The Logarithmic Limit-Set of an Algebraic Variety

Bergman, George M.

doi:10.2307/1995858

Cited by 70 publications

(127 citation statements)

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“…Bergman shows in [1] that the second and third descriptions are equivalent, and it follows from Bieri and Groves [2] that all descriptions are equivalent:…”

Section: Define V (B)mentioning

confidence: 99%

“…If the ideal J is principal, defined by an element f = 0, then Bergman states in [1] that the set V (f ) ∞ is precisely Sph(f ). This will now be proved using the following lemma.…”

Section: Spherical Dualsmentioning

confidence: 99%

“…The construction starting after "(4)" on p. 465 in [1] is followed word-by-word, up to the sentence "Hence the points u y log |y| ∈ B m will approach (0, . .…”

Section: Lemma 6 Let V Be An Algebraic Variety Inmentioning

confidence: 99%

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Boundary slopes and the logarithmic limit set

Tillmann

2005

Topology

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The A-polynomial of a manifold whose boundary consists of a single torus is generalised to an eigenvalue variety of a manifold whose boundary consists of a finite number of tori, and the set of strongly detected boundary curves is determined by Bergman's logarithmic limit set, which describes the exponential behaviour of the eigenvalue variety at infinity. This enables one to read off the detected boundary curves of a multi-cusped manifold in a similar way to the 1-cusped case, where the slopes are encoded in the Newton polygon of the A-polynomial.

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“…Bergman shows in [1] that the second and third descriptions are equivalent, and it follows from Bieri and Groves [2] that all descriptions are equivalent:…”

Section: Define V (B)mentioning

confidence: 99%

“…If the ideal J is principal, defined by an element f = 0, then Bergman states in [1] that the set V (f ) ∞ is precisely Sph(f ). This will now be proved using the following lemma.…”

Section: Spherical Dualsmentioning

confidence: 99%

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Boundary slopes and the logarithmic limit set

Tillmann

2005

Topology

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“…In [1], Bergman defined the logarithmic limit-set of an algebraic variety in order to study its exponential behavior at infinity. We follow [15] in calling this set the Bergman complex of the variety.…”

Section: Introductionmentioning

confidence: 99%

The Bergman complex of a matroid and phylogenetic trees

Ardila

Klivans

2006

Journal of Combinatorial Theory, Series B

225

365

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We study the Bergman complex B(M) of a matroid M: a polyhedral complex which arises in algebraic geometry, but which we describe purely combinatorially. We prove that a natural subdivision of the Bergman complex of M is a geometric realization of the order complex of the proper part of its lattice of flats. In addition, we show that the Bergman fan B(K n ) of the graphical matroid of the complete graph K n is homeomorphic to the space of phylogenetic trees T n × R. This leads to a proof that the link of the origin in T n is homeomorphic to the order complex of the proper part of the partition lattice n .

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“…Throughout this paper, familiarity with the fundamental notions of matroid theory will be very useful; we refer the reader to [10,Chapters 1,2] for the basic definitions.…”

Section: Introductionmentioning

confidence: 99%

Subdominant Matroid Ultrametrics

Ardila

2005

Ann. Comb.

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Given a matroid M on the ground set E, the Bergman fan B(M), or space of Multrametrics, is a polyhedral complex in R E which arises in several different areas, such as tropical algebraic geometry, dynamical systems, and phylogenetics. Motivated by the phylogenetic situation, we study the following problem: Given a point ω in R E , we wish to find an M-ultrametric which is closest to it in the ∞ -metric.The solution to this problem follows easily from the existence of the subdominant M-ultrametric: a componentwise maximum M-ultrametric which is componentwise smaller than ω. A procedure for computing it is given, which brings together the points of view of matroid theory and tropical geometry.When the matroid in question is the graphical matroid of the complete graph K n , the Bergman fan B(K n ) parameterizes the equidistant phylogenetic trees with n leaves. In this case, our results provide a conceptual explanation for Chepoi and Fichet's method for computing the tree that most closely matches measured data.

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The Logarithmic Limit-Set of an Algebraic Variety

Abstract: Abstract. Let C be the field of complex numbers and V a subvariety of (C-{0})".

Cited by 70 publications

References 1 publication

Boundary slopes and the logarithmic limit set

Boundary slopes and the logarithmic limit set

The Bergman complex of a matroid and phylogenetic trees

Subdominant Matroid Ultrametrics

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