Tropical Combinatorics

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

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

(Co-)Cellular Resolutions II Proposition If for every b ∈ Z d the subcomplex P≤b is acyclic over K, then F P • is a free resolution of the ideal I generated by all monomials corresponding to the vertex labels of P. Moreover, if tF = tG for all cells F ⊃ G then the cellular resolution is minimal. cellular: tH = lcm {tG : for G ⊂ H a face} cocellular: t H = lcm {tG : for G ⊃ H a face} reverse arrows: F • P [Bayer & Sturmfels 1998] [Miller 1998]
201 111 021 300 210 030 chain complex with img φ1 = I 003 ideal 012 An Example I = 〈x 3 1, x 2 1x2, x 3 2, x 2 1x3, x1x2x3, x 2 2x3, x2x 2 3, x 3 3〉 lcm condition labels for all cells f(P) = (8, 11, 4) check condition for each b ∈ Z d e.g., b = (2,2,2) minimal resolution F P 4 φ3 11 • : 0 → S −→ S φ2 8 −→ S φ1 −→ S → 0
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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 and 14:
Tropical Hyperplanes F(x) = (α1
Page 15 and 16:
duality between min and max: Max-Tr
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: (Co-)Cellular Resolutions II Propos
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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