Hilbert's seventeenth problem asks whether every polynomial with real coefficients
that is nonnegative on all real inputs is a sum of
squares of rational functions. Artin (1927)
answered the question affirmatively. Equivalently, for each such polynomial 
in a fixed finite number of variables, there is a nonzero polynomial 
such that 
is a sum of polynomial squares.
A dimension-uniform extension fails for stable symmetric functions. Write 
for the unnormalized power
sums and set
 |
Debus and Schabert (2026) proved that this homogeneous octic is positive on every nonzero finite real vector, but admits no nonzero
bounded-degree formal multiplier 
for which both 
and 
specialize to sums of polynomial squares in every dimension.
For any polynomial sum-of-squares multiplier of the specialization to 
variables, its polynomial
degree exceeds 
.
The obstruction persists for nonsymmetric multipliers and does not contradict Artin's
theorem for each fixed dimension. Debus and Schabert (2026) credit GPT-6 Astra with
initial proofs of their principal results, subsequently
verified and strengthened by the authors.
