Tropical Combinatorics

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

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

general tropical conic <strong>Tropical</strong> Conics (a200 ⊙ x ⊙2 1 ) ⊕ (a110 ⊙ x1 ⊙ x2) ⊕ (a101 ⊙ x1 ⊙ x3) ⊕ (a020 ⊙ x ⊙2 2 ) ⊕ (a011 ⊙ x2 ⊙ x3) ⊕ (a002 ⊙ x ⊙2 3 ) Example (a200, a110, a101, a020, a011, a002) = (6, 5, 5, 6, 5, 7) 4 3 2 1 0 -1 -2 -3 -4 -4 -3 -2 -1 3 2 0 4 1 1 2 3 4 (2,0,0) 1 (1,1,0) (1,0,1) 2 3 4 (0,2,0) (0,1,1) (0,0,2)
duality between min and max: Max-<strong>Tropical</strong> Hyperplanes max(−x, −y) = −min(x, y) Remark T is min-tropical hyperplane ⇐⇒ −T is max-tropical hyperplane −α α min/max
Page 1 and 2: Tropical Combinatorics Michael Josw
Page 3 and 4: 1 Tropical Hypersurfaces The tropic
Page 5 and 6: tropical semi-ring: (R ∪ {+∞},
Page 7 and 8: Tropical Polynomials read ordinary
Page 9 and 10: tropical semi-module (R d , ⊕,
Page 11 and 12: The Newton Polytope of a Tropical P
Page 13: Tropical Hyperplanes F(x) = (α1
Page 17 and 18: valuation map val : C{{z}} → Q
Page 19 and 20: A Lifting Theorem I Theorem (Einsie
Page 21 and 22: tropicalization of (homogeneous) po
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 and 80:
A = A(V ): max-tropical hyperplane
Page 81 and 82:
n = d = 3, ⎛ ⎞ 0 1 0 V = ⎝0 1
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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