  Artin solved Hilbert's 17th problem, proving that a real polynomial in $n$
variables that is positive semidefinite is a sum of squares of rational
functions, and Pfister showed that only $2^n$ squares are needed. In this
paper, we investigate situations where Pfister's theorem may be improved. We
show that a real polynomial of degree $d$ in $n$ variables that is positive
semidefinite is a sum of $2^n-1$ squares of rational functions if $d\leq 2n-2$.
If $n$ is even, or equal to $3$ or $5$, this result also holds for $d=2n$.
