library -- library of interesting nonnegative forms

Usage:

library(name,R)

library(name,var)
Inputs:
- name, a string, either "Motzkin", "Robinson", "Schmuedgen", "Lax-Lax", "Choi-Lam", "Scheiderer", "Harris"
- R, a ring, a polynomial ring
- var, a list, of variables
Outputs:
- a ring element, a nonnegative form

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.

Ways to use library:

library(String,List)
library(String,Ring)

For the programmer

The object library is a method function.

The source of this document is in SumsOfSquares/SOSdoc.m2:777:0.