<Text-field style="Heading 1" layout="Heading 1">Plucker Tree Ideal Second Secant Dimension</Text-field>

#Proves that the dimension of the second secant of the Plucker tree ideals is at least 4n-10 by applying the tropical secant dimension lemma. This establishes the base cases required in the proof of Lemma 4.7.

<Text-field style="Heading 2" layout="Heading 2">6-leaf Snowflake Tree (Splits: 12|3456, 34|1256, 56|1234)</Text-field>

TropSec6 := proc(v1,v2) local VT, M1, M2, i; # The exponent vectors below are indexed by the trivial splits and then by 12|3456, 34|1256, and 56|1234 VT := [Vector([1,1,0,0,0,0, 0,0,0 ]), Vector([1,0,1,0,0,0, 1,1,0 ]), Vector([1,0,0,1,0,0, 1,1,0 ]), Vector([1,0,0,0,1,0, 1,0,1 ]), Vector([1,0,0,0,0,1, 1,0,1 ]), Vector([0,1,1,0,0,0, 1,1,0 ]), Vector([0,1,0,1,0,0, 1,1,0 ]), Vector([0,1,0,0,1,0, 1,0,0 ]), Vector([0,1,0,0,0,1, 1,0,1 ]), Vector([0,0,1,1,0,0, 0,0,0 ]), Vector([0,0,1,0,1,0, 0,1,1 ]), Vector([0,0,1,0,0,1, 0,1,1 ]), Vector([0,0,0,1,1,0, 0,1,1 ]), Vector([0,0,0,1,0,1, 0,1,1 ]), Vector([0,0,0,0,1,1, 0,0,0 ])]: M1:= <0,0,0,0,0,0,0,0,0>: M2:= <0,0,0,0,0,0,0,0,0>: for i from 1 to nops(VT) do if DotProduct(VT[i],v1) < DotProduct(VT[i],v2) then M1:= <M1|VT[i]>; elif DotProduct(VT[i],v2) < DotProduct(VT[i],v1) then M2:= <M2|VT[i]>; end if; end do; return # M1,M2,Rank(M1), Rank(M2); [Rank(M1) + Rank(M2),Rank(M1),Rank(M2),M1,M2,v1,v2]; end proc: v1 := Vector(9, {(1) = 8, (2) = 5, (3) = 10, (4) = 6, (5) = 2, (6) = 2, (7) = 9, (8) = 10, (9) = 5}): v2 := Vector(9, {(1) = 5, (2) = 9, (3) = 4, (4) = 9, (5) = 2, (6) = 10, (7) = 3, (8) = 8, (9) = 10}): TropSec6(v1,v2);

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<Text-field style="Heading 2" layout="Heading 2">8-leaf Tree with Four Cherries (Splits: 17|234568, 28|134567, 34|125678, 56|123478, 1278|3456)</Text-field>

TropSec8:= proc(v1,v2) local VT, M1, M2, i; # The exponent vectors below are indexed by the trivial # then by 17|234568, 28|134567, 34|125678, 56|123478, and 1278|3456 VT := [Vector([1,1,0,0,0,0,0,0, 1,0,0,0,0]), Vector([1,0,1,0,0,0,0,0, 1,1,0,0,1]), Vector([1,0,0,1,0,0,0,0, 1,0,1,0,1]), Vector([1,0,0,0,1,0,0,0, 1,0,1,1,1]), Vector([1,0,0,0,0,1,0,0, 1,0,0,1,1]), Vector([1,0,0,0,0,0,1,0, 1,0,0,0,0]), Vector([1,0,0,0,0,0,0,1, 1,1,0,0,0]), Vector([0,1,1,0,0,0,0,0, 0,1,1,0,1]), Vector([0,1,0,1,0,0,0,0, 0,1,1,0,1]), Vector([0,1,0,0,1,0,0,0, 0,1,0,1,1]), Vector([0,1,0,0,0,1,0,0, 0,1,0,1,1]), Vector([0,1,0,0,0,0,1,0, 1,1,0,0,0]), Vector([0,1,0,0,0,0,0,1, 0,1,0,0,0]), Vector([0,0,1,1,0,0,0,0, 0,0,1,0,0]), Vector([0,0,1,0,1,0,0,0, 0,0,1,1,0]), Vector([0,0,1,0,0,1,0,0, 0,0,1,1,0]), Vector([0,0,1,0,0,0,1,0, 1,0,1,0,1]), Vector([0,0,1,0,0,0,0,1, 0,1,1,0,1]), Vector([0,0,0,1,1,0,0,0, 0,0,1,1,0]), Vector([0,0,0,1,0,1,0,0, 0,0,1,1,0]), Vector([0,0,0,1,0,0,1,0, 1,0,1,1,1]), Vector([0,0,0,1,0,0,0,1, 0,1,1,1,1]), Vector([0,0,0,0,1,1,0,0, 0,0,0,1,0]), Vector([0,0,0,0,1,0,1,0, 1,0,0,1,1]), Vector([0,0,0,0,1,0,0,1, 0,1,0,1,1]), Vector([0,0,0,0,0,1,1,0, 1,0,0,0,1]), Vector([0,0,0,0,0,1,0,1, 0,1,0,0,1]), Vector([0,0,0,0,0,0,1,1, 1,1,0,0,0])]: M1:= <0,0,0,0,0,0,0,0,0,0,0,0,0>: M2:= <0,0,0,0,0,0,0,0,0,0,0,0,0>: for i from 1 to nops(VT) do if DotProduct(VT[i],v1) < DotProduct(VT[i],v2) then M1:= <M1|VT[i]>; elif DotProduct(VT[i],v2) < DotProduct(VT[i],v1) then M2:= <M2|VT[i]>; end if; end do; return # M1,M2,Rank(M1), Rank(M2); [Rank(M1) + Rank(M2),Rank(M1),Rank(M2),M1,M2,v1,v2]; end proc: v1 := Vector(13, {(1) = 5, (2) = 8, (3) = 10, (4) = 6, (5) = 10, (6) = 9, (7) = 6, (8) = 4, (9) = 4, (10) = 5, (11) = 2, (12) = 3, (13) = 2}): v2 := Vector(13, {(1) = 3, (2) = 6, (3) = 3, (4) = 5, (5) = 3, (6) = 1, (7) = 4, (8) = 6, (9) = 4, (10) = 7, (11) = 3, (12) = 8, (13) = 7}): TropSec8(v1,v2);

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