Fixed harmonic parts: exact low-degree breakpoints and a deformation criterion

Definitions and mathematical statement

Fix integers $n,d\ge1$. Interpret the stated types $(d,d)$ and $(d-1,d-1)$ as bihomogeneous degrees. Let $S(z,\bar z)$ be a real-valued Hermitian bihomogeneous polynomial of degree $(d,d)$ on ${\mathbb{C}}^n$, harmonic in the sense \[ \sum_{i=1}^n\frac{\partial^2S}{\partial z_i\partial\bar z_i}=0. \] Put $s=\sum_i|z_i|^2$. For $k\in{\mathbb{Z}}_{\ge0}$, determine the number of Hermitian bihomogeneous polynomials $A$ of degree $(d-1,d-1)$ for which \[ S+sA=\sum_{j=1}^k|f_j(z)|^2 \] with holomorphic homogeneous degree-$d$ polynomials $f_j$. Here zero components are allowed, so the condition is intrinsic SOS rank at most $k$. If the intention is exactly $k$ linearly independent components, that is a different rank stratum and must be specified. Which properties of $S$ determine emptiness, finiteness, and infinitude? In particular determine the finiteness breakpoint \[ \kappa(S)=\min\{k:\text{infinitely many such }A\text{ exist}\}, \] with $+\infty$ if no such $k$ exists.

More explicitly, choose monomial column vectors $v_d,v_{d-1}$ of lengths \[ N=\binom{n+d-1}{d},\qquad q=\binom{n+d-2}{d-1}. \] Write $S=v_d^*C_Sv_d$ and $A=v_{d-1}^*Hv_{d-1}$, and let $L:\operatorname{Herm}_q\to\operatorname{Herm}_N$ be the injective real-linear coefficient map for multiplication by $s$. The set to count is the affine determinantal slice \[ \{H\in\operatorname{Herm}_q:C_S+L(H)\succeq0,\ \operatorname{rank}(C_S+L(H))\le k\}. \] This counts multipliers $A$, not their nonunique squared-norm representations.

The supplied aid reports two exact special cases. For $S=0$, Huang's lemma gives only $A=0$ below $k=n$; at and above $n$, the family $A=t|z_1|^{2(d-1)}$, $t\ge0$, gives infinitely many solutions, so $\kappa(0)=n$. For $d=1$, write $S=z^*Mz$ with $\operatorname{tr}M=0$. If the least eigenvalue $\lambda_{\min}$ has multiplicity $\ell$, the multipliers are real constants $a$, feasibility is $a\ge-\lambda_{\min}$, the endpoint has rank $n-\ell$, and every larger $a$ has rank $n$. Thus there are no solutions for $k<n-\ell$, one for $n-\ell\le k<n$, and infinitely many for $k\ge n$.

The aid also reports the following local deformation criterion. For feasible $C_0=C_S+L(H_0)\succeq0$ of rank $r$ and kernel $K$, a nonconstant two-sided affine line $C_0+tD$ of PSD matrices of the same rank exists for all sufficiently small positive and negative $t$ precisely when \[ \operatorname{im}L\cap\{D\in\operatorname{Herm}_N:Dv=0\text{ for every }v\in K\}\ne\{0\}. \] In particular $q^2+r^2>N^2$ forces such a line and rules out isolation at that point. This criterion concerns affine constant-rank deformations; its failure does not by itself prove isolation against all nonlinear or rank-changing deformations. The original general-$S$ remark contains a corrupted bound printed as $k<n$ followed by a separated $>2$; its intended uniqueness range cannot be recovered reliably from this record.

Short English statement

Count squared-norm completions with a fixed harmonic part by studying a positive semidefinite matrix slice with a rank bound. The supplied aid gives exact answers for zero harmonic part and degree one, plus a criterion for constant-rank affine deformations; a corrupted uniqueness bound in the source remains unresolved.

Sources

• P. Ebenfelt, Problem 3 and its remarks in the squared-norm portion of the AIM workshop Emerging applications of complexity for CR mappings, https://aimath.org/WWN/crmappings/crmappings.pdf. The remark attributes the zero-harmonic-part restriction to a lemma of Huang, without a full citation. The trailing "Plurisubharmonic polynomials" is a subsequent heading.
• Local UnsolvedMath registry, permanent ID 20002869, exact code catalogue entry; canonical location aim-several-complex-variables-notes.json, notes[26], dated 14 August 2026; statement labeled exact despite the extracted remark corruption.
• Research attempt 1: partial result, problem open at the run, rigor label proved, novelty confidence low; registry status partially solved. The matrix criterion and special-case classifications are supplied machine-generated research aids, not peer reviewed, requiring independent verification before citation.