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Stiefel tropical linear spaces
- JOURNAL OF COMBINATORIAL THEORY, SERIES A 135 (2015), 291–331. POLYMATROID SUBDIVISION 27
, 2013
"... The tropical Stiefel map associates to a tropical matrix A its tropical Plücker vector of maximal minors, and thus a tropical linear space L(A). We call the L(A)s obtained in this way Stiefel tropical linear spaces. We prove that they are dual to certain matroid subdivisions of polytopes of trans ..."
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Cited by 11 (1 self)
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The tropical Stiefel map associates to a tropical matrix A its tropical Plücker vector of maximal minors, and thus a tropical linear space L(A). We call the L(A)s obtained in this way Stiefel tropical linear spaces. We prove that they are dual to certain matroid subdivisions of polytopes of transversal matroids, and we relate their combinatorics to a canonically associated tropical hyperplane arrangement. We also explore a broad connection with the secondary fan of the Newton polytope of the product of all maximal minors of a matrix. In addition, we investigate the natural parametrization of L(A) arising from the tropical linear map defined by A.
Log-concavity of characteristic polynomials and the Bergman fan of matroids
- MATHEMATISCHE ANNALEN
, 2012
"... In a recent paper, the first author proved the log-concavity of the coefficients of the characteristic polynomial of a matroid realizable over a field of characteristic 0, answering a long-standing conjecture of Read in graph theory. We extend the proof to all realizable matroids, making progress t ..."
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Cited by 11 (1 self)
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In a recent paper, the first author proved the log-concavity of the coefficients of the characteristic polynomial of a matroid realizable over a field of characteristic 0, answering a long-standing conjecture of Read in graph theory. We extend the proof to all realizable matroids, making progress towards a more general conjecture of Rota–Heron–Welsh. Our proof follows from an identification of the coefficients of the reduced characteristic polynomial as answers to particular intersec-tion problems on a toric variety. The log-concavity then follows from an inequality of Hodge type.
Tropical cycles and Chow polytopes
- BEITR. ALGEBRA GEOM
, 2010
"... The Chow polytope of an algebraic cycle in a torus depends only on its tropicalisation. Generalising this, we associate a Chow polytope to any abstract tropical variety in a tropicalised toric variety. Several significant poly-hedra associated to tropical varieties are special cases of our Chow po ..."
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Cited by 1 (0 self)
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The Chow polytope of an algebraic cycle in a torus depends only on its tropicalisation. Generalising this, we associate a Chow polytope to any abstract tropical variety in a tropicalised toric variety. Several significant poly-hedra associated to tropical varieties are special cases of our Chow polytope. The Chow polytope of a tropical variety X is given by a simple combinatorial construction: its normal subdivision is the Minkowski sum of X and a reflected skeleton of the fan of the ambient toric variety.
RESEARCH STATEMENT
"... I study the interplay between combinatorics and algebraic geometry, with applications to number theory. I use ideas from tropical geometry which transforms questions in algebraic geometry into questions in combinatorics through the combinatorial study of degenerations and stratifications. A brief in ..."
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I study the interplay between combinatorics and algebraic geometry, with applications to number theory. I use ideas from tropical geometry which transforms questions in algebraic geometry into questions in combinatorics through the combinatorial study of degenerations and stratifications. A brief introduction to tropical geometry can be found on my website at
