Description
Title: An efficient sum of squares certificate for 4ary 4icAbstract: Hilbert's 17th problem (resolved by Artin and Schreier) can be
formulated as the existence, for any globally nonnegative nary
homogeneous polynomial (a.k.a. form) f, of a sum of squares (s.o.s., for
short) form q so that fq is a sum of squares (thus f is nonnegative).
Note that once the degree of q is known, finding it can be done by
solving a semidefinite optimisation (SDP) feasibility problem.
For n=3, Hilbert has shown a quadratic, in the degree of f, bound on the
degree of q. In general, the best known degree bounds are huge. The next
interesting case is n=4, and f of degree 4. We show that in this case
there exists a product of two nonnegative quadrics q so that qf is an
s.o.s. of quartics. As a step towards deciding whether it is suﬃcient
to use a quadratic multiplier q, we show that there exist nons.o.s.
nonnegative 3ary sextics ac−b2, with a, b, c of degrees 2, 3, 4,
respectively (this gives a new class of nonnegative 3ary sextics which
are not s.o.s.).
Period  9 Mar 2020 

Visiting from  Pembrooke College, University of Oxford (United Kingdom) 
Visitor degree  PhD 
Degree of Recognition  Local 
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