Description
This method contains a library of some interesting nonnegative forms.
The Motzkin polynomial is a ternary sextic that is nonnegative, but is not a sum of squares. It was the first such example found.
i1 : R = QQ[x,y,z];
|
i2 : library("Motzkin", R)
4 2 2 4 2 2 2 6
o2 = x y + x y - 3x y z + z
o2 : R
|
The Robinson and Schmüdgen polynomials are also ternary sextics that are not sums of squares.
i3 : library("Robinson", R)
6 4 2 2 4 6 4 2 2 2 2 4 2 2 4 2 4 6
o3 = x - x y - x y + y - x z + 3x y z - y z - x z - y z + z
o3 : R
|
i4 : library("Schmuedgen", R)
6 4 2 2 4 6 3 2 4 4 2 2 2 2
o4 = 199x - x y + 2x y + 200y - 4x y z + 4x*y z - 1588x z - 12x y z -
------------------------------------------------------------------------
4 2 3 3 2 3 2 4 2 4
1600y z + 16x z - 16x*y z + 3200x z + 3200y z
o4 : R
|
The Lax-Lax and Choi-Lam polynomials are quaternary quartics that are not sums of squares.
i5 : R = QQ[x,y,z,w];
|
i6 : library("Lax-Lax", R)
4 3 3 4 3 2 2 3 2 3 3
o6 = x - x y - x*y + y - x z + x y*z + x*y z - y z + x*y*z - x*z - y*z
------------------------------------------------------------------------
4 3 2 2 3 2 2 2
+ z - x w + x y*w + x*y w - y w + x z*w - 3x*y*z*w + y z*w + x*z w +
------------------------------------------------------------------------
2 3 2 2 2 3 3 3 4
y*z w - z w + x*y*w + x*z*w + y*z*w - x*w - y*w - z*w + w
o6 : R
|
i7 : library("Choi-Lam", R)
2 2 2 2 2 2 4
o7 = x y + x z + y z - 4x*y*z*w + w
o7 : R
|
The Scheiderer polynomial is a sum of squares over the reals, but not over the rationals.
i8 : R = QQ[x,y,z];
|
i9 : library("Scheiderer", R)
4 3 4 2 2 2 2 3 3 4
o9 = x + x*y + y - 3x y*z - 4x*y z + 2x z + x*z + y*z + z
o9 : R
|
The Harris polynomial is a ternary form of degree 10 with 30 projective zeros (the largest number known in August 2018).
i10 : library("Harris", R)
10 8 2 6 4 4 6 2 8 10 8 2 6 2 2
o10 = 16x - 36x y + 20x y + 20x y - 36x y + 16y - 36x z + 57x y z -
-----------------------------------------------------------------------
4 4 2 2 6 2 8 2 6 4 4 2 4 2 4 4 6 4
38x y z + 57x y z - 36y z + 20x z - 38x y z - 38x y z + 20y z +
-----------------------------------------------------------------------
4 6 2 2 6 4 6 2 8 2 8 10
20x z + 57x y z + 20y z - 36x z - 36y z + 16z
o10 : R
|
References: Some concrete aspects of Hilbert's 17th problem. B. Reznick. Contemporary mathematics (2000), 253, pp. 251-272.