Tropical Combinatorics

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$Banach J. Math. Anal. 4 (2010), no. 2, 37–44 FUNCTIONAL ... - EMIS$

$Banach J. Math. Anal. 2 (2008), no. 1, 16–20 A STUDY ON ... - EMIS$

A = A(V ): max-tropical hyperplane arrangement in T d−1 Key Observation Proposition For p, q ∈ T d−1 let r ∈ tconv max {p, q} be an arbitrary point on the max-tropical line segment between p and q. Then for arbitrary k ∈ [d]: tA(r)k ≤ max{tA(p)k,tA(q)k} Corollary For arbitrary b ∈ Nd : (CA)≤b := p ∈ T d−1 : tA(p) ≤ b = {C ∈ CA : tA(C) ≤ b} max-tropically convex and hence contractible.
n = d = 3, ⎛ ⎞ 0 1 0 V = ⎝0 1 1⎠ , 0 0 1 b = (2, 2, 2) 2 1 0 (0,0,3) (0,1,2) v3 (2,0,1) (1,1,1) -1 -1 0 1 2 v2 v1 (0,2,1) (3,0,0) (2,1,0) (0,3,0) Example
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Tropical Combinatorics Michael Josw
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1 Tropical Hypersurfaces The tropic
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tropical semi-ring: (R ∪ {+∞},
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Tropical Polynomials read ordinary
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tropical semi-module (R d , ⊕,
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The Newton Polytope of a Tropical P
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Tropical Hyperplanes F(x) = (α1
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duality between min and max: Max-Tr
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valuation map val : C{{z}} → Q
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A Lifting Theorem I Theorem (Einsie
Page 21 and 22:
tropicalization of (homogeneous) po
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Page 25 and 26:
Page 27 and 28:
Tropical Convexity for x, y ∈ R d
Page 29 and 30: Tropical Convexity for x, y ∈ R d
Page 31 and 32: Example: Tropical Line Segment in T
Page 33 and 34: Example: Tropical Line Segment in T
Page 35 and 36: consider V = (v1, v2, . . .,vn) in
Page 37 and 38: consider V = (v1, v2, . . .,vn) in
Page 39 and 40: duality between min and max: Recall
Page 41 and 42: tconv{v1, . . .,vn} ⊂ T d−1 dua
Page 43 and 44: P, Q : polytopes in R d P + Q = {p
Page 45 and 46: P, Q : polytopes in R d P + Q = {p
Page 47 and 48: e1, e2, . . .,en : affine basis of
Page 49 and 50: Cayley Trick for Products of Simpli
Page 51 and 52: 4 3 2 1 0 (1,0,3) (2,0,2) (3,0,1) (
Page 53 and 54: 4 3 2 1 0 (1,0,3) (2,0,2) (3,0,1) (
Page 55 and 56: A Tropical Hypersurface point vi
Page 57 and 58: A Tropical Hypersurface point vi
Page 59 and 60: Conclusion II n points in T d−1
Page 61 and 62: Conclusion II n points in T d−1
Page 63 and 64: The Coarse Type Ideal Definition Le
Page 65 and 66: Resolutions via Coarse Tropical Con
Page 67 and 68: Minimal Free Resolutions S = K[x1,
Page 69 and 70: (Co-)Cellular Resolutions II Propos
Page 71 and 72: 201 111 021 300 210 030 chain compl
Page 73 and 74: 201 111 021 300 210 030 chain compl
Page 75 and 76: Proposition Each tropically convex
Page 77 and 78: Proposition Each tropically convex
Page 79: A = A(V ): max-tropical hyperplane
Page 83 and 84: 4 3 2 1 0 (1,0,3) (2,0,2) (3,0,1) (
Page 85 and 86: Conclusion and Remarks III tropical
Page 87: Conclusion and Remarks III tropical
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Tropical Combinatorics

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