# On tightness of degree bounds for sums of squares nonnegativity certificates for multivariate polynomials.

Dmitrii Pasechnik

Hilbert 17th conjecture (proved by Artin and Schrier using a
non-constructive method) states that any globally nonnegative
\$n\$-variate  degree \$d\$ polynomial \$f\$ can be expressed as a sum of
squares (s.o.s.) of rational functions, or, equavalently, as a ratio
\$p/q\$, where \$p\$ and \$q\$ are s.o.s. of polynomials.

The best known, recently obtained, general upper bound on degrees of \$p\$
and \$q\$ is elementary recursive, but still a height 5 tower of
exponents.  Much of the interest in such bounds stems from the fact that
they allow an efficient check of nonnegativity of \$f\$ using semidefinite
optimisation (SDP).

Not much is known for small values of \$n\$ and \$d>2\$. A
quadratic bound for \$n=2\$ has been obtained by Hilbert in 1893, and
slightly improved for \$d<12\$ in the past 10 years.

We show that for \$n=3\$ a nonnegative \$f\$ of degree 4 is
the ratio \$p/q\$ of s.o.s. of polynomials \$p\$, \$q\$ of degrees 8 and 4,
resp.  We conjecture that there are "special"  \$f=ax^2+2bx+c\$, where
\$a\$, \$b\$, \$c\$ in \$R[y,z]\$, for which the "8/4"- degree bound is sharp.
Towards this, we show that \$h:=Disc(f)=ac-b^2\$ cannot be s.o.s. in such
a case, and we construct such \$h\$---this is new and of independent
interest.  We discuss further progress towards settling the question
completely.