text] and agrees with the potential of the affine Toda theory of the A N type. We also obtain Lax operators for various Fano manifolds; Grassmannians, rational surfaces, etc. In these examples the number of variables of the Lax operators is the same as the dimension of the original manifold. Our result shows that Fano manifolds exhibit a new type of mirror phenomenon where mirror partner is a noncompact Calabi–Yau manifold of the type of an algebraic torus C *N equipped with a specific superpotential.

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