Action of the complex Monge-Ampère operator on piecewise-linear functions and exponential tropical varieties

Kazarnovskii, Boris

doi:10.1070/im2014v078n05abeh002712

Cited by 4 publications

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“…Assume that the vertices of X are numbered in ascending values of functional v. If γ is a simplex then the set P(p, q, γ, v) consists of one element and (2) turns to (1). The formulas similar to (2) are used in the geometry of polyhedra, in tropical geometry, in the theory of toric varieties and in complex analysis [4][5][6][7][8][9][10].…”

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On product of cocycles in a polyhedral complex

Kazarnovskii¹

2015

Preprint

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A product of cochains in a polyhedral complex is constructed. The multiplication algorithm depends on the choice of a parameter. The parameter is a linear functional on the ambient space. Cocycles form a subring of the ring of cochains, cobounds form an ideal of the ring of cocycles, and the quotient ring is a ring of cohomologies. If the complex is simplicial, then the algorithm reduces to the well known algorithm of Cech. Similar algorithms are used in geometry of polyhedra for multiplying of cocycles taking values in the exterior algebra of the ambient space. Therefore, we assume the ring of values to be supercommutative. This text contains the basic definitions, the statements of main theorems and some of their applications in convex geometry. The full text will be offered to "Izvestiya: Mathematics". * The research was carried out at the IITP RAS at the expense of the Russian Foundation for Sciences (project № 14-50-00150)

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“…This publication has two direct sources. The first is the construction of the intersection of tropical varieties as the intersection of cycles with values in exterior algebra [7,9]. The second source is the unpublished text of A.…”

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On product of cocycles in a polyhedral complex

Kazarnovskii¹

2015

Preprint

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Exponential sums equations and tropical geometry

Gallinaro¹

2023

Sel. Math. New Ser.

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Zilber’s Exponential-Algebraic Closedness Conjecture states that algebraic varieties in $${{\mathbb {C}}}^n \times ({{\mathbb {C}}}^\times )^n$$ C n × ( C × ) n intersect the graph of complex exponentiation, unless that contradicts the algebraic and transcendence properties of $${\textrm{exp}}$$ exp . We establish a case of the conjecture, showing that it holds for varieties which split as the product of a linear subspace of the additive group and an algebraic subvariety of the multiplicative group. This amounts to solving certain systems of exponential sums equations, and it generalizes old results of Zilber, which required the linear subspace to either be defined over a generic subfield of the real numbers, or it to be any subspace defined over the reals assuming unproved conjectures from Diophantine geometry and transcendence theory. The proofs use the theory of amoebas and tropical geometry.

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On the product of cocycles in a polyhedral complex

Казарновский¹,

Kazarnovskii²

2017

Известия Российской академии наук. Серия математическая

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Построен алгоритм умножения коцепей полиэдрального комплекса, зависящий от выбора линейного функционала на содержащем комплекс пространстве. Коциклы образуют подкольцо кольца коцепей, кограницы - идеал кольца коциклов, а фактор-кольцо является кольцом когомологий. Алгоритм умножения зависит от геометрии клеток комплекса. Если комплекс симплициален, т. е. геометрия клеток наиболее проста, то правило умножения коцепей сводится к известному алгоритму Чеха. Алгоритм имеет геометрическое происхождение. Например, он применяется для вычисления смешанных объемов многогранников и для построения стабильного пересечения тропических многообразий. В геометрии обычно перемножаются коциклы, принимающие значения во внешней алгебре пространства. Поэтому мы предполагаем, что кольцо значений суперкоммутативно. Библиография: 18 наименований.

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Action of the complex Monge-Ampère operator on piecewise-linear functions and exponential tropical varieties

Cited by 4 publications

References 10 publications

On product of cocycles in a polyhedral complex

On product of cocycles in a polyhedral complex

Exponential sums equations and tropical geometry

On the product of cocycles in a polyhedral complex

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