The \(E_6\) Pezzotope and the tropical positive Yoshida prevariety
[1]:
using Oscar
----- ----- ----- - -----
| | | | | | | | | |
| | | | | | | |
| | ----- | | | |-----
| | | | |-----| | |
| | | | | | | | | |
----- ----- ----- - - - -
...combining (and extending) ANTIC, GAP, Polymake and Singular
Version 0.12.0-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 introduce the 15 vertices of the E6 pezzotope as combinatorially described in Hacking-Keel-Tevelev 2007. These are vectors with entries 0 and 1 and have 36 coordinates. For the coordinates we use the lexicographic ordering: 12, 13, …, 123, 124, …, 456, 123456=7. For the ordering of the points, we follow the order introduced in Section 8 of our paper “Positive del Pezzo Geometry”.
[8]:
Vertices = matrix(QQ,[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 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 0 0 0 0 0 0 0 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 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 0 0 0 0 0 0 0 0;
0 0 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 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 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;
1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0;
0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0;
0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0;
0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 1 0 0 0 1 0 0 0 1 0 0;
0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 1 0]);
Now we consider the linear map \(M_6: \mathbb{R}^{36} \rightarrow \mathbb{R}^{40}\) defined in Section 9 of our paper “Positive del Pezzo Geometry”. This is the map used by Bernd Sturmfels and co-authors to map the Bergman fan of \(E_6\) into the tropical Yoshida variety. More details on this can be found in the paper “Tropicalization of del Pezzo surfaces”.
The ordering on the coorindates of \(\mathbb{R}^{40}\) is the one used in defining the ideal of the Yoshida variety below.
[9]:
M6 = matrix(QQ, [1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0;
0 0 0 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0;
0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 1 0;
0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 1;
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 1 0 0 1 0 1 1;
0 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1;
0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 0 1;
0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 0 1 0 1 0;
0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 1 0 0;
1 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 0 0 0 0 1 1 1 0 0 0 0 1;
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 1 0 1 0 0 1 0;
0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0;
0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 1 1 0 0;
0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 0;
1 0 0 1 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 1 1 0 0 1 0 0 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 0 0 0 1 0 0 0 0 0 0 0 0 0;
1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0;
0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0;
0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 1 0 0 0 0 0 0;
0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0;
0 0 0 0 1 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0;
0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0;
0 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 0 0 1 0 0;
0 1 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0;
1 0 0 1 0 0 0 1 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1;
0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1;
0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 1 0;
0 0 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0;
0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0;
0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0;
1 0 0 0 1 1 1 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0;
1 1 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0;
1 0 1 0 1 0 0 1 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0;
0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 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 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 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 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1]);
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 \(F_1\) and \(F_2\). We follow the notation of the article “Tropicalization of Classical Moduli Spaces”.
The rays of type \(F_1\) are just 36 unit vectors in \(\mathbb{R}^{36}\).
The rays of type \(F_2\) corresponds to 120 root subsystems of type \(A_2\).
WARNING: We are not following the coordinate ordering used by Q. Ren.
[10]:
F2 = matrix(QQ, [1 0 1 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; 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 1 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 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 1 0 0 0 0 0 1 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 1 0 1 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 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 1 0 0 0 0 0 0 0 0 0 0 1 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 1 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; 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 1 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 1 1 0 0 0 0 0 0 0 0 0 0 0 0; 1 1 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 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 1 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 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 1 1 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 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 1 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 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 1 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 1 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 0 0 0 0 0; 0 0 1 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 0 0 0 0; 0 0 0 1 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 1 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 1 0 0 1 0; 0 0 0 1 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 1 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 0 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 1 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 1 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 1 0 1 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 1 0 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 1 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 1 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1; 0 0 0 1 1 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 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 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 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 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 1 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 1 0 0 0 1 0; 0 0 0 0 1 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 1 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 1 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 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1; 0 0 0 0 0 0 0 0 0 0 1 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 1 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 1 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 1 1 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 1 0 1 0 0 0 0 0 0; 0 0 1 1 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 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 1 0; 0 0 0 0 0 0 0 0 0 0 0 0 0 1 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 1 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; 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 0 1 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 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0; 0 0 0 0 0 0 0 0 0 0 1 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 0 0 0 0 0 0 0 0 0 1 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 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 1 0 0; 0 0 0 0 0 1 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 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 0 1 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 0 0 0 0 1 0 0 0 0 0 0 0 1 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 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 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0; 1 0 0 0 1 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 1 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 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 1 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 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 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 0 0 1; 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 1 0 0 0 0 0 1 0; 0 0 0 0 0 0 0 0 0 0 0 0 1 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 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 1 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 1 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 1 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 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 0 0 0 0 0 0 0 1 0 0 0 0 0 0 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 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 0 1 0 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 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 1 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 1 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 1 0 0; 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 1 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 1 0 0 0 0 1 0 0 0; 0 0 0 0 0 1 0 0 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 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 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 1 1 0 0 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 1 0 0 0 0 0 0 0 0 0 0 0 1 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 1 0 0 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 0 1 0 1 0 0 0 0 0 0 0 0; 0 1 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 1 0 0 0 0 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 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 1 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0; 0 0 0 0 0 0 1 0 0 0 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0; 0 0 1 0 1 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 1 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 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1; 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 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 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 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 1 0 0 1 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 1 0 0 1 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 1 0 0 0 0 0 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 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 1 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 0 0 1 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 1 0 0 1 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 1 0 0 0 1 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 1 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 1 0 0 0 0 1 0 0 0 0 0 0 0 1; 0 0 0 0 0 0 1 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 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1; 0 1 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 0 0 0 0 0 0 1 0 0 0 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 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 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 0 0 0 0 0 0 1 0 0 0 0; 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1]);
Using the matrix \(M_6\), we map the rays to the tropical Yoshida variety.
[11]:
F1R = M6;
[12]:
size(F1R)
[12]:
(36, 40)
[13]:
F2R = F2*M6;
[14]:
size(F2R)
[14]:
(120, 40)
The Yoshida Variety
The next step is to introduce the Yoshida variety. In order to do this, we need to introduce the defining ideal which lives in a polynomial ring with \(40\) variables.
[15]:
S, (y1,y2,y3,y4,y5,y6,y7,y8,y9,y10,y11,y12,y13,y14,y15,y16,y17,y18,y19,y20,y21,y22,y23,y24,y25,y26,y27,y28,y29,y30,y31,y32,y33,y34,y35,y36,y37,y38,y39,y40) = PolynomialRing(QQ,["y1","y2","y3","y4","y5","y6","y7","y8","y9","y10","y11","y12","y13","y14","y15","y16","y17","y18","y19","y20","y21","y22","y23","y24","y25","y26","y27","y28","y29","y30","y31","y32","y33","y34","y35","y36","y37","y38","y39","y40"])
[15]:
(Multivariate Polynomial Ring in 40 variables y1, y2, y3, y4, ..., y40 over Rational Field, QQMPolyRingElem[y1, y2, y3, y4, y5, y6, y7, y8, y9, y10 … y31, y32, y33, y34, y35, y36, y37, y38, y39, y40])
Each variable corresponds to a polynomial of degree 9 in the linear forms of the \(E_6\) root system. The variable ordering is important.
[16]:
L = ideal([y1+y10-y11-y32, y1+y10-y14+y17, y1+y10-y2+y34, y1-y11-y15+y16, y1-y11+y3-y33,y1+y12-y13-y32, y1+y12-y15+y17, y1-y13-y14+y16, y1-y13-y35+y7, y1-y14+y24+y6, y1-y15+y27+y8, y1+y16-y28-y5, y1+y17-y21-y9, y1-y18+y31-y32, y1-y18-y33+y34,y1-y18-y35+y40, y1-y2+y3+y31, y1-y2+y6-y9, y1-y21+y24+y34, y1-y21+y27+y40, y1+y24-y28-y35,y1+y27-y28-y33, y1+y31-y4+y7, y1+y12-y4+y40, y1-y5+y6+y7, y1+y3-y5+y8,y10-y11-y12+y13, y10-y11+y18-y31, y10-y12-y14+y15, y10-y12+y22-y25, y10+y13-y16+y17,y10+y13-y19+y30, y10-y14+y21+y9, y10+y15-y16-y32, y10+y15-y20-y38, y10-y16+y23+y37,y10+y17-y24-y6, y10+y18-y19+y22, y10+y18-y2+y33, y10-y19+y23-y24,y10-y2+y21-y24, y10-y20+y21+y22, y10-y20+y23+y33, y10-y25+y30-y31, y10-y25-y38+y9,y10-y3-y31+y34, y10-y3-y32+y33, y10-y3+y37-y38, y10+y30+y37-y6, y10+y34-y6+y9,y11+y12-y16+y17, y11+y12-y19+y30, y11-y13-y14+y15, y11-y13+y22-y25, y11-y14+y17+y32, y11-y14+y29+y36, y11+y15-y28-y5, y11-y16+y27+y8, y11+y17-y26-y39,y11-y18-y25+y30, y11-y18-y3+y34, y11-y19+y22+y31, y11-y19-y39+y8, y11-y2+y31+y33,y11-y2+y32+y34, y11-y2+y36-y39, y11+y22+y36-y5, y11-y25-y28+y29, y11-y26+y27+y30,y11-y26+y29+y34, y11+y27-y28-y3, y11+y33-y5+y8, y12-y13+y18-y31, y12+y14-y16-y32,y12+y14-y20-y38, y12-y15+y21+y9, y12-y16+y26+y39, y12+y17-y27-y8, y12+y18-y19+y25,y12+y18+y35-y4, y12-y19+y26-y27, y12-y20+y21+y25, y12-y20+y26+y35,y12+y21-y27-y4, y12-y22+y30-y31, y12-y22-y38+y9, y12+y30+y39-y8, y12-y31+y40-y7,y12-y32+y35-y7, y12-y38+y39-y7, y12+y40-y8+y9, y13+y14-y28-y5, y13-y15+y17+y32,y13-y15+y29+y36, y13-y16+y24+y6, y13+y17-y23-y37, y13-y18-y22+y30, y13-y18+y40-y7,y13-y19+y25+y31, y13-y19-y37+y6, y13-y22-y28+y29, y13-y23+y24+y30, y13-y23+y29+y40,y13+y24-y28-y7, y13+y25+y36-y5, y13+y31+y35-y4, y13+y32-y4+y40, y13+y35-y5+y6,y13+y36-y37-y4, y14-y15+y22-y25, y14-y16-y35+y7, y14-y17-y2+y34,y14-y2-y24-y9, y14-y2-y29-y39, y14-y20+y22-y9, y14-y20-y39+y7, y14-y21-y25-y38,y14-y21+y34-y6, y14+y22-y29-y5, y14-y24-y5+y7, y14-y25-y28-y36, y14-y26-y32-y39,y14-y26+y34-y36, y14-y26-y35-y38, y14-y28-y35-y6, y15-y16+y3-y33, y15-y17-y4+y40,y15-y20+y25-y9, y15-y20+y3-y37, y15-y21-y22-y38, y15-y21+y40-y8, y15-y22-y28-y36,y15-y23-y32-y37, y15-y23-y33-y38, y15-y23-y36+y40, y15+y25-y29-y5, y15-y27+y3-y5,y15-y27-y4-y9, y15-y28-y33-y8, y15-y29-y37-y4, y16-y17-y19+y30, y16-y19-y24-y37,y16-y19-y27-y39, y16-y20+y32-y38, y16-y20+y33-y37, y16-y20+y35-y39, y16-y23-y3-y38,y16-y23+y30-y6, y16-y24+y35-y5, y16-y26+y30-y8, y16-y26-y38-y7, y16-y27+y33-y5,y16-y28-y3-y8, y16-y28-y6-y7, y17+y19-y23-y6, y17+y19-y26-y8, y17+y2-y21-y6,y17+y2-y26-y36, y17-y21+y4-y8, y17-y23-y36+y4, y17-y24-y30-y37, y17-y24-y34-y9,y17-y27-y30-y39, y17-y27-y40-y9, y17-y29+y32-y36, y17-y29-y34-y39, y17-y29-y37-y40,y18-y19+y20-y21, y18-y2+y20-y23, y18-y2+y3+y32, y18+y20-y26-y4, y18-y21+y24+y33, y18-y21+y27+y35, y18+y22-y23+y24, y18-y23+y29+y7,y18+y24-y28-y40, y18+y25-y26+y27, y18-y26+y29+y3, y18+y27-y28-y34, y18-y28+y29-y30,y18+y32-y4+y7, y19-y2+y21-y23, y19-y2-y22+y33, y19-y2+y36-y8, y19-y20+y24+y33,y19-y20+y27+y35, y19+y21-y26-y4, y19-y25+y35-y4, y19-y31+y36-y5, y19+y33+y39-y5,y19+y35+y37-y5, y19+y36-y4-y6, y2-y20+y22+y24, y2-y20+y29+y7, y2+y22+y39-y5,y2+y23-y26-y4, y2-y26+y29-y32, y2-y3-y4+y7, y2-y31-y5+y8, y2-y4-y6+y8, y2-y5+y7+y9,y20-y23-y3-y32, y20-y24+y39-y5, y20-y25-y27-y4, y20-y26-y32-y7, y20-y29-y3-y4,y20-y29-y5+y9, y21+y22-y23-y33, y21-y23-y36+y8, y21+y25-y26-y35, y21-y26-y36+y6,y21-y28-y33-y40, y21-y28-y34-y35, y21-y28-y36+y38, y22-y23+y28+y40, y22+y24-y29-y7,y22-y30+y40-y7, y22+y31+y39-y8, y22-y33+y36-y8, y22+y39-y7-y9, y23-y26+y3-y7,y23-y26+y6-y8, y23-y28-y30-y7, y23-y28+y38-y8, y23-y29+y32-y4, y24-y27+y33-y35,y24-y27+y37-y39, y24-y27+y34-y40, y24-y29+y30-y40,y24-y29-y39+y9, y25-y26+y28+y34,y25+y27-y29-y3, y25-y3-y30+y34, y25-y3+y37-y9,y25+y31+y37-y6, y25+y34+y38-y6,y25-y35+y36-y6, y26-y28-y3-y30, y26-y28+y38-y6,y20-y27+y37-y5, y27-y29+y30-y34,y27-y29-y37+y9, y3+y30+y38-y6, y3+y31-y6+y9, y3+y4-y5+y9, y3-y6-y7+y8, y30-y31+y38-y9,y30-y34+y37-y9, y30+y38+y7-y8, y30+y39-y40-y9, y31-y32+y33-y34, y31-y32+y35-y40,y31+y33-y36+y39, y31-y34+y37-y38,y31+y35-y36+y37, y31-y38+y39-y40, y31-y4+y5-y6,y31+y7-y8+y9, y32-y33+y37-y38,y32+y34-y36+y39, y32-y35-y38+y39, y32-y36+y37+y40,y33-y34-y35+y40, y33-y35-y37+y39, y33-y36+y38+y40, y34+y35-y36+y38, y34-y37+y39-y40,y25+y37+y4-y5, y22+y38+y40-y8, y1-y4+y8-y9]);
[17]:
G = gens(L);
[18]:
size(G)
[18]:
(270,)
Tropicalization
Now we introduce the tropical semiring, the trivial valuation and a function that check whether a tropical polynomial vanishes at a point.
[19]:
T = TropicalSemiring();
val = TropicalSemiringMap(QQ);
[20]:
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
[20]:
vanishes (generic function with 1 method)
We first check that the \(F_1\) and \(F_2\) rays belong to the tropical Yoshida prevariety.
[21]:
for i in 1:36
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
[22]:
for i in 1:120
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
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.
[23]:
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 1:40)
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
[23]:
positive_vanishes (generic function with 1 method)
[24]:
for i in 1:36
check = [positive_vanishes(g,F1R[i,:]) for g in G]
checkmat= matrix(QQ, [check])
zvec = zero_matrix(QQ,1,270)
if checkmat==zvec
println(F1R[i,:])
end
end
[1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 1 0 0 1 0 1 1]
[0 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1]
[1 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 0 0 0 0 1 1 1 0 0 0 0 1]
[0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 1 1 0 0]
[1 0 0 1 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 1 1 0 0 1 0 0 0 0 1]
[0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 1 0 0 0 0 0 0]
[0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1]
[0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 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 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1]
Out of the 36 rays in \(\texttt{F1R}\) we see that 10 of them belong to the tropical positive Yoshida prevariety.
[25]:
for i in 1:120
check = [positive_vanishes(g,F2R[i,:]) for g in G]
checkmat= matrix(QQ, [check])
zvec = zero_matrix(QQ,1,270)
if checkmat==zvec
println(F2R[i,:])
end
end
[1 1 1 0 0 1 0 0 1 1 1 0 0 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1]
[1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 1 3 1 1 1 1 1 1 1 1 1]
[1 1 1 1 0 0 1 0 0 1 1 1 1 0 0 0 0 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1]
[1 1 1 1 0 0 1 0 0 1 1 1 1 0 0 0 0 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1]
[1 0 0 1 0 0 1 1 1 0 0 1 1 0 1 0 1 1 0 0 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 3]
[1 0 0 1 0 0 1 1 1 0 0 1 1 0 1 0 1 1 0 0 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 3]
[0 0 1 0 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 1 1 1 1 3 1 0 0 1 0 0 1 1 1 1]
[1 1 1 0 0 1 0 0 1 1 1 0 0 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1]
[1 1 1 1 0 0 1 0 0 1 1 1 1 0 0 0 0 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1]
[1 1 1 0 0 1 0 0 1 1 1 0 0 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1]
[1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 1 3 1 1 1 1 1 1 1 1 1]
[0 0 1 0 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 1 1 1 1 3 1 0 0 1 0 0 1 1 1 1]
[1 0 0 1 0 0 1 1 1 0 0 1 1 0 1 0 1 1 0 0 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 3]
[0 0 1 0 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 1 1 1 1 3 1 0 0 1 0 0 1 1 1 1]
[1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 1 3 1 1 1 1 1 1 1 1 1]
Out of the \(120\) rays in \(\texttt{F2R}\) we see \(5\) distinct rays in the tropical pisitive Yoshida prevariety: 1, 2, 3, 5, 7.
We collect them together we the previous \(10\).
[26]:
Rays = matrix(QQ, [
1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 1 0 0 1 0 1 1;
0 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1;
1 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 0 0 0 0 1 1 1 0 0 0 0 1;
0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 1 1 0 0;
1 0 0 1 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 1 1 0 0 1 0 0 0 0 1;
0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 1 0 0 0 0 0 0;
0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1;
0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 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 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 1 1 0 0 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1;
1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 1 3 1 1 1 1 1 1 1 1 1;
1 1 1 1 0 0 1 0 0 1 1 1 1 0 0 0 0 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1;
1 0 0 1 0 0 1 1 1 0 0 1 1 0 1 0 1 1 0 0 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 3;
0 0 1 0 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 1 1 1 1 3 1 0 0 1 0 0 1 1 1 1]);
We run another check to see that they are indeed positive.
[27]:
for i in 1:15
check = [positive_vanishes(g,Rays[i,:]) for g in G]
checkmat= matrix(QQ, [check])
zvec = zero_matrix(QQ,1,270)
if checkmat==zvec
println(Rays[i,:])
end
end
[1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 1 0 0 1 0 1 1]
[0 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1]
[1 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 0 0 0 0 1 1 1 0 0 0 0 1]
[0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 1 1 0 0]
[1 0 0 1 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 1 1 0 0 1 0 0 0 0 1]
[0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 1 0 0 0 0 0 0]
[0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1]
[0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 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 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 1 1 0 0 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1]
[1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 1 3 1 1 1 1 1 1 1 1 1]
[1 1 1 1 0 0 1 0 0 1 1 1 1 0 0 0 0 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1]
[1 0 0 1 0 0 1 1 1 0 0 1 1 0 1 0 1 1 0 0 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 3]
[0 0 1 0 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 1 1 1 1 3 1 0 0 1 0 0 1 1 1 1]
Now we take sums of two, three and four rays, respectively, and check which of these sums lie in the positive part. The computations confirm the f-vector (15,60,90,45).
[28]:
k=0
for i in 1:15
for j in i+1:15
R = Rays[i,:]+Rays[j,:]
check = [positive_vanishes(g,R) for g in G]
checkmat= matrix(QQ, [check])
zvec = zero_matrix(QQ,1,270)
if checkmat==zvec
println(R)
println([i,j])
k=k+1
end
end
end
println(k)
[2 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 2 0 0 1 0 0 1 0 0 1 1 0 0 1 1 2 2 1 0 0 0 0 1]
[1, 4]
[1 1 2 0 0 1 0 0 1 2 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 2 1 1 2 0 0 1 1 0 0]
[1, 5]
[2 1 1 1 0 0 1 0 0 1 1 1 1 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 2 2 1 1 1 0 0 0 0 1]
[1, 6]
[1 1 2 0 0 0 0 0 0 1 2 0 0 0 0 0 0 2 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 2 0 0 0 0 0 0]
[1, 7]
[1 1 1 0 0 0 0 0 0 2 2 1 1 0 0 0 0 2 1 0 0 1 0 0 1 0 0 0 0 1 2 1 1 1 0 0 0 0 0 0]
[1, 9]
[1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 1 1 1 1 1 1]
[1, 10]
[2 2 2 0 0 1 0 0 1 2 2 0 0 1 0 0 1 2 0 0 1 0 0 1 1 1 1 1 1 1 2 2 2 4 1 1 1 1 1 1]
[1, 11]
[2 2 2 1 1 1 1 1 1 2 2 1 1 0 0 0 0 2 1 0 0 1 0 0 1 0 0 0 0 1 4 2 2 2 1 1 1 1 1 1]
[1, 12]
[2 2 2 1 0 0 1 0 0 2 2 1 1 0 0 0 0 4 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 1 0 0 0 0 1]
[1, 13]
[0 0 0 0 0 0 1 1 2 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 1 0 1 2 1 0 0 1 0 0 1 1 2 2]
[2, 3]
[1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 2 0 0 2 1 1 1 0 0 1 2 1 0 1 0 1 2]
[2, 4]
[0 0 1 0 0 1 0 0 2 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 0 1 2 1 0 0 2 0 0 2 1 1 1]
[2, 5]
[0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 1 1 2 1 2 2 0 0 0 2 0 0 1 0 1 1]
[2, 7]
[0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 1 0 0 0 1 1 2 0 0 1 1 2 2 0 0 0 1 0 0 1 0 1 2]
[2, 8]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 1 1 2 1 1 2 1 2 2]
[2, 10]
[1 1 1 0 0 1 0 0 2 1 1 0 0 1 0 0 2 1 0 0 1 0 0 2 1 1 2 1 2 2 1 1 1 4 1 1 2 1 2 2]
[2, 11]
[1 0 0 1 0 0 1 1 2 0 0 1 1 0 1 0 2 1 0 0 1 1 1 2 0 0 2 1 2 2 1 1 1 2 1 1 2 1 2 4]
[2, 14]
[0 0 1 0 0 1 1 1 2 1 1 1 1 0 0 1 2 1 1 0 0 1 1 2 1 1 2 1 2 4 1 0 0 2 0 0 2 1 2 2]
[2, 15]
[0 0 1 0 0 1 1 1 2 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 2 2 0 0 1 0 0 1 2 1 1]
[3, 5]
[1 0 0 1 0 0 2 1 1 0 0 2 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 2 1 0 0 1 0 0 1 1 2]
[3, 6]
[0 0 0 0 0 0 2 1 1 0 0 1 1 0 0 0 0 1 0 0 0 2 1 1 0 0 0 1 1 2 1 0 0 0 0 0 0 1 1 2]
[3, 8]
[0 0 0 0 0 0 1 1 1 1 1 2 1 0 0 0 0 1 1 0 0 2 0 0 1 0 0 0 0 2 2 0 0 0 0 0 0 1 1 1]
[3, 9]
[0 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 2 1 1 1 1 1 1 2 2 2]
[3, 10]
[1 1 1 1 1 1 2 2 2 1 1 2 1 0 0 0 0 1 1 0 0 2 0 0 1 0 0 0 0 2 4 1 1 1 1 1 1 2 2 2]
[3, 12]
[1 0 0 1 0 0 2 2 2 0 0 2 1 0 1 0 1 1 0 0 1 2 1 1 0 0 1 1 1 2 2 1 1 1 1 1 1 2 2 4]
[3, 14]
[0 0 1 0 0 1 2 2 2 1 1 2 1 0 0 1 1 1 1 0 0 2 1 1 1 1 1 1 1 4 2 0 0 1 0 0 1 2 2 2]
[3, 15]
[2 0 0 1 0 0 1 0 0 0 0 1 1 0 0 0 0 2 0 0 1 0 0 1 0 0 1 1 0 0 1 1 1 1 2 0 0 0 0 2]
[4, 6]
[1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 1 0 0 1 1 1 2 2 1 1 0 0 1 2 1 0 0 0 0 1]
[4, 7]
[1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 2 0 0 1 1 1 2 0 0 1 2 1 1 0 0 1 1 1 0 0 0 0 2]
[4, 8]
[1 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 0 0 1 1 2 2 2 1 1 1 1 2]
[4, 10]
[2 1 1 0 0 1 0 0 1 1 1 0 0 1 0 0 1 2 0 0 2 0 0 2 1 1 2 2 1 1 1 1 2 4 2 1 1 1 1 2]
[4, 11]
[2 1 1 1 0 0 1 0 0 1 1 1 1 0 0 0 0 4 1 1 2 1 1 2 1 1 2 2 1 1 1 1 2 2 2 0 0 0 0 2]
[4, 13]
[2 0 0 1 0 0 1 1 1 0 0 1 1 0 1 0 1 2 0 0 2 1 1 2 0 0 2 2 1 1 1 1 2 2 2 1 1 1 1 4]
[4, 14]
[0 0 2 0 0 1 0 0 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 2 1 1 1 1 2 1 0 0 2 0 0 1 1 0 0]
[5, 7]
[0 0 1 0 0 1 0 0 1 2 1 1 1 0 0 0 0 1 1 0 0 1 0 0 2 0 0 0 0 2 2 0 0 1 0 0 1 1 0 0]
[5, 9]
[0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 2 1 1 2 1 1 2 2 1 1]
[5, 10]
[1 1 2 0 0 2 0 0 2 2 1 0 0 1 0 0 1 1 0 0 1 0 0 1 2 1 1 1 1 2 2 1 1 4 1 1 2 2 1 1]
[5, 11]
[1 1 2 1 1 2 1 1 2 2 1 1 1 0 0 0 0 1 1 0 0 1 0 0 2 0 0 0 0 2 4 1 1 2 1 1 2 2 1 1]
[5, 12]
[0 0 2 0 0 2 1 1 2 2 1 1 1 0 0 1 1 1 1 0 0 1 1 1 2 1 1 1 1 4 2 0 0 2 0 0 2 2 1 1]
[5, 15]
[1 0 0 1 0 0 2 0 0 0 0 1 2 0 0 0 0 2 0 0 0 1 1 1 0 0 0 1 1 1 1 1 0 0 1 0 0 0 0 2]
[6, 8]
[1 0 0 1 0 0 1 0 0 1 1 2 2 0 0 0 0 2 1 0 0 1 0 0 1 0 0 0 0 1 2 1 0 0 1 0 0 0 0 1]
[6, 9]
[1 0 0 1 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 2 2 1 1 2 1 1 1 1 2]
[6, 10]
[2 1 1 2 1 1 2 1 1 1 1 2 2 0 0 0 0 2 1 0 0 1 0 0 1 0 0 0 0 1 4 2 1 1 2 1 1 1 1 2]
[6, 12]
[2 1 1 2 0 0 2 0 0 1 1 2 2 0 0 0 0 4 1 1 1 1 1 1 1 1 1 1 1 1 2 2 1 1 2 0 0 0 0 2]
[6, 13]
[2 0 0 2 0 0 2 1 1 0 0 2 2 0 1 0 1 2 0 0 1 1 1 1 0 0 1 1 1 1 2 2 1 1 2 1 1 1 1 4]
[6, 14]
[0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 0 2 0 0 0 1 1 1 1 1 1 2 2 2 0 0 0 1 0 0 0 0 0 1]
[7, 8]
[0 0 1 0 0 0 0 0 0 1 2 1 1 0 0 0 0 2 1 0 0 1 0 0 2 1 1 1 1 2 1 0 0 1 0 0 0 0 0 0]
[7, 9]
[1 1 2 0 0 1 0 0 1 1 2 0 0 1 0 0 1 2 0 0 1 0 0 1 2 2 2 2 2 2 1 1 1 4 1 1 1 1 1 1]
[7, 11]
[1 1 2 1 0 0 1 0 0 1 2 1 1 0 0 0 0 4 1 1 1 1 1 1 2 2 2 2 2 2 1 1 1 2 1 0 0 0 0 1]
[7, 13]
[0 0 2 0 0 1 1 1 1 1 2 1 1 0 0 1 1 2 1 0 0 1 1 1 2 2 2 2 2 4 1 0 0 2 0 0 1 1 1 1]
[7, 15]
[0 0 0 0 0 0 1 0 0 1 1 1 2 0 0 0 0 2 1 0 0 2 1 1 1 0 0 1 1 2 1 0 0 0 0 0 0 0 0 1]
[8, 9]
[1 1 1 1 0 0 2 0 0 1 1 1 2 0 0 0 0 4 1 1 1 2 2 2 1 1 1 2 2 2 1 1 1 1 1 0 0 0 0 2]
[8, 13]
[1 0 0 1 0 0 2 1 1 0 0 1 2 0 1 0 1 2 0 0 1 2 2 2 0 0 1 2 2 2 1 1 1 1 1 1 1 1 1 4]
[8, 14]
[0 0 1 0 0 1 2 1 1 1 1 1 2 0 0 1 1 2 1 0 0 2 2 2 1 1 1 2 2 4 1 0 0 1 0 0 1 1 1 2]
[8, 15]
[1 1 1 1 1 1 1 1 1 2 2 2 2 0 0 0 0 2 2 0 0 2 0 0 2 0 0 0 0 2 4 1 1 1 1 1 1 1 1 1]
[9, 12]
[1 1 1 1 0 0 1 0 0 2 2 2 2 0 0 0 0 4 2 1 1 2 1 1 2 1 1 1 1 2 2 1 1 1 1 0 0 0 0 1]
[9, 13]
[0 0 1 0 0 1 1 1 1 2 2 2 2 0 0 1 1 2 2 0 0 2 1 1 2 1 1 1 1 4 2 0 0 1 0 0 1 1 1 1]
[9, 15]
[1 1 1 0 0 1 0 0 1 1 1 0 0 1 0 0 1 1 0 0 1 0 0 1 1 1 1 1 1 1 2 2 2 4 2 2 2 2 2 2]
[10, 11]
[1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 1 4 2 2 2 2 2 2 2 2 2]
[10, 12]
[1 0 0 1 0 0 1 1 1 0 0 1 1 0 1 0 1 1 0 0 1 1 1 1 0 0 1 1 1 1 2 2 2 2 2 2 2 2 2 4]
[10, 14]
60
[29]:
k=0
for i in 1:15
for j in (i+1):15
for l in (j+1):15
R = Rays[i,:]+Rays[j,:]+Rays[l,:]
check = [positive_vanishes(g,R) for g in G]
checkmat= matrix(QQ, [check])
zvec = zero_matrix(QQ,1,270)
if checkmat==zvec
println(R)
println([i,j,l])
k=k+1
end
end
end
end
println(k)
[3 1 1 1 0 0 1 0 0 1 1 1 1 0 0 0 0 3 0 0 1 0 0 1 0 0 1 1 0 0 2 2 2 2 2 0 0 0 0 2]
[1, 4, 6]
[2 1 2 0 0 0 0 0 0 1 2 0 0 0 0 0 0 3 0 0 1 0 0 1 1 1 2 2 1 1 1 1 2 3 1 0 0 0 0 1]
[1, 4, 7]
[2 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 2 0 0 1 0 0 1 0 0 1 1 0 0 2 2 3 3 2 1 1 1 1 2]
[1, 4, 10]
[3 2 2 0 0 1 0 0 1 2 2 0 0 1 0 0 1 3 0 0 2 0 0 2 1 1 2 2 1 1 2 2 3 5 2 1 1 1 1 2]
[1, 4, 11]
[3 2 2 1 0 0 1 0 0 2 2 1 1 0 0 0 0 5 1 1 2 1 1 2 1 1 2 2 1 1 2 2 3 3 2 0 0 0 0 2]
[1, 4, 13]
[1 1 3 0 0 1 0 0 1 2 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 1 1 1 1 2 2 1 1 3 0 0 1 1 0 0]
[1, 5, 7]
[1 1 2 0 0 1 0 0 1 3 2 1 1 0 0 0 0 2 1 0 0 1 0 0 2 0 0 0 0 2 3 1 1 2 0 0 1 1 0 0]
[1, 5, 9]
[1 1 2 0 0 1 0 0 1 2 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 3 2 2 3 1 1 2 2 1 1]
[1, 5, 10]
[2 2 3 0 0 2 0 0 2 3 2 0 0 1 0 0 1 2 0 0 1 0 0 1 2 1 1 1 1 2 3 2 2 5 1 1 2 2 1 1]
[1, 5, 11]
[2 2 3 1 1 2 1 1 2 3 2 1 1 0 0 0 0 2 1 0 0 1 0 0 2 0 0 0 0 2 5 2 2 3 1 1 2 2 1 1]
[1, 5, 12]
[2 1 1 1 0 0 1 0 0 2 2 2 2 0 0 0 0 3 1 0 0 1 0 0 1 0 0 0 0 1 3 2 1 1 1 0 0 0 0 1]
[1, 6, 9]
[2 1 1 1 0 0 1 0 0 1 1 1 1 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 3 3 2 2 2 1 1 1 1 2]
[1, 6, 10]
[3 2 2 2 1 1 2 1 1 2 2 2 2 0 0 0 0 3 1 0 0 1 0 0 1 0 0 0 0 1 5 3 2 2 2 1 1 1 1 2]
[1, 6, 12]
[3 2 2 2 0 0 2 0 0 2 2 2 2 0 0 0 0 5 1 1 1 1 1 1 1 1 1 1 1 1 3 3 2 2 2 0 0 0 0 2]
[1, 6, 13]
[1 1 2 0 0 0 0 0 0 2 3 1 1 0 0 0 0 3 1 0 0 1 0 0 2 1 1 1 1 2 2 1 1 2 0 0 0 0 0 0]
[1, 7, 9]
[2 2 3 0 0 1 0 0 1 2 3 0 0 1 0 0 1 3 0 0 1 0 0 1 2 2 2 2 2 2 2 2 2 5 1 1 1 1 1 1]
[1, 7, 11]
[2 2 3 1 0 0 1 0 0 2 3 1 1 0 0 0 0 5 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 3 1 0 0 0 0 1]
[1, 7, 13]
[2 2 2 1 1 1 1 1 1 3 3 2 2 0 0 0 0 3 2 0 0 2 0 0 2 0 0 0 0 2 5 2 2 2 1 1 1 1 1 1]
[1, 9, 12]
[2 2 2 1 0 0 1 0 0 3 3 2 2 0 0 0 0 5 2 1 1 2 1 1 2 1 1 1 1 2 3 2 2 2 1 0 0 0 0 1]
[1, 9, 13]
[2 2 2 0 0 1 0 0 1 2 2 0 0 1 0 0 1 2 0 0 1 0 0 1 1 1 1 1 1 1 3 3 3 5 2 2 2 2 2 2]
[1, 10, 11]
[2 2 2 1 1 1 1 1 1 2 2 1 1 0 0 0 0 2 1 0 0 1 0 0 1 0 0 0 0 1 5 3 3 3 2 2 2 2 2 2]
[1, 10, 12]
[0 0 1 0 0 1 1 1 3 1 0 1 0 0 0 0 1 0 0 0 0 1 0 1 1 0 1 0 1 3 2 0 0 2 0 0 2 2 2 2]
[2, 3, 5]
[0 0 0 0 0 0 2 1 2 0 0 1 1 0 0 0 1 1 0 0 0 2 1 2 0 0 1 1 2 3 1 0 0 1 0 0 1 1 2 3]
[2, 3, 8]
[0 0 0 0 0 0 1 1 2 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 1 0 1 2 2 1 1 2 1 1 2 2 3 3]
[2, 3, 10]
[1 0 0 1 0 0 2 2 3 0 0 2 1 0 1 0 2 1 0 0 1 2 1 2 0 0 2 1 2 3 2 1 1 2 1 1 2 2 3 5]
[2, 3, 14]
[0 0 1 0 0 1 2 2 3 1 1 2 1 0 0 1 2 1 1 0 0 2 1 2 1 1 2 1 2 5 2 0 0 2 0 0 2 2 3 3]
[2, 3, 15]
[1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 2 0 0 1 0 0 2 1 1 3 2 2 2 0 0 1 3 1 0 1 0 1 2]
[2, 4, 7]
[1 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 2 0 0 1 1 1 3 0 0 2 2 2 2 0 0 1 2 1 0 1 0 1 3]
[2, 4, 8]
[1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 2 0 0 2 1 1 1 1 1 2 3 2 1 2 1 2 3]
[2, 4, 10]
[2 1 1 0 0 1 0 0 2 1 1 0 0 1 0 0 2 2 0 0 2 0 0 3 1 1 3 2 2 2 1 1 2 5 2 1 2 1 2 3]
[2, 4, 11]
[2 0 0 1 0 0 1 1 2 0 0 1 1 0 1 0 2 2 0 0 2 1 1 3 0 0 3 2 2 2 1 1 2 3 2 1 2 1 2 5]
[2, 4, 14]
[0 0 2 0 0 1 0 0 2 1 1 0 0 0 0 0 1 1 0 0 0 0 0 1 2 1 2 1 2 3 1 0 0 3 0 0 2 1 1 1]
[2, 5, 7]
[0 0 1 0 0 1 0 0 2 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 0 1 2 2 1 1 3 1 1 3 2 2 2]
[2, 5, 10]
[1 1 2 0 0 2 0 0 3 2 1 0 0 1 0 0 2 1 0 0 1 0 0 2 2 1 2 1 2 3 2 1 1 5 1 1 3 2 2 2]
[2, 5, 11]
[0 0 2 0 0 2 1 1 3 2 1 1 1 0 0 1 2 1 1 0 0 1 1 2 2 1 2 1 2 5 2 0 0 3 0 0 3 2 2 2]
[2, 5, 15]
[0 0 1 0 0 0 1 0 1 0 1 0 1 0 0 0 1 2 0 0 0 1 1 2 1 1 2 2 3 3 0 0 0 2 0 0 1 0 1 2]
[2, 7, 8]
[1 1 2 0 0 1 0 0 2 1 2 0 0 1 0 0 2 2 0 0 1 0 0 2 2 2 3 2 3 3 1 1 1 5 1 1 2 1 2 2]
[2, 7, 11]
[0 0 2 0 0 1 1 1 2 1 2 1 1 0 0 1 2 2 1 0 0 1 1 2 2 2 3 2 3 5 1 0 0 3 0 0 2 1 2 2]
[2, 7, 15]
[1 0 0 1 0 0 2 1 2 0 0 1 2 0 1 0 2 2 0 0 1 2 2 3 0 0 2 2 3 3 1 1 1 2 1 1 2 1 2 5]
[2, 8, 14]
[0 0 1 0 0 1 2 1 2 1 1 1 2 0 0 1 2 2 1 0 0 2 2 3 1 1 2 2 3 5 1 0 0 2 0 0 2 1 2 3]
[2, 8, 15]
[1 1 1 0 0 1 0 0 2 1 1 0 0 1 0 0 2 1 0 0 1 0 0 2 1 1 2 1 2 2 2 2 2 5 2 2 3 2 3 3]
[2, 10, 11]
[1 0 0 1 0 0 1 1 2 0 0 1 1 0 1 0 2 1 0 0 1 1 1 2 0 0 2 1 2 2 2 2 2 3 2 2 3 2 3 5]
[2, 10, 14]
[0 0 1 0 0 1 1 1 2 2 1 2 1 0 0 0 0 1 1 0 0 2 0 0 2 0 0 0 0 3 3 0 0 1 0 0 1 2 1 1]
[3, 5, 9]
[0 0 1 0 0 1 1 1 2 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 2 3 1 1 2 1 1 2 3 2 2]
[3, 5, 10]
[1 1 2 1 1 2 2 2 3 2 1 2 1 0 0 0 0 1 1 0 0 2 0 0 2 0 0 0 0 3 5 1 1 2 1 1 2 3 2 2]
[3, 5, 12]
[0 0 2 0 0 2 2 2 3 2 1 2 1 0 0 1 1 1 1 0 0 2 1 1 2 1 1 1 1 5 3 0 0 2 0 0 2 3 2 2]
[3, 5, 15]
[1 0 0 1 0 0 3 1 1 0 0 2 2 0 0 0 0 2 0 0 0 2 1 1 0 0 0 1 1 2 2 1 0 0 1 0 0 1 1 3]
[3, 6, 8]
[1 0 0 1 0 0 2 1 1 1 1 3 2 0 0 0 0 2 1 0 0 2 0 0 1 0 0 0 0 2 3 1 0 0 1 0 0 1 1 2]
[3, 6, 9]
[1 0 0 1 0 0 2 1 1 0 0 2 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 3 2 1 1 2 1 1 2 2 3]
[3, 6, 10]
[2 1 1 2 1 1 3 2 2 1 1 3 2 0 0 0 0 2 1 0 0 2 0 0 1 0 0 0 0 2 5 2 1 1 2 1 1 2 2 3]
[3, 6, 12]
[2 0 0 2 0 0 3 2 2 0 0 3 2 0 1 0 1 2 0 0 1 2 1 1 0 0 1 1 1 2 3 2 1 1 2 1 1 2 2 5]
[3, 6, 14]
[0 0 0 0 0 0 2 1 1 1 1 2 2 0 0 0 0 2 1 0 0 3 1 1 1 0 0 1 1 3 2 0 0 0 0 0 0 1 1 2]
[3, 8, 9]
[1 0 0 1 0 0 3 2 2 0 0 2 2 0 1 0 1 2 0 0 1 3 2 2 0 0 1 2 2 3 2 1 1 1 1 1 1 2 2 5]
[3, 8, 14]
[0 0 1 0 0 1 3 2 2 1 1 2 2 0 0 1 1 2 1 0 0 3 2 2 1 1 1 2 2 5 2 0 0 1 0 0 1 2 2 3]
[3, 8, 15]
[1 1 1 1 1 1 2 2 2 2 2 3 2 0 0 0 0 2 2 0 0 3 0 0 2 0 0 0 0 3 5 1 1 1 1 1 1 2 2 2]
[3, 9, 12]
[0 0 1 0 0 1 2 2 2 2 2 3 2 0 0 1 1 2 2 0 0 3 1 1 2 1 1 1 1 5 3 0 0 1 0 0 1 2 2 2]
[3, 9, 15]
[1 1 1 1 1 1 2 2 2 1 1 2 1 0 0 0 0 1 1 0 0 2 0 0 1 0 0 0 0 2 5 2 2 2 2 2 2 3 3 3]
[3, 10, 12]
[1 0 0 1 0 0 2 2 2 0 0 2 1 0 1 0 1 1 0 0 1 2 1 1 0 0 1 1 1 2 3 2 2 2 2 2 2 3 3 5]
[3, 10, 14]
[2 0 0 1 0 0 2 0 0 0 0 1 2 0 0 0 0 3 0 0 1 1 1 2 0 0 1 2 1 1 1 1 1 1 2 0 0 0 0 3]
[4, 6, 8]
[2 0 0 1 0 0 1 0 0 0 0 1 1 0 0 0 0 2 0 0 1 0 0 1 0 0 1 1 0 0 2 2 2 2 3 1 1 1 1 3]
[4, 6, 10]
[3 1 1 2 0 0 2 0 0 1 1 2 2 0 0 0 0 5 1 1 2 1 1 2 1 1 2 2 1 1 2 2 2 2 3 0 0 0 0 3]
[4, 6, 13]
[3 0 0 2 0 0 2 1 1 0 0 2 2 0 1 0 1 3 0 0 2 1 1 2 0 0 2 2 1 1 2 2 2 2 3 1 1 1 1 5]
[4, 6, 14]
[1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 0 3 0 0 1 1 1 2 1 1 2 3 2 2 0 0 1 2 1 0 0 0 0 2]
[4, 7, 8]
[2 1 2 0 0 1 0 0 1 1 2 0 0 1 0 0 1 3 0 0 2 0 0 2 2 2 3 3 2 2 1 1 2 5 2 1 1 1 1 2]
[4, 7, 11]
[2 1 2 1 0 0 1 0 0 1 2 1 1 0 0 0 0 5 1 1 2 1 1 2 2 2 3 3 2 2 1 1 2 3 2 0 0 0 0 2]
[4, 7, 13]
[2 1 1 1 0 0 2 0 0 1 1 1 2 0 0 0 0 5 1 1 2 2 2 3 1 1 2 3 2 2 1 1 2 2 2 0 0 0 0 3]
[4, 8, 13]
[2 0 0 1 0 0 2 1 1 0 0 1 2 0 1 0 1 3 0 0 2 2 2 3 0 0 2 3 2 2 1 1 2 2 2 1 1 1 1 5]
[4, 8, 14]
[2 1 1 0 0 1 0 0 1 1 1 0 0 1 0 0 1 2 0 0 2 0 0 2 1 1 2 2 1 1 2 2 3 5 3 2 2 2 2 3]
[4, 10, 11]
[2 0 0 1 0 0 1 1 1 0 0 1 1 0 1 0 1 2 0 0 2 1 1 2 0 0 2 2 1 1 2 2 3 3 3 2 2 2 2 5]
[4, 10, 14]
[0 0 2 0 0 1 0 0 1 2 2 1 1 0 0 0 0 2 1 0 0 1 0 0 3 1 1 1 1 3 2 0 0 2 0 0 1 1 0 0]
[5, 7, 9]
[1 1 3 0 0 2 0 0 2 2 2 0 0 1 0 0 1 2 0 0 1 0 0 1 3 2 2 2 2 3 2 1 1 5 1 1 2 2 1 1]
[5, 7, 11]
[0 0 3 0 0 2 1 1 2 2 2 1 1 0 0 1 1 2 1 0 0 1 1 1 3 2 2 2 2 5 2 0 0 3 0 0 2 2 1 1]
[5, 7, 15]
[1 1 2 1 1 2 1 1 2 3 2 2 2 0 0 0 0 2 2 0 0 2 0 0 3 0 0 0 0 3 5 1 1 2 1 1 2 2 1 1]
[5, 9, 12]
[0 0 2 0 0 2 1 1 2 3 2 2 2 0 0 1 1 2 2 0 0 2 1 1 3 1 1 1 1 5 3 0 0 2 0 0 2 2 1 1]
[5, 9, 15]
[1 1 2 0 0 2 0 0 2 2 1 0 0 1 0 0 1 1 0 0 1 0 0 1 2 1 1 1 1 2 3 2 2 5 2 2 3 3 2 2]
[5, 10, 11]
[1 1 2 1 1 2 1 1 2 2 1 1 1 0 0 0 0 1 1 0 0 1 0 0 2 0 0 0 0 2 5 2 2 3 2 2 3 3 2 2]
[5, 10, 12]
[1 0 0 1 0 0 2 0 0 1 1 2 3 0 0 0 0 3 1 0 0 2 1 1 1 0 0 1 1 2 2 1 0 0 1 0 0 0 0 2]
[6, 8, 9]
[2 1 1 2 0 0 3 0 0 1 1 2 3 0 0 0 0 5 1 1 1 2 2 2 1 1 1 2 2 2 2 2 1 1 2 0 0 0 0 3]
[6, 8, 13]
[2 0 0 2 0 0 3 1 1 0 0 2 3 0 1 0 1 3 0 0 1 2 2 2 0 0 1 2 2 2 2 2 1 1 2 1 1 1 1 5]
[6, 8, 14]
[2 1 1 2 1 1 2 1 1 2 2 3 3 0 0 0 0 3 2 0 0 2 0 0 2 0 0 0 0 2 5 2 1 1 2 1 1 1 1 2]
[6, 9, 12]
[2 1 1 2 0 0 2 0 0 2 2 3 3 0 0 0 0 5 2 1 1 2 1 1 2 1 1 1 1 2 3 2 1 1 2 0 0 0 0 2]
[6, 9, 13]
[2 1 1 2 1 1 2 1 1 1 1 2 2 0 0 0 0 2 1 0 0 1 0 0 1 0 0 0 0 1 5 3 2 2 3 2 2 2 2 3]
[6, 10, 12]
[2 0 0 2 0 0 2 1 1 0 0 2 2 0 1 0 1 2 0 0 1 1 1 1 0 0 1 1 1 1 3 3 2 2 3 2 2 2 2 5]
[6, 10, 14]
[0 0 1 0 0 0 1 0 0 1 2 1 2 0 0 0 0 3 1 0 0 2 1 1 2 1 1 2 2 3 1 0 0 1 0 0 0 0 0 1]
[7, 8, 9]
[1 1 2 1 0 0 2 0 0 1 2 1 2 0 0 0 0 5 1 1 1 2 2 2 2 2 2 3 3 3 1 1 1 2 1 0 0 0 0 2]
[7, 8, 13]
[0 0 2 0 0 1 2 1 1 1 2 1 2 0 0 1 1 3 1 0 0 2 2 2 2 2 2 3 3 5 1 0 0 2 0 0 1 1 1 2]
[7, 8, 15]
[1 1 2 1 0 0 1 0 0 2 3 2 2 0 0 0 0 5 2 1 1 2 1 1 3 2 2 2 2 3 2 1 1 2 1 0 0 0 0 1]
[7, 9, 13]
[0 0 2 0 0 1 1 1 1 2 3 2 2 0 0 1 1 3 2 0 0 2 1 1 3 2 2 2 2 5 2 0 0 2 0 0 1 1 1 1]
[7, 9, 15]
[1 1 1 1 0 0 2 0 0 2 2 2 3 0 0 0 0 5 2 1 1 3 2 2 2 1 1 2 2 3 2 1 1 1 1 0 0 0 0 2]
[8, 9, 13]
[0 0 1 0 0 1 2 1 1 2 2 2 3 0 0 1 1 3 2 0 0 3 2 2 2 1 1 2 2 5 2 0 0 1 0 0 1 1 1 2]
[8, 9, 15]
90
[30]:
k=0
for i in 1:15
for j in i+1:15
for l in j+1:15
for m in l+1:15
R = Rays[i,:]+Rays[j,:]+Rays[l,:]+Rays[m,:]
check = [positive_vanishes(g,R) for g in G]
checkmat= matrix(QQ, [check])
zvec = zero_matrix(QQ,1,270)
if checkmat==zvec
println(R)
println([i,j,l,m])
k=k+1
end
end
end
end
end
println(k)
[3 1 1 1 0 0 1 0 0 1 1 1 1 0 0 0 0 3 0 0 1 0 0 1 0 0 1 1 0 0 3 3 3 3 3 1 1 1 1 3]
[1, 4, 6, 10]
[4 2 2 2 0 0 2 0 0 2 2 2 2 0 0 0 0 6 1 1 2 1 1 2 1 1 2 2 1 1 3 3 3 3 3 0 0 0 0 3]
[1, 4, 6, 13]
[3 2 3 0 0 1 0 0 1 2 3 0 0 1 0 0 1 4 0 0 2 0 0 2 2 2 3 3 2 2 2 2 3 6 2 1 1 1 1 2]
[1, 4, 7, 11]
[3 2 3 1 0 0 1 0 0 2 3 1 1 0 0 0 0 6 1 1 2 1 1 2 2 2 3 3 2 2 2 2 3 4 2 0 0 0 0 2]
[1, 4, 7, 13]
[3 2 2 0 0 1 0 0 1 2 2 0 0 1 0 0 1 3 0 0 2 0 0 2 1 1 2 2 1 1 3 3 4 6 3 2 2 2 2 3]
[1, 4, 10, 11]
[1 1 3 0 0 1 0 0 1 3 3 1 1 0 0 0 0 3 1 0 0 1 0 0 3 1 1 1 1 3 3 1 1 3 0 0 1 1 0 0]
[1, 5, 7, 9]
[2 2 4 0 0 2 0 0 2 3 3 0 0 1 0 0 1 3 0 0 1 0 0 1 3 2 2 2 2 3 3 2 2 6 1 1 2 2 1 1]
[1, 5, 7, 11]
[2 2 3 1 1 2 1 1 2 4 3 2 2 0 0 0 0 3 2 0 0 2 0 0 3 0 0 0 0 3 6 2 2 3 1 1 2 2 1 1]
[1, 5, 9, 12]
[2 2 3 0 0 2 0 0 2 3 2 0 0 1 0 0 1 2 0 0 1 0 0 1 2 1 1 1 1 2 4 3 3 6 2 2 3 3 2 2]
[1, 5, 10, 11]
[2 2 3 1 1 2 1 1 2 3 2 1 1 0 0 0 0 2 1 0 0 1 0 0 2 0 0 0 0 2 6 3 3 4 2 2 3 3 2 2]
[1, 5, 10, 12]
[3 2 2 2 1 1 2 1 1 3 3 3 3 0 0 0 0 4 2 0 0 2 0 0 2 0 0 0 0 2 6 3 2 2 2 1 1 1 1 2]
[1, 6, 9, 12]
[3 2 2 2 0 0 2 0 0 3 3 3 3 0 0 0 0 6 2 1 1 2 1 1 2 1 1 1 1 2 4 3 2 2 2 0 0 0 0 2]
[1, 6, 9, 13]
[3 2 2 2 1 1 2 1 1 2 2 2 2 0 0 0 0 3 1 0 0 1 0 0 1 0 0 0 0 1 6 4 3 3 3 2 2 2 2 3]
[1, 6, 10, 12]
[2 2 3 1 0 0 1 0 0 3 4 2 2 0 0 0 0 6 2 1 1 2 1 1 3 2 2 2 2 3 3 2 2 3 1 0 0 0 0 1]
[1, 7, 9, 13]
[0 0 1 0 0 1 1 1 3 1 0 1 0 0 0 0 1 0 0 0 0 1 0 1 1 0 1 0 1 3 3 1 1 3 1 1 3 3 3 3]
[2, 3, 5, 10]
[0 0 2 0 0 2 2 2 4 2 1 2 1 0 0 1 2 1 1 0 0 2 1 2 2 1 2 1 2 6 3 0 0 3 0 0 3 3 3 3]
[2, 3, 5, 15]
[1 0 0 1 0 0 3 2 3 0 0 2 2 0 1 0 2 2 0 0 1 3 2 3 0 0 2 2 3 4 2 1 1 2 1 1 2 2 3 6]
[2, 3, 8, 14]
[0 0 1 0 0 1 3 2 3 1 1 2 2 0 0 1 2 2 1 0 0 3 2 3 1 1 2 2 3 6 2 0 0 2 0 0 2 2 3 4]
[2, 3, 8, 15]
[1 0 0 1 0 0 2 2 3 0 0 2 1 0 1 0 2 1 0 0 1 2 1 2 0 0 2 1 2 3 3 2 2 3 2 2 3 3 4 6]
[2, 3, 10, 14]
[1 0 1 0 0 0 1 0 1 0 1 0 1 0 0 0 1 3 0 0 1 1 1 3 1 1 3 3 3 3 0 0 1 3 1 0 1 0 1 3]
[2, 4, 7, 8]
[2 1 2 0 0 1 0 0 2 1 2 0 0 1 0 0 2 3 0 0 2 0 0 3 2 2 4 3 3 3 1 1 2 6 2 1 2 1 2 3]
[2, 4, 7, 11]
[2 0 0 1 0 0 2 1 2 0 0 1 2 0 1 0 2 3 0 0 2 2 2 4 0 0 3 3 3 3 1 1 2 3 2 1 2 1 2 6]
[2, 4, 8, 14]
[2 1 1 0 0 1 0 0 2 1 1 0 0 1 0 0 2 2 0 0 2 0 0 3 1 1 3 2 2 2 2 2 3 6 3 2 3 2 3 4]
[2, 4, 10, 11]
[2 0 0 1 0 0 1 1 2 0 0 1 1 0 1 0 2 2 0 0 2 1 1 3 0 0 3 2 2 2 2 2 3 4 3 2 3 2 3 6]
[2, 4, 10, 14]
[1 1 3 0 0 2 0 0 3 2 2 0 0 1 0 0 2 2 0 0 1 0 0 2 3 2 3 2 3 4 2 1 1 6 1 1 3 2 2 2]
[2, 5, 7, 11]
[0 0 3 0 0 2 1 1 3 2 2 1 1 0 0 1 2 2 1 0 0 1 1 2 3 2 3 2 3 6 2 0 0 4 0 0 3 2 2 2]
[2, 5, 7, 15]
[1 1 2 0 0 2 0 0 3 2 1 0 0 1 0 0 2 1 0 0 1 0 0 2 2 1 2 1 2 3 3 2 2 6 2 2 4 3 3 3]
[2, 5, 10, 11]
[0 0 2 0 0 1 2 1 2 1 2 1 2 0 0 1 2 3 1 0 0 2 2 3 2 2 3 3 4 6 1 0 0 3 0 0 2 1 2 3]
[2, 7, 8, 15]
[1 1 2 1 1 2 2 2 3 3 2 3 2 0 0 0 0 2 2 0 0 3 0 0 3 0 0 0 0 4 6 1 1 2 1 1 2 3 2 2]
[3, 5, 9, 12]
[0 0 2 0 0 2 2 2 3 3 2 3 2 0 0 1 1 2 2 0 0 3 1 1 3 1 1 1 1 6 4 0 0 2 0 0 2 3 2 2]
[3, 5, 9, 15]
[1 1 2 1 1 2 2 2 3 2 1 2 1 0 0 0 0 1 1 0 0 2 0 0 2 0 0 0 0 3 6 2 2 3 2 2 3 4 3 3]
[3, 5, 10, 12]
[1 0 0 1 0 0 3 1 1 1 1 3 3 0 0 0 0 3 1 0 0 3 1 1 1 0 0 1 1 3 3 1 0 0 1 0 0 1 1 3]
[3, 6, 8, 9]
[2 0 0 2 0 0 4 2 2 0 0 3 3 0 1 0 1 3 0 0 1 3 2 2 0 0 1 2 2 3 3 2 1 1 2 1 1 2 2 6]
[3, 6, 8, 14]
[2 1 1 2 1 1 3 2 2 2 2 4 3 0 0 0 0 3 2 0 0 3 0 0 2 0 0 0 0 3 6 2 1 1 2 1 1 2 2 3]
[3, 6, 9, 12]
[2 1 1 2 1 1 3 2 2 1 1 3 2 0 0 0 0 2 1 0 0 2 0 0 1 0 0 0 0 2 6 3 2 2 3 2 2 3 3 4]
[3, 6, 10, 12]
[2 0 0 2 0 0 3 2 2 0 0 3 2 0 1 0 1 2 0 0 1 2 1 1 0 0 1 1 1 2 4 3 2 2 3 2 2 3 3 6]
[3, 6, 10, 14]
[0 0 1 0 0 1 3 2 2 2 2 3 3 0 0 1 1 3 2 0 0 4 2 2 2 1 1 2 2 6 3 0 0 1 0 0 1 2 2 3]
[3, 8, 9, 15]
[3 1 1 2 0 0 3 0 0 1 1 2 3 0 0 0 0 6 1 1 2 2 2 3 1 1 2 3 2 2 2 2 2 2 3 0 0 0 0 4]
[4, 6, 8, 13]
[3 0 0 2 0 0 3 1 1 0 0 2 3 0 1 0 1 4 0 0 2 2 2 3 0 0 2 3 2 2 2 2 2 2 3 1 1 1 1 6]
[4, 6, 8, 14]
[3 0 0 2 0 0 2 1 1 0 0 2 2 0 1 0 1 3 0 0 2 1 1 2 0 0 2 2 1 1 3 3 3 3 4 2 2 2 2 6]
[4, 6, 10, 14]
[2 1 2 1 0 0 2 0 0 1 2 1 2 0 0 0 0 6 1 1 2 2 2 3 2 2 3 4 3 3 1 1 2 3 2 0 0 0 0 3]
[4, 7, 8, 13]
[0 0 3 0 0 2 1 1 2 3 3 2 2 0 0 1 1 3 2 0 0 2 1 1 4 2 2 2 2 6 3 0 0 3 0 0 2 2 1 1]
[5, 7, 9, 15]
[2 1 1 2 0 0 3 0 0 2 2 3 4 0 0 0 0 6 2 1 1 3 2 2 2 1 1 2 2 3 3 2 1 1 2 0 0 0 0 3]
[6, 8, 9, 13]
[1 1 2 1 0 0 2 0 0 2 3 2 3 0 0 0 0 6 2 1 1 3 2 2 3 2 2 3 3 4 2 1 1 2 1 0 0 0 0 2]
[7, 8, 9, 13]
[0 0 2 0 0 1 2 1 1 2 3 2 3 0 0 1 1 4 2 0 0 3 2 2 3 2 2 3 3 6 2 0 0 2 0 0 1 1 1 2]
[7, 8, 9, 15]
45