sosdecTernary -- sum of squares decomposition for ternary forms

Synopsis

Usage:

(p,q) = sosdecTernary(f)
Inputs:
- f, a ring element, a homogeneous polynomial in 3 variables
Optional inputs:
- RoundTol => ..., default value 3, tolerance for rational rounding
- Solver => ..., default value "CSDP", picking a semidefinite programming solver
- Verbosity => ..., default value 0, control the level of information printed
Outputs:
- p, a list, of SOSPolys
- q, a list, of SOSPolys

Description

Given a nonnegative ternary form $f$, this method uses Hilbert's algorithm to compute a decomposition of $f$ as $f=\frac{\prod_ip_i}{\prod_iq_i}$ where each factor $p_i$ and $q_i$ is a sum of squares. The method returns null if $f$ is not nonnegative. As an example, consider the homogeneous Motzkin polynomial.

i1 : R = RR[x,y,z];

i2 : f = library ("Motzkin", {x,y,z});

i3 : (p,q) = sosdecTernary (f);
-- warning: experimental computation over inexact field begun
--          results not reliable (one warning given per session)

The result, in this case, is a quotient of two sums of squares.

i4 : (#p, #q)

o4 = (1, 1)

o4 : Sequence

i5 : clean(1e-4, f * value q#0 - value p#0)

o5 = 0

o5 : R

References: Products of positive forms, linear matrix inequalities, and Hilbert 17th problem for ternary forms, E. de Klerk, and D.V. Pasechnik, European J. Oper. Res. (2004), pp. 39-45.

Caveat

$\bullet$ This implementation only works with the solvers "CSDP" and "MOSEK".

$\bullet$ Due to the iterative nature of the algorithm, it could happen that some of the output sums of squares are rational while some are real.

Ways to use sosdecTernary :

sosdecTernary(RingElement)

For the programmer

The object sosdecTernary is a method function with options.