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

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.