The \(E_7\) Pezzotope and the tropical positive Göpel prevariety
[2]:
using Oscar
[ Info: Precompiling Oscar [f1435218-dba5-11e9-1e4d-f1a5fab5fc13]
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...combining (and extending) ANTIC, GAP, Polymake and Singular
Version 0.14.0-DEV ...
... which comes with absolutely no warranty whatsoever
Type: '?Oscar' for more information
(c) 2019-2023 by The OSCAR Development Team
Combinatorial Data
We begin by collecting some data.
We first introdue the 34 vertices of the E7 pezzotope as combinatorially described in Hacking-Keel-Tevelev 2007. These are vectors with entries 0 and 1 and have 63 coordinates. For the coordinates we use the lexicographic ordering: 12, 13, …, 123, 124, …, 456, 234567=1, …, 123456=7. For the ordering of the points we follow the order introduced in Section 8 of our paper “Positive del Pezzo Geometry”.
[4]:
Vertices = matrix(QQ,[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
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1 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 0 1;
0 0 1 1 0 0 1 0 0 1 1 0 0 1 1 1 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 1 1 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 0 1 1 1 0 0 1 0 0 1 1 0 0
]);
Now we consider the linear map \(M_7: \mathbb{R}^{63} \rightarrow \mathbb{R}^{135}\) defined in Section 9 of our paper “Positive del Pezzo Geometry”. This is the map used by Bernd and co-authors to map the Bergman fan of \(E_7\) into the tropical Göpel variety. More details on this can be found in the paper “Tropicalization of del Pezzo surfaces”.
The ordering on the coorindates of \(\mathbb{R}^{135}\) is the one used in defining the ideal of the Göpel variety below.
[5]:
M7 = matrix(QQ, [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 ;
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0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 ;
0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 ;
1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 ;
0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 ;
0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 ;
0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 ;
0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 ;
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
]);
size(M7)
[5]:
(63, 135)
We use the data computed by Q. Ren for “Tropicalization of Classical Moduli Spaces”, in particular those rays of the Bergman fan that are of type \(F_1\), \(F_2\), \(F_4\) and \(F_{24}\). We follow the notation of the article “Tropicalization of Classical Moduli Spaces”.
The rays of type \(F_1\) are just 63 unit vectors in \(\mathbb{R}^{63}\).
The rays of type \(F_2\) corresponds to \(336\) root subsystems of type \(A_2\).
The rays of type \(F_4\) correspond to \(1260\) root subsystems of type \(A_3\).
The raus of type \(F_{24}\) correspond to \(288\) root subszstems of type \(A_6\).
WARNING: We are not following the coordinate ordering used by Q. Ren.
[6]:
include("E7_F2.jl");
[7]:
include("E7_F4.jl");
[8]:
include("E7_F24.jl");
Using the matrix \(M_7\) we map them to the tropical Göpel variety.
[9]:
F1R = M7;
[10]:
size(F1R)
[10]:
(63, 135)
[11]:
F2R = F2*M7;
[12]:
size(F2R)
[12]:
(336, 135)
[13]:
F4R = F4*M7;
[14]:
size(F4R)
[14]:
(1260, 135)
[15]:
F24R = F24*M7;
[16]:
size(F24R)
[16]:
(288, 135)
The Göpel Variety
The next step is to introduce the Göpel variety. In order to do this, we need to introduce the defining ideal which lives a polynomial ring with \(135\) variables.
[17]:
include("gopelprime.jl");
Each variable corresponds to a polynomial of degree 15 in the linear forms of the \(E_7\) root system. The ordering is important.
[18]:
linrel
[18]:
ideal(-f1236754 + f1243576 + g5172436, f1236574 - f1236754 + g1263457, -f1234657 + f1235647 - g3172645, f1235764 - f1236754 + g3142756, f1234567 - f1234657 + g1243756, -f1234657 + f1236457 - g7132546, f1236457 - f1236547 + g1263745, f1236745 - f1236754 - g2163745, -f1235467 + f1236457 - g3172456, g1253746 - g3172546 + g7132546, -g7132456 + g7132546 - g7132645, f1234657 - f1237654 + g6152347, g1234756 - g5162347 + g6152347, f1235467 - f1237465 + g4162357, -g6152347 + g6152437 - g6152734, g2173456 - g3172456 + g4172356, g3152467 - g3162457 + g3172456, -g1263457 + g1263547 - g1263745, -g1263745 + g2163745 - g6123745, -g1263745 + g4152637 - g5142637, -f1235467 + f1243576 + g6142735, -f1235467 + f1235764 + g5123647, -f1235467 + f1235476 + g2153467, g1234657 - g1253746 + g1273546, g1253746 - g2153746 + g5123746, g1253746 - g4162537 + g6142537, g1253467 - g1253647 + g1253746, g2173546 - g3172546 + g5172346, -g3124657 + g3152746 - g3172546, g7123546 - g7132546 + g7152346, g4132567 - g6132547 + g7132546, g7132645 - g7142635 + g7152634, g7123645 - g7132645 + g7162345, g4132657 - g5132647 + g7132645, g2134567 - g6132745 + g7132645, -g7132456 + g7152436 - g7162435, g7123456 - g7132456 + g7142356, g2134756 - g4132756 + g7132456, -g5132467 + g6132457 - g7132456, g3172546 - g4172536 + g6172534, g4152367 - g6152347 + g7152346, -f1237456 + f1237654 + g7123546, -f1236754 + f1237654 - g4132567, f1243567 - f1243657 + g1234756, g1234756 - g1253647 + g1263547, g1234756 - g2134756 + g3124756, -g1234756 + g4172356 - g7142356, -g1234567 + g1234657 - g1234756, -g5162347 + g5162437 - g5162734, g2163547 - g3162547 + g5162347, g4162357 - g5162347 + g7162345, g5142367 - g5162347 + g5172346, g1273456 - g5162734 + g6152734, g6132745 - g6142735 + g6152734, g6123457 - g6152734 + g6172534, -g3152746 + g4152736 - g6152734, g6132457 - g6152437 + g6172435, -g5123647 + g5132647 - g5162347, -g4152367 + g4162357 - g4172356, -g4152367 + g4152637 - g4152736, g4132567 - g4162537 + g4172536, -g4132567 + g4132657 - g4132756, -g2153467 + g3152467 - g4152367, g1234567 - g4152367 + g5142367, g1253467 - g3142567 + g4132567, g2134567 - g4132567 + g5132467, f1234756 - f1237456 + g6132547, -f1234756 + f1243756 + g5172634, -f1234756 + f1234765 + g2143756, -f1237456 + f1237465 + g2173456, -f1237456 + f1243765 + g5142637, -g2134756 + g5132647 - g6132547, -g6132547 + g6142537 - g6172534, f1243576 - f1243675 - g3124756, f1234576 - f1234756 + g1243657, -f1234756 + f1235746 - g3162745, f1235476 - f1237456 + g3162457, -f1243657 + f1243756 + g3124567, f1236475 - f1243657 - g7142536, g3162745 - g4162735 + g5162734, g5123467 - g5162734 + g5172634, g5132746 - g5142736 + g5162734, g2163457 - g5162734 + g7162534, g1273546 - g4162735 + g6142735, g2163547 - g4162735 + g7162435, g4162357 - g4162537 + g4162735, -g4132756 + g4152736 - g4162735, -g2173546 + g4172635 - g6172435, -g1263547 + g4172635 - g7142635, g4172356 - g4172536 + g4172635, -g1243567 + g1263547 - g1273546, -g1243567 + g6172435 - g7162435, -g1243567 + g3152467 - g5132467, g1234567 - g1243567 + g1253467, f1234675 - f1234765 + g1243567, g2143567 - g2163547 + g2173546, g6123745 - g6142537 + g6152437, -g3152467 + g6152437 - g7152436, f1235476 - f1243567 - g7142635, -f1243567 + f1243765 + g3124657, f1236745 - f1243567 - g4172536, -f1243567 + f1243576 + g2134567, -g3162457 + g5162437 - g7162435, -g5123746 + g5142637 - g5162437, g2163745 - g4162537 + g5162437, g5132467 - g5162437 + g5172436, g1273456 - g1273546 + g1273645, -g1273456 + g2173456 - g7123456, g1253467 - g1263457 + g1273456, -f1237546 + f1237564 + g2173546, -f1237465 + f1237564 + g7123645, f1235764 - f1237564 + g4132657, -f1236574 + f1237564 - g3142567, -f1235746 + f1243657 + g4172635, g1273456 - g3142756 + g4132756, g2143567 - g6142735 + g7142635, -g2134567 + g2143567 - g2153467, g2143567 - g3142567 + g5142367, g2143567 - g2143657 + g2143756, f1237465 - f1237645 + g1273546, g1273546 - g3152746 + g5132746, g2173456 - g2173546 + g2173645, -g2134657 + g2153746 - g2173546, -g3142657 + g5142637 - g7142635, g7142356 - g7142536 + g7142635, -g6172345 + g6172435 - g6172534, -f1235764 + f1243675 + g6172435, -f1235674 + f1243765 + g7162435, g7162345 - g7162435 + g7162534, -g6142357 + g6142537 - g6142735, -g3142756 + g5142736 - g6142735, g5123467 - g5123647 + g5123746, -g5123467 + g6123457 - g7123456, g1253467 - g2153467 + g5123467, g5123467 - g5132467 + g5142367, -g1263547 + g3152647 - g5132647, -f1236475 + f1236745 - g1263547, g3124657 - g4123657 + g6123457, g4123657 - g5123647 + g7123645, -g4123657 + g4132657 - g4162357, -g4123657 + g4152736 - g4172536, g1243657 - g2143657 + g4123657, -g2134756 + g2153647 - g2163547, g2163457 - g2163547 + g2163745, g3124567 - g3124657 + g3124756, -g3124567 + g6123745 - g7123645, g3124567 - g3142567 + g3152467, -g1234567 + g2134567 - g3124567, g4123756 - g5123746 + g6123745, -g3124756 + g4123756 - g7123456, g4123756 - g4132756 + g4172356, g4123756 - g4152637 + g4162537, -f1234576 + f1234675 + g4123756, -g3124657 + g3142657 - g3162457, g1234657 - g2134657 + g3124657, -g5123746 + g5132746 - g5172346, -f1237546 + f1237645 + g7123456, f1237456 - f1237546 + g1273645, g1253647 - g3162547 + g6132547, g6132457 - g6132547 + g6132745, f1234756 - f1236754 + g7152346, g1234657 - g5172346 + g7152346, g7152346 - g7152436 + g7152634, g2153746 - g3152746 + g7152346, -g7142356 + g7152346 - g7162345, -f1234657 + f1234675 + g2143657, -g2153647 + g3152647 - g6152347, -g6142357 + g6152347 - g6172345, f1235674 - f1237654 + g3142657, -f1237645 + f1237654 + g2173645, f1236457 - f1236475 - g2163457, f1236457 - f1243675 - g5142736, g7132546 - g7142536 + g7162534, -g2134657 + g5132746 - g7132546, -g2153746 + g4152637 - g6152437, g3142567 - g3162547 + g3172546, -f1235476 + f1235746 - g1253647, f1235746 - f1237546 + g6132457, f1235746 - f1235764 - g2153746, g3162457 - g3162547 + g3162745, g1273645 - g3162745 + g6132745, g3142756 - g3152746 + g3162745, g2163745 - g3162745 + g7162345, -g7123456 + g7152634 - g7162534, g1273645 - g2173645 + g7123645, g7123645 - g7142536 + g7152436, -g1253647 + g2153647 - g5123647, -g5123647 + g5142736 - g5172436, g3124756 - g3152647 + g3162547, g6123745 - g6132745 + g6172345, -f1236475 + f1236574 + g6123745, -g6123457 + g6132457 - g6142357, g1263457 - g2163457 + g6123457, f1235476 - f1236475 + g4172356, -g4172356 + g5172346 - g6172345, g2163457 - g3162457 + g4162357, -g1234657 + g4162357 - g6142357, -g1273645 + g4152736 - g5142736, -g2153647 + g4152736 - g7152436, -f1237546 + f1243675 + g4152637, -g3152647 + g4152637 - g7152634, g3162547 - g4162537 + g7162534, f1234675 - f1243675 - g7162534, f1234675 - f1236475 + g5132746, f1234675 - f1235674 + g6172345, f1234675 - f1237645 + g3152647, g2134657 - g2143657 + g2163457, -g2143657 + g5142736 - g7142536, g2143657 - g3142657 + g6142357, -g2143657 + g2153647 - g2173645, f1243756 - f1243765 - g2134756, f1234765 - f1235764 + g7162345, -f1243657 + f1243675 + g2134657, -f1235746 + f1236745 - g7142356, g1234567 - g1263745 + g1273645, g4132657 - g4152637 + g4172635, f1235476 - f1235674 - g5123746, -f1237564 + f1243657 + g6152437, g2153467 - g6152734 + g7152634, g6123547 - g6132547 + g6152347, g1243756 - g3172456 + g7132456, g1243756 - g1253746 + g1263745, -g1234756 + g1243756 - g1273456, g1243756 - g5162437 + g6152437, g1243567 - g1243657 + g1243756, g1243756 - g2143756 + g4123756, -f1234567 + f1236547 - g3172546, f1234567 - f1235467 + g7132645, -f1234567 + f1237564 - g5162347, f1234567 - f1243567 - g6152734, -f1234567 + f1234576 + g2143567, f1234567 - f1234765 - g4123657, g3172456 - g3172546 + g3172645, -g1263745 + g3172645 - g7132645, -g3172645 + g4172635 - g5172634, g3124567 - g3162745 + g3172645, g3142657 - g3152647 + g3172645, -g2173645 + g3172645 - g6172345, f1235647 - f1236547 + g7132456, -f1237645 + f1243576 + g4162537, g1253647 - g4172536 + g7142536, -g2173645 + g4172536 - g5172436, -g2134657 + g4132657 - g6132457, g1263457 - g3142657 + g4132657, -g4132756 + g5132746 - g6132745, -f1235674 + f1236574 - g4132756, g2153467 - g2163457 + g2173456, g2153467 - g2153647 + g2153746, g3152467 - g3152647 + g3152746, f1243675 - f1243765 + g1234567, g1234567 - g6172345 + g7162345, g5142367 - g5142637 + g5142736, g5142367 - g6142357 + g7142356, -g3142567 + g6142537 - g7142536, g3142567 - g3142657 + g3142756, g1253467 - g6172534 + g7162534, f1235674 - f1235764 + g1253467, g2134567 - g2134657 + g2134756, -f1237654 + f1243567 + g5162437, f1237564 - f1237654 + g1273456, f1234657 - f1243657 - g5162734, g4123567 - g6123547 + g7123546, f1234657 - f1234756 - g4123567, g4123567 - g4162735 + g4172635, -g1243567 + g2143567 - g4123567, g1273546 - g2173546 + g7123546, g7123546 - g7142635 + g7162435, g6123547 - g6142735 + g6172435, g3124567 - g4123567 + g5123467, g1263547 - g2163547 + g6123547, g4123567 - g4123657 + g4123756, g3124657 - g5123746 + g7123546, f1236475 - f1237465 + g3152467, -g7123456 + g7123546 - g7123645, g3124756 - g5123647 + g6123547, g6123457 - g6123547 + g6123745, g4123567 - g4132567 + g4152367, -f1235467 + f1235647 - g1253746, f1235647 - f1243756 - g4162735, f1235647 - f1235746 - g5123467, f1235647 - f1235674 - g2153647, f1235647 - f1237645 + g6142357, f1236457 - f1236754 - g6123547, f1236457 - f1237456 + g4152367, f1236547 - f1236574 - g2163547, f1236547 - f1236745 - g6123457, f1236547 - f1243765 - g4152736, f1236547 - f1237546 + g5142367, g3172456 - g5172436 + g6172435, g3124756 - g3142756 + g3172456, -g2134567 + g2163745 - g2173645, g5132467 - g5132647 + g5132746, f1236745 - f1237645 + g5132467, -f1237465 + f1243756 + g6142537, f1243576 - f1243756 + g1234657, -f1236574 + f1243756 + g7152436, g2173456 - g5172634 + g6172534, g5132647 - g5142637 + g5172634, g5172346 - g5172436 + g5172634, -g1263457 + g5172634 - g7152634, g2134756 - g2143756 + g2173456, g2143756 - g5142637 + g6142537, g2143756 - g2153746 + g2163745, g2143756 - g3142756 + g7142356, f1234765 - f1237465 + g5132647, f1234765 - f1243765 - g6172534, -f1234765 + f1236745 - g3152746, f1234576 - f1235476 + g6132745, f1234576 - f1237546 + g3162547, -f1234576 + f1243576 + g7152634, g1243657 - g3162457 + g6132457, g1243657 - g1253647 + g1273645, g1234657 - g1243657 + g1263457, g1243657 - g5172436 + g7152436, f1234576 - f1236574 + g5172346)
[19]:
G = gens(linrel)
[19]:
315-element Vector{QQMPolyRingElem}:
-f1236754 + f1243576 + g5172436
f1236574 - f1236754 + g1263457
-f1234657 + f1235647 - g3172645
f1235764 - f1236754 + g3142756
f1234567 - f1234657 + g1243756
-f1234657 + f1236457 - g7132546
f1236457 - f1236547 + g1263745
f1236745 - f1236754 - g2163745
-f1235467 + f1236457 - g3172456
g1253746 - g3172546 + g7132546
-g7132456 + g7132546 - g7132645
f1234657 - f1237654 + g6152347
g1234756 - g5162347 + g6152347
⋮
g2143756 - g3142756 + g7142356
f1234765 - f1237465 + g5132647
f1234765 - f1243765 - g6172534
-f1234765 + f1236745 - g3152746
f1234576 - f1235476 + g6132745
f1234576 - f1237546 + g3162547
-f1234576 + f1243576 + g7152634
g1243657 - g3162457 + g6132457
g1243657 - g1253647 + g1273645
g1234657 - g1243657 + g1263457
g1243657 - g5172436 + g7152436
f1234576 - f1236574 + g5172346
[20]:
size(G)
[20]:
(315,)
Tropicalization
Now we introduce the tropical semiring, the trivial valuation and a function that check whether a tropical polynomial vanishes at a point.
[21]:
T = TropicalSemiring();
val = TropicalSemiringMap(QQ);
[22]:
function min_achieved_twice(v::Vector)
m=minimum(v)
c = 0
for w in v
if w == m
c=c+1
end
end
return c >= 2
end
function vanishes(f::AbstractAlgebra.Generic.MPoly{Oscar.TropicalSemiringElem{typeof(min)}},x::Matrix{Oscar.TropicalSemiringElem{typeof(min)}})
ter = collect(terms(f))
v = [evaluate(ter[i],vec(x)) for i in 1:length(ter)]
return min_achieved_twice(v)
end
[22]:
vanishes (generic function with 1 method)
We first check that the \(F_1\), \(F_2\), \(F_4\) and \(F_{24}\) rays belong to the tropical Göpel prevariety.
[23]:
for i in 1:63
V = F1R[i,:]
TV = [T(v) for v in V]
for g in G
boo = vanishes(tropical_polynomial(g,val),TV)
if boo == false
println("Warning:")
println(i)
end
end
end
[24]:
for i in 1:336
V = F2R[i,:]
TV = [T(v) for v in V]
for g in G
boo = vanishes(tropical_polynomial(g,val),TV)
if boo == false
println("Warning:")
println(i)
end
end
end
[25]:
for i in 1:1260
V = F4R[i,:]
TV = [T(v) for v in V]
for g in G
boo = vanishes(tropical_polynomial(g,val),TV)
if boo == false
println("Warning:")
println(i)
end
end
end
[26]:
for i in 1:288
V = F24R[i,:]
TV = [T(v) for v in V]
for g in G
boo = vanishes(tropical_polynomial(g,val),TV)
if boo == false
println("Warning:")
println(i)
end
end
end
No warnings!
Now we move to the tropical positive part. The following function takes as input a polynomial and a vector and returns 0 if the polynomial positively vanishes at the vector and 1 otherwise.
[27]:
function positive_vanishes(f::MPolyRingElem,V::QQMatrix)
C= coefficients_and_exponents(f)
LV = [];
RV = [];
for i in C
s = sum(V[j]*i[2][j] for j in 1:135)
if i[1]>0
push!(LV,s)
else
push!(RV,s)
end
end
if minimum(LV)==minimum(RV)
return 0
else
return 1
end
end
[27]:
positive_vanishes (generic function with 1 method)
[28]:
posF1R = []
for i in 1:63
check = [positive_vanishes(g,F1R[i,:]) for g in G]
checkmat= matrix(QQ, [check])
zvec = zero_matrix(QQ,1,315)
if checkmat==zvec
println(F1R[i,:])
push!(posF1R,F1R[i,:]);
end
end
println(length(posF1R))
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]
10
We need to check that all 10 elements in \(\texttt{posF1R}\) are distinct.
[29]:
z = zero_matrix(QQ,1,135);
c=10;
for i in 1:9
for j in i+1:10
if z==posF1R[i]-posF1R[j]
c = c-1
end
end
end
println(c)
10
So we see that out of the 63 rays in \(\texttt{F1R}\) ten rays belong to the tropical positive Yoshida prevariety.
[30]:
posF2R = []
for i in 1:336
check = [positive_vanishes(g,F2R[i,:]) for g in G]
checkmat= matrix(QQ, [check])
zvec = zero_matrix(QQ,1,315)
if checkmat==zvec
println(F2R[i,:])
push!(posF2R,F2R[i,:]);
end
end
println(length(posF2R))
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1]
[0 1 0 1 1 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1]
[0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]
[1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1]
[1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0]
12
[31]:
z = zero_matrix(QQ,1,135);
c=12;
for i in 1:c-1
for j in i+1:c
if z==posF2R[i]-posF2R[j]
c = c-1
end
end
end
println(c)
12
Out of the \(336\) rays in \(\texttt{F2R}\) we see 12 distinct rays in the tropical pisitive Yoshida prevariety.
We collect them together with the previous 10 and obtain 22 rays. These numbers fit with the numbers or \(A_1\) and \(A_2\) root systems from Hacking, Keel and Tevelev.
[32]:
posF4R=[]
for i in 1:1260
check = [positive_vanishes(g,F4R[i,:]) for g in G]
checkmat= matrix(QQ, [check])
zvec = zero_matrix(QQ,1,315)
if checkmat==zvec
println(F4R[i,:])
push!(posF4R,F4R[i,:]);
end
end
println(length(posF4R))
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 2 1 1 2 1 1 2 1 1 1 1 1 1 1 1 2 1 1 2 1 1 2 1 1 1 1 1 1 1 1 2 1 1 2 1 1 2 1 1 1 1 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1]
[1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0 0 1 1 1 1 1 1 2 1 1 1 0 0 1 0 0 0 0 0 0 0 0 2 1 1 1 0 0 1 0 0 0 0 0 0 0 0 2 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 2 1 1 2 1 1 1 1 1 1 2 1 1 1 1 2 1 1 2 1 1 1 1 1 1 2 1 1]
[1 1 1 1 1 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 1 1 1 1 1 1 2 2 2 2 2 2 2 0 0 2 0 0 2 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1]
[1 1 1 1 1 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 1 1 1 1 1 1 2 2 2 2 2 2 2 0 0 2 0 0 2 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1]
[1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0 0 1 1 1 1 1 1 2 1 1 1 0 0 1 0 0 0 0 0 0 0 0 2 1 1 1 0 0 1 0 0 0 0 0 0 0 0 2 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 2 1 1 2 1 1 1 1 1 1 2 1 1 1 1 2 1 1 2 1 1 1 1 1 1 2 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 2 1 1 2 1 1 2 1 1 1 1 1 1 1 1 2 1 1 2 1 1 2 1 1 1 1 1 1 1 1 2 1 1 2 1 1 2 1 1 1 1 1 1 1 1]
[1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 2 1 1 2 1 1 2 1 1 1 1 1 1 1 1 2 1 1 2 1 1 2 1 1 1 1 1 1 1 1 2 1 1 0 0 0 0 0 0 0 0 1 0 0 1 2 1 1 0 0 0 0 0 0 0 0 1 0 0 1 2 1 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1]
[1 2 1 2 1 1 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 0 0 1 2 1 2 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 2 0 0 1 0 0 0 0 0 0 1 0 0 1 1 2 0 0 1 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 2 1 1 2]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0]
[1 2 1 2 1 1 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 0 0 1 2 1 2 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 2 0 0 1 0 0 0 0 0 0 1 0 0 1 1 2 0 0 1 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 2 1 1 2]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 1 2 1 1 2 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 1 2 1 1 2 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2]
[0 1 0 1 1 1 0 1 0 1 1 1 0 1 0 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 0 0 1 0 0 1 0 0 0 0 0 0 2 1 1 0 0 1 0 0 1 0 0 0 0 0 0 2 1 1 0 0 1 0 0 1 0 0 0 0 0 0 2 1 1 1 1 1 1 1 1 2 1 1 2 1 1 2 1 1]
[0 1 0 1 1 1 0 1 0 1 1 1 0 1 0 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 0 0 1 0 0 1 0 0 0 0 0 0 2 1 1 0 0 1 0 0 1 0 0 0 0 0 0 2 1 1 0 0 1 0 0 1 0 0 0 0 0 0 2 1 1 1 1 1 1 1 1 2 1 1 2 1 1 2 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1]
[1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 2 1 1 2 1 1 2 1 1 1 1 1 1 1 1 2 1 1 2 1 1 2 1 1 1 1 1 1 1 1 2 1 1 0 0 0 0 0 0 0 0 1 0 0 1 2 1 1 0 0 0 0 0 0 0 0 1 0 0 1 2 1 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0]
18
[33]:
z = zero_matrix(QQ,1,135);
c=18;
b = bases(uniform_matroid(2,18));
for a in b
if z == posF4R[a[1]]-posF4R[a[2]]
c=c-1
println(a)
end
end
println(c)
[1, 7]
[2, 16]
[3, 6]
[4, 5]
[8, 17]
[9, 11]
[10, 18]
[12, 13]
[14, 15]
9
We need only to keep the distinct rays.
[34]:
h=[1,2,3,4,8,9,10,12,14]
H = [posF4R[i] for i in h];
z = zero_matrix(QQ,1,135);
c=length(H);
for i in 1:c-1
for j in i+1:c
if z==H[i]-H[j]
c = c-1
end
end
end
println(c)
posF4R=H;
9
[35]:
posF24R = []
for i in 1:288
check = [positive_vanishes(g,F24R[i,:]) for g in G]
checkmat= matrix(QQ, [check])
zvec = zero_matrix(QQ,1,315)
if checkmat==zvec
println(F24R[i,:])
push!(posF24R,F24R[i,:]);
end
end
println(length(posF24R))
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3]
[3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 0 0 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3]
[3 3 3 3 3 3 3 3 0 3 0 3 3 3 0 3 0 3 3 3 0 3 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3]
[3 3 3 3 3 3 3 3 0 3 0 3 3 3 0 3 0 3 3 3 0 3 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3]
[3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 0 0 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3]
[3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 0 0 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3]
[3 3 3 3 3 3 3 3 0 3 0 3 3 3 0 3 0 3 3 3 0 3 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3]
[3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 0 0 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3]
[3 3 3 3 3 3 3 3 0 3 0 3 3 3 0 3 0 3 3 3 0 3 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3]
[3 3 3 3 3 3 3 3 0 3 0 3 3 3 0 3 0 3 3 3 0 3 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3]
[3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 0 0 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3]
[3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 0 0 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3]
[3 3 3 3 3 3 3 3 0 3 0 3 3 3 0 3 0 3 3 3 0 3 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3]
[3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 0 0 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3]
[3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 0 0 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3]
[3 3 3 3 3 3 3 3 0 3 0 3 3 3 0 3 0 3 3 3 0 3 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3]
[3 3 3 3 3 3 3 3 0 3 0 3 3 3 0 3 0 3 3 3 0 3 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3]
24
[36]:
z = zero_matrix(QQ,1,135);
H = [posF24R[1]];
c=24;
for i in 2:24
boo = false
for j in 1:i-1
boo = boo || (z==H[j]-posF24R[i])
end
if !boo
push!(H,posF24R[i])
end
end
println(c)
println(length(H))
posF24R=H
24
3
[36]:
3-element Vector{QQMatrix}:
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3]
[3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 0 0 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3]
[3 3 3 3 3 3 3 3 0 3 0 3 3 3 0 3 0 3 3 3 0 3 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 0 0 3 0 0 3 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3]
We observe 10+12+9+3=34 distinct rays with positive vanishing. This recovers the number of 34 vertices of the \(E_7\) pezzotope.
[37]:
Rays=[]
Rays=vcat(posF1R[1],posF1R[2])
for i in 3:length(posF1R)
Rays = vcat(Rays,posF1R[i])
end
for i in 1:length(posF2R)
Rays = vcat(Rays,posF2R[i])
end
for i in 1:length(posF4R)
Rays = vcat(Rays,posF4R[i])
end
for i in 1:3
Rays = vcat(Rays,posF24R[i])
end
println(size(Rays))
(34, 135)
We run another check to see that they are indeed positive.
[38]:
for i in 1:34
check = [positive_vanishes(g,Rays[i,:]) for g in G]
checkmat= matrix(QQ, [check])
zvec = zero_matrix(QQ,1,315)
if !(checkmat==zvec)
println("Warning: ",Rays[i,:])
end
end
Now we take sums of two, three, four, five and six of rays, respectively, and check which of these sums lie in the positive part. This confirms the f-vector (34,297,1105,2000,1737,579).
[39]:
k=0
Edges=[]
for i in 1:34
for j in i+1:34
R = Rays[i,:]+Rays[j,:]
check = [positive_vanishes(g,R) for g in G]
checkmat= matrix(QQ, [check])
zvec = zero_matrix(QQ,1,315)
if checkmat==zvec
push!(Edges,[i,j])
k=k+1
end
end
end
println(k)
297
We obtain 297 edges from the 34 vertices from \(\texttt{posF1R}\) and \(\texttt{posF2R}\). This verifies the second entry in the f-vector (34,297,1105,2000,1737,579). The following cells verify the remaining 4 entries of the f-vector: The numbers and that the corresponding cones have indeed the correct dimension.
WARNING: These computations are very slow! They will run for several hours! The output files are available as downloads on the MathRepo page.
[164]:
k=0
cones3=[]
for i in 1:34
for j in (i+1):34
for l in (j+1):34
R = Rays[i,:]+Rays[j,:]+Rays[l,:]
check = [positive_vanishes(g,R) for g in G]
checkmat= matrix(QQ, [check])
zvec = zero_matrix(QQ,1,315)
if checkmat==zvec
push!(cones3,[i,j,l])
k=k+1
end
end
end
end
println(k)
1105
[175]:
k=0
cones4 =[]
for i in 1:34
for v in cones3
if !(i in v)
R = Rays[i,:]+Rays[v[1],:]+Rays[v[2],:]+Rays[v[3],:]
check = [positive_vanishes(g,R) for g in G]
checkmat= matrix(QQ, [check])
zvec = zero_matrix(QQ,1,315)
if checkmat==zvec
h=sort(vcat(v,[i]) )
if !(h in cones4)
push!(cones4,h)
k=k+1
end
end
end
end
end
println(k)
2000
[176]:
k=0
cones5=[]
for i in 1:34
for v in cones4
if !(i in v)
R = Rays[i,:]+Rays[v[1],:]+Rays[v[2],:]+Rays[v[3],:]+Rays[v[4],:]
check = [positive_vanishes(g,R) for g in G]
checkmat= matrix(QQ, [check])
zvec = zero_matrix(QQ,1,315)
if checkmat==zvec
h=sort(vcat(v,[i]) )
if !(h in cones5)
push!(cones5,h)
k=k+1
end
end
end
end
end
println(k)
1737
[177]:
k=0
cones6=[]
for i in 1:34
for v in cones5
if !(i in v)
R = Rays[i,:]+Rays[v[1],:]+Rays[v[2],:]+Rays[v[3],:]+Rays[v[4],:]+Rays[v[5],:]
check = [positive_vanishes(g,R) for g in G]
checkmat= matrix(QQ, [check])
zvec = zero_matrix(QQ,1,315)
if checkmat==zvec
h=sort(vcat(v,[i]) )
if !(h in cones6)
push!(cones6,h)
k=k+1
end
end
end
end
end
println(k)
579
We check that the cones we computed really have the dimension we claim. The following code shows this. It also confirms that the arising fan is simplicial.
[182]:
boo = true
for e in Edges
boo = boo && (rank(Rays[e,:]) == 2)
end
println(boo)
true
[183]:
boo = true
for e in cones3
boo = boo && (rank(Rays[e,:]) == 3)
end
println(boo)
true
[187]:
boo = true
for e in cones4
boo = boo && (rank(Rays[e,:]) == 4)
end
println(boo)
true
[185]:
boo = true
for e in cones5
boo = boo && (rank(Rays[e,:]) == 5)
end
println(boo)
true
[186]:
boo = true
for e in cones6
boo = boo && (rank(Rays[e,:]) == 6)
end
println(boo)
true