Tropical Combinatorics

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

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

Resolutions via Coarse <strong>Tropical</strong> Convexity Theorem (Dochtermann, Sanyal & J. 2009+) Let A be an arrangement of n tropical hyperplanes in T d−1 . The colabeled complex CA supports a minimal cocellular resolution of the coarse type ideal I t(A). Eliahou-Kervaire resolution of 〈x1, x2, . . .,xn〉 d in the sufficiently generic case
Minimal Free Resolutions S = K[x1, . . .,xd] polynomial ring with Z d -grading deg x a = a free Z d -graded resolution F• of monomial ideal I ⊆ S is exact (algebraic) complex of Z d -graded S-modules: F• : · · · φk+1 −→ Fk φk −→ · · · φ2 −→ F1 φ1 −→ F0 → 0 Fi ∼ = a∈Z d S(−a)βi,a free Z d -graded S-modules maps φi homogeneous F0 = S and img φ1 = I i-th syzygy module img φi+1 ⊂ Fi resolution minimal if βi,a = dimK Tor S i (I, K)a fine graded Betti numbers
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 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: Resolutions via Coarse Tropical Con
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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