Tropical Combinatorics

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

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

<strong>Tropical</strong> Polynomials read ordinary (Laurent) polynomial with real coefficients as function replace operations “+” and “·” by “⊕” and “⊙” Example F(x) := (7 ⊙ x ⊙3 ) ⊕ (3 ⊙ x) ⊕ 4 = min(7 + 3x,3 + x,4) Definition tropical polynomial F vanishes at p :⇔ there are at least two terms where the minimum F(p) is attained F(1) = min(7 + 3 · 1, 3+1 , 4 ) = 4
tropical semi-module (R d , ⊕, ⊙) componentwise addition tropical scalar multiplication <strong>Tropical</strong> Hypersurfaces Definition tropical hypersurface T (F) := vanishing locus of (multi-variate) tropical polynomial F Example F(x) = (7 ⊙ x ⊙3 ) ⊕ (3 ⊙ x) ⊕ 4 T (F) = {−2, 1} ⊂ R 1 (−2,1) −2 1 7 + 3x (1,4) 3 + x 4
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: Tropical Polynomials read ordinary
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 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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