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 Tropical varieties for exponential sums

2019 ◽  
Vol 377 (3-4) ◽  
pp. 863-882
Author(s):  
Alperen A. Ergür ◽  
Grigoris Paouris ◽  
J. Maurice Rojas
Keyword(s):  
Exponential Sums ◽  
Tropical Varieties

scholarly journals The Relationship Between Bulb Yield and Allicin Concentration in Garlic Varieties

Proceedings ◽  
2019 ◽  
Vol 36 (1) ◽  
pp. 28
Author(s):  
Nguyen ◽  
Wehr ◽  
O’Hare ◽  
Hong ◽  
Menzies ◽  
...  
Keyword(s):  
Block Design ◽  
Health Benefits ◽  
Field Trials ◽  
Organosulfur Compounds ◽  
Potential Health ◽  
Agronomic Practices ◽  
Negative Effect ◽  
Increasing Demand ◽  
Tropical Varieties ◽  
The Relationship

Garlic (Allium sativum L.) is used as a vegetable and medicinal plant. It is a rich source of organosulfur compounds, in particular allicin, which contributes to the flavour and health benefits. Although worldwide garlic production is increasing, demand continues to outstrip supply. Improving the yield of garlic will address the increasing demand, while increasing allicin concentration will improve its potential health benefits and flavour. It is unknown if increasing garlic bulb size (yield) has a negative effect on allicin concentration of garlic. Therefore, a fundamental understanding of the relationship between yield and allicin concentration is required. A field experiment was conducted at Gatton (QLD, Australia) with 32 varieties, of which 29 varieties were sourced from World Vegetable Centre and 3 varieties from existing Queensland sub-tropical varieties. The garlic cloves were planted in March 2018 with 4 replicates in a randomised complete block design and harvested when garlic had 70% senescence. Varieties showed large variation in bulb size, ranging from about 35 to 120 g, with fresh yield ranging from about 5.5 to 16 t/ha. The allicin concentration ranged from 3.5 to 6.6 mg g−1 fresh weight (FW) between varieties and more than 50% of varieties were under the minimum pharmaceutical standard for allicin concentration (>4.5 mg g−1 in FW). Across varieties there was an inverse relationship between yield and allicin concentration. However, there were some varieties which had both high allicin concentration and yield. Current field trials investigate the effect of agronomic practices on yield and allicin concentration in garlic varieties.

scholarly journals On Rational Equivalence in Tropical Geometry

2016 ◽  
Vol 68 (2) ◽  
pp. 241-257 ◽  
Author(s):  
Lars Allermann ◽  
Simon Hampe ◽  
Johannes Rau
Keyword(s):  
Tropical Geometry ◽  
Chow Groups ◽  
Independent Interest ◽  
Main Step ◽  
Rational Equivalence ◽  
Basic Properties ◽  
Boundary Points ◽  
Intersection Ring ◽  
Tropical Varieties

AbstractThis article discusses the concept of rational equivalence in tropical geometry (and replaces an older, imperfect version). We give the basic definitions in the context of tropical varieties without boundary points and prove some basic properties. We then compute the “bounded” Chow groups of Rn by showing that they are isomorphic to the group of fan cycles. The main step in the proof is of independent interest. We show that every tropical cycle in Rn is a sum of (translated) fan cycles. This also proves that the intersection ring of tropical cycles is generated in codimension 1 (by hypersurfaces).

scholarly journals Three notions of tropical rank for symmetric matrices

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Dustin Cartwright ◽  
Melody Chan
Keyword(s):  
Convex Hull ◽  
Symmetric Matrices ◽  
Star Tree ◽  
Close Study ◽  
International Audience ◽  
Donnent Lieu ◽  
Tropical Varieties

International audience We introduce and study three different notions of tropical rank for symmetric matrices and dissimilarity matrices in terms of minimal decompositions into rank 1 symmetric matrices, star tree matrices, and tree matrices. Our results provide a close study of the tropical secant sets of certain nice tropical varieties, including the tropical Grassmannian. In particular, we determine the dimension of each secant set, the convex hull of the variety, and in most cases, the smallest secant set which is equal to the convex hull. Nous introduisons et étudions trois notions différentes de rang tropical pour des matrices symétriques et des matrices de dissimilarité, en utilisant des décompositions minimales en matrices symétriques de rang 1, en matrices d'arbres étoiles, et en matrices d'arbres. Nos résultats donnent lieu à une étude détaillée des ensembles des sécantes tropicales de certaines jolies variétés tropicales, y compris la grassmannienne tropicale. En particulier, nous déterminons la dimension de chaque ensemble des sécantes, l'enveloppe convexe de la variété, ainsi que, dans la plupart des cas, le plus petit ensemble des sécantes qui est égal à l'enveloppe convexe.

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