GRÖBNER THEORY AND TROPICAL GEOMETRY ON SPHERICAL VARIETIES

AbstractEvery tropical ideal in the sense of Maclagan–Rincón has an associated tropical variety, a finite polyhedral complex equipped with positive integral weights on its maximal cells. This leads to the realisability question, ubiquitous in tropical geometry, of which weighted polyhedral complexes arise in this manner. Using work of Las Vergnas on the non-existence of tensor products of matroids, we prove that there is no tropical ideal whose variety is the Bergman fan of the direct sum of the Vámos matroid and the uniform matroid of rank two on three elements and in which all maximal cones have weight one.

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Algebraic singularities of scattering amplitudes from tropical geometry

Journal of High Energy Physics ◽

10.1007/jhep04(2021)002 ◽

2021 ◽

Vol 2021 (4) ◽

Author(s):

James Drummond ◽

Jack Foster ◽

Ömer Gürdoğan ◽

Chrysostomos Kalousios

Keyword(s):

Scattering Amplitudes ◽

Natural Language ◽

Tropical Geometry ◽

Cluster Algebras ◽

Finite Alphabet ◽

Square Root ◽

Finite Sets ◽

Yang Mills ◽

Square Roots ◽

Algebraic Singularities

Abstract We address the appearance of algebraic singularities in the symbol alphabet of scattering amplitudes in the context of planar $$ \mathcal{N} $$ N = 4 super Yang-Mills theory. We argue that connections between cluster algebras and tropical geometry provide a natural language for postulating a finite alphabet for scattering amplitudes beyond six and seven points where the corresponding Grassmannian cluster algebras are finite. As well as generating natural finite sets of letters, the tropical fans we discuss provide letters containing square roots. Remarkably, the minimal fan we consider provides all the square root letters recently discovered in an explicit two-loop eight-point NMHV calculation.

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Tropical Geometry and Machine Learning

Proceedings of the IEEE ◽

10.1109/jproc.2021.3065238 ◽

2021 ◽

pp. 1-28

Author(s):

Petros Maragos ◽

Vasileios Charisopoulos ◽

Emmanouil Theodosis

Keyword(s):

Machine Learning ◽

Tropical Geometry

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Tropical geometry and the motivic nearby fiber

Compositio Mathematica ◽

10.1112/s0010437x11005446 ◽

2011 ◽

Vol 148 (1) ◽

pp. 269-294 ◽

Cited By ~ 6

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.

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