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# library -- library of interesting nonnegative forms

## Synopsis

• Usage:
library(name,R)
library(name,var)
• Inputs:
• name, , either "Motzkin", "Robinson", "Schmuedgen", "Lax-Lax", "Choi-Lam", "Scheiderer", "Harris"
• R, a ring, a polynomial ring
• var, a list, of variables
• Outputs:
• , 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 .