Diophantine and tropical geometry, and uniformity of rational points on curves

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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Counting and equidistribution in quaternionic Heisenberg groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004121000426 ◽

2021 ◽

pp. 1-38

Author(s):

JOUNI PARKKONEN ◽

FRÉDÉRIC PAULIN

Keyword(s):

Hyperbolic Geometry ◽

Quaternion Algebra ◽

Rational Points ◽

Hyperbolic Spaces ◽

Heisenberg Groups ◽

Arithmetic Groups ◽

Hermitian Forms ◽

Dimension 2 ◽

Counting Formula ◽

The Relationship

Abstract We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.

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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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Arithmetic of higher-dimensional orbifolds and a mixed Waring problem

Mathematische Zeitschrift ◽

10.1007/s00209-021-02695-w ◽

2021 ◽

Author(s):

Tim Browning ◽

Shuntaro Yamagishi

Keyword(s):

Circle Method ◽

Rational Points ◽

Mean Value ◽

Mean Value Theorem ◽

Waring’S Problem ◽

Waring Problem ◽

Waring's Problem ◽

Main Conjecture ◽

Higher Dimensional ◽

Thin Set

AbstractWe study the density of rational points on a higher-dimensional orbifold $$(\mathbb {P}^{n-1},\Delta )$$ ( P n - 1 , Δ ) when $$\Delta $$ Δ is a $$\mathbb {Q}$$ Q -divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational points on such an orbifold constitutes a thin set. Our approach relies on the Hardy–Littlewood circle method to first study an asymptotic version of Waring’s problem for mixed powers. In doing so we make crucial use of the recent resolution of the main conjecture in Vinogradov’s mean value theorem, due to Bourgain–Demeter–Guth and Wooley.

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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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Le complémentaire des puissances -ièmes dans un corps de nombres est un ensemble diophantien

Compositio Mathematica ◽

10.1112/s0010437x15007368 ◽

2015 ◽

Vol 151 (10) ◽

pp. 1965-1980 ◽

Cited By ~ 2

Author(s):

Jean-Louis Colliot-Thélène ◽

Jan Van Geel

Keyword(s):

Prime Number ◽

Rational Points ◽

Parameter Family ◽

Symmetric Power ◽

Symmetric Powers

For $n=2$ the statement in the title is a theorem of B. Poonen (2009). He uses a one-parameter family of varieties together with a theorem of Coray, Sansuc and one of the authors (1980), on the Brauer–Manin obstruction for rational points on these varieties. For $n=p$, $p$ any prime number, A. Várilly-Alvarado and B. Viray (2012) considered analogous families of varieties. Replacing this family by its $(2p+1)$th symmetric power, we prove the statement in the title using a theorem on the Brauer–Manin obstruction for rational points on such symmetric powers. The latter theorem is based on work of one of the authors with Swinnerton-Dyer (1994) and with Skorobogatov and Swinnerton-Dyer (1998), work generalising results of Salberger (1988).

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