Recall here that diagonal entries are set to 0 and ignored in generating the polytrope.
Case 1: [[0, 15, 16], [9, 0, 8], [14, 18, 0]]
Matrix whose columns correspond to the different h_{P, w}^{*} coefficients, one for each monomial weight w, in this case, x_1 and x_2.
[ -153 -2269]
[ -592 -8408]
[ -143 -1947]
Kernel (the set of vectors orthogonal to all h_{P, w}^{*}):
Free module of degree 3 and rank 1 over Integer Ring
Echelon basis matrix:
[ 6215 -3322 7103]
This means that the set of all h_{P, w}^{*}(z) = c_1 * z + c_2 * z^2 + c_3 * z^3 for P above are on the hyperplane 6215 * c_1 -3322 * c_2 + 7103 * c_3 = 0, implying the existence of all sign patterns of the coefficients of h_{P, w}^{*}(z) EXCEPT for the alternating sign pattern. Extensive search finds a case where the alternating sign pattern is possible:
Case 2: [[0, 14, 17], [3, 0, 6], [-3, 10, 0]]
Matrix whose columns correspond to the different h_{P, w}^{*} coefficients, one for each monomial weight w, in this case, x_1 and x_2.
[1367 36]
[4663 123]
[ 988 26]
Kernel (the set of vectors orthogonal to all h_{P, w}^{*}):
Free module of degree 3 and rank 1 over Integer Ring
Echelon basis matrix:
[ 22 -2 -21]
which implies the existence of a weighted h* polynomial h_{P, w}^{*} whose coefficients are alternating.
For the following cases, we list out two cases of polytopes: one whose set of all h_{P, w}^{*} is orthogonal to a vector with an alternating sign pattern, and one whose set of all h_{P, w}^{*} is orthogonal to a vector NOT of an alternating sign pattern. These are enough to show the existence of all sign patterns.
Dimension 2, Degree 2
Case 1: [[0, 12, 18], [-8, 0, 10], [-10, -2, 0]]
Matrix whose columns correspond to the different h_{P, w}^{*} coefficients, one for each monomial weight w.
[ 7900 2815 1100]
[66028 23263 8828]
[52484 18325 6804]
[ 3380 1165 420]
Kernel (the set of vectors orthogonal to all h_{P, w}^{*}):
Free module of degree 4 and rank 1 over Integer Ring
Echelon basis matrix:
[ 77 -35 45 -195]
Case 2: [[0, 16, 11], [-5, 0, -3], [-2, 11, 0]]
Matrix whose columns correspond to the different h_{P, w}^{*} coefficients, one for each monomial weight w.
[ 2669 -2289 3134]
[ 24048 -21458 28506]
[ 20589 -18884 24574]
[ 1510 -1433 1818]
Kernel (the set of vectors orthogonal to all h_{P, w}^{*}):
Free module of degree 4 and rank 1 over Integer Ring
Echelon basis matrix:
[ 198677 245405 -584827 3714725]
Dimension 2, Degree 3
Case 1: [[0, 14, 5], [-12, 0, -8], [-1, 11, 0]]
Matrix whose columns correspond to the different h_{P, w}^{*} coefficients, one for each monomial weight w.
[ 540 -1222 3174 -10204]
[ 8785 -20986 55994 -173716]
[ 16049 -39545 106715 -322937]
[ 4439 -11201 30341 -89308]
[ 91 -234 624 -1729]
Kernel (the set of vectors orthogonal to all h_{P, w}^{*}):
Free module of degree 5 and rank 1 over Integer Ring
Echelon basis matrix:
[ 14239901 -1267864 -782782 4734236 -54986446]
Case 2: [[0, 3, 7], [11, 0, 17], [0, -4, 0]]
Matrix whose columns correspond to the different h_{P, w}^{*} coefficients, one for each monomial weight w.
[ 10605 18872 40586 109172]
[ 221747 407218 893424 2406985]
[ 487260 914601 2034621 5489184]
[ 165201 317458 716808 1937323]
[ 4921 9863 22869 62042]
Kernel (the set of vectors orthogonal to all h_{P, w}^{*}):
Free module of degree 5 and rank 1 over Integer Ring
Echelon basis matrix:
[ 2148065793 401859318 -750980867 1940222004 -13512500295]
Dimension 2, Degree 4
Case 1: [[0, 14, 15], [-10, 0, 5], [-9, 1, 0]]
Matrix whose columns correspond to the different h_{P, w}^{*} coefficients, one for each monomial weight w.
[ 741438 90777 22077 5307 1454]
[ 32814420 3804153 838665 188499 47812]
[149521236 16764296 3469722 746084 179652]
[130230136 14157968 2758930 568364 130152]
[ 20746894 2171015 391545 76265 16366]
[ 257044 24783 3733 633 116]
Kernel (the set of vectors orthogonal to all h_{P, w}^{*}):
Free module of degree 6 and rank 1 over Integer Ring
Echelon basis matrix:
[ 1463 -385 231 -273 663 -4641]
Case 2: [[0, 17, 7], [-5, 0, -4], [-1, 13, 0]]
Matrix whose columns correspond to the different h_{P, w}^{*} coefficients, one for each monomial weight w.
[ 36340 -47267 75941 -159533 482415]
[ 1644925 -2240216 3714544 -7881530 22975779]
[ 7638196 -10688423 18049785 -38501261 109981944]
[ 6794528 -9742738 16723726 -35831200 100580262]
[ 1116260 -1648300 2885108 -6212266 17091189]
[ 14975 -23452 42660 -92686 245667]
Kernel (the set of vectors orthogonal to all h_{P, w}^{*}):
Free module of degree 6 and rank 1 over Integer Ring
Echelon basis matrix:
[ 1639434890017 642221712235 -626466698675 898597295245 -2409726629339 16923457899883]
Dimension 2, Degree 5
Case 1: [[0, 9, 18], [13, 0, 9], [4, -3, 0]]
Matrix whose columns correspond to the different h_{P, w}^{*} coefficients, one for each monomial weight w.
[ 20235244 10479382 5726278 3489670 2172754 2669428]
[ 1983284006 1031744019 565995347 345001875 216623441 263658942]
[17739028706 9250414737 5084613005 3099170547 1953529271 2365564962]
[33081603756 17285222230 9515866398 5799410434 3666235626 4420618132]
[14964861856 7835139888 4320130192 2632380600 1668708016 2003099472]
[ 1348111342 707730807 391002319 238166235 151473637 180742974]
[ 8818130 4657225 2583773 1572127 1006055 1184170]
Kernel (the set of vectors orthogonal to all h_{P, w}^{*}):
Free module of degree 7 and rank 1 over Integer Ring
Echelon basis matrix:
[ 623645 -124729 55913 -46189 67925 -191425 1416545]
Case 2: [[0, 10, 0], [12, 0, -4], [19, 10, 0]]
Matrix whose columns correspond to the different h_{P, w}^{*} coefficients, one for each monomial weight w.
[ -51305810 -24336951 -12547237 -7156893 -4733077 -4393882]
[ -5288003170 -2520921573 -1299942597 -739123791 -486386877 -442472369]
[-48600611422 -23225520909 -11976104859 -6796741863 -4459120431 -4012889078]
[-92706344862 -44386199159 -22885039387 -12966792545 -8484216433 -7567501098]
[-42940662092 -20596785234 -10617457518 -6005274774 -3917614710 -3462197188]
[ -3992959712 -1919521248 -989121228 -558065754 -362537946 -316499641]
[ -28087012 -13559734 -6977878 -3913216 -2516206 -2137424]
Kernel (the set of vectors orthogonal to all h_{P, w}^{*}):
Free module of degree 7 and rank 1 over Integer Ring
Echelon basis matrix:
[ 2191227205520746365754 -197225569917331128695 42426877509433888336 -10692173333807458241 -11803340790359655539 100037598732416149969 -1169079858339276966665]
Dimension 3, Degree 1
Case 2: [[0, 13, -6, 11], [6, 0, -9, 3], [15, 19, 0, 17], [9, 14, -5, 0]]
Matrix whose columns correspond to the different h_{P, w}^{*} coefficients, one for each monomial weight w.
[ 50 -4991 13325]
[ 493 -44232 119779]
[ 451 -37401 102464]
[ 36 -2696 7512]
Kernel (the set of vectors orthogonal to all h_{P, w}^{*}):
Free module of degree 4 and rank 1 over Integer Ring
Echelon basis matrix:
[ 3814246 -1262114 1233726 -3469459]
Case 2: [[0, -9, -3, -2], [16, 0, 6, 7], [14, 5, 0, 5], [9, 0, 2, 0]]
Matrix whose columns correspond to the different h_{P, w}^{*} coefficients, one for each monomial weight w.
[ -1665 1062 489]
[-12385 7990 3585]
[ -9015 5874 2577]
[ -514 340 144]
Kernel (the set of vectors orthogonal to all h_{P, w}^{*}):
Free module of degree 4 and rank 1 over Integer Ring
Echelon basis matrix:
[ 70 -7 -15 205]