Literal and boundary counterexamples for Hermitian sums of squares

Definitions and mathematical statement

Problem 1 (John D'Angelo), with equality locus distinguished. Let $\Omega\subseteq{\mathbb{C}}^n$ be strongly pseudoconvex with compact algebraic boundary, and let $R(z,\bar z)$ be a real polynomial positive on $b\Omega$. Does it equal a finite Hermitian sum $\sum_j|p_j(z)|^2$ of holomorphic polynomials? The literal source says equality "on $\Omega$", yet cites a positive unit-ball result [CD]. The aid separates that open-domain wording from the likely intended boundary equality, rather than silently changing the question.

Reported literal obstruction. On the unit ball, for every $c>1$, \[ R(z,\bar z)=c-\|z\|^2 \] is positive even on the closed ball. Equality with a polynomial Hermitian sum on the open ball would be a polynomial identity everywhere. Its Hermitian coefficient matrix $\operatorname{diag}(c,-1,\ldots,-1)$ is not positive semidefinite; equivalently its negative Levi form is incompatible with a holomorphic sum of squares. Thus the literal statement is false even in the case described as affirmative by the source.

Reported negative answer to the boundary interpretation. The aid attributes to Putinar and Scheiderer (2010) smooth strongly pseudoconvex counterexamples and gives the following explicit family in ${\mathbb{C}}^2$: \[ G_\epsilon=\left\{(z,u): \tfrac12|z^2-1|^2+|u|^2+ \epsilon|z(z^2-1)|^2<1\right\}. \] For every sufficiently small $\epsilon>0$, it reports that this is a bounded strongly pseudoconvex domain with smooth compact algebraic boundary. The exact admissible epsilon threshold is not supplied. The fixed polynomial \[ R_*(z,u)=4-|z|^2-|u|^2 \] satisfies $R_*\geq2-\sqrt2>0$ on the closure, uniformly in that range.

The aid's boundary obstruction uses $a=(1,1)$, $b=(-1,1)$, which lie on $bG_\epsilon$. Let $r$ be the displayed defining polynomial minus one, and polarize by replacing conjugate coordinates with independent ones. Its polarization vanishes at all pairs from $\{a,b\}$. The boundary-ideal divisibility argument for this defining equation therefore forces any identity $R_*|_{bG_\epsilon}=\sum_j|p_j|^2|_{bG_\epsilon}$ to yield the Gram matrix \[ \left(R_*(x,\bar y)\right)_{x,y\in\{a,b\}} =\begin{pmatrix}2&4\\4&2\end{pmatrix}. \] Its determinant is $-12$, contradicting positive semidefiniteness. The divisibility/polarized-boundary step is essential: positivity of an arbitrary polarized matrix at arbitrary boundary pairs is not a general consequence of boundary equality without that step.

Thus the aid reports failure both of literal open-domain equality and of the intended general boundary version, while preserving the separate unit-ball boundary result cited as [CD]. The family and its ideal argument require primary-source verification; no rational-function, weighted-sum or other modified positivity theorem is inferred.

Short English statement

Open-domain equality is already false on the ball for $c-\|z\|^2$. The aid also reports a boundary counterexample on strongly pseudoconvex algebraic domains: one fixed positive polynomial forces a non-positive polarized Gram matrix. The intended boundary interpretation and the source's unit-ball citation are kept separate.

Sources

• Local UnsolvedMath registry, permanent ID $20002883$, catalogue entry, aim-several-complex-variables-notes.json, notes[40], Complexity of mappings in CR geometry, Problem 1, John D'Angelo: https://aimath.org/WWN/complexitycrmap/complexitycrmap.pdf.
• Original [CD] reference for the unit-ball assertion, without its full bibliography in the supplied record. Supplied research attribution: Putinar and Scheiderer (2010), negative boundary answer; no title/publication details are supplied.
• Supplied explicit $G_\epsilon,R_*$ sharpening: attempt 1, counterexample, status at run "refuted", rigor "proved", low novelty confidence; registry solved/SOLVED-IN-LITERATURE and reconstructed_unverified. The machine-generated aid is not peer reviewed and requires independent verification before citation. The supplied verification reports PDF/bibliography inspection; this draft has not visited it.