Affine hyperplane rank bounds and an explicit Veronese benchmark

Definitions and mathematical statement

For a real-valued polynomial $p(z,\overline z)$ on ${\mathbb{C}}^n$, its Hermitian rank is the smallest $N\geq0$ for which \[ p(z,\overline z)=\sum_{j=1}^N\varepsilon_j|p_j(z)|^2,\qquad \varepsilon_j\in\{1,-1\}, \] with holomorphic polynomials $p_j$; equivalently it is the rank of the Hermitian coefficient matrix in the monomial expansion. For real-analytic Hermitian functions, the analogous local analytic rank uses holomorphic germs, or the rank, possibly infinite, of the convergent Hermitian coefficient matrix. The original question moves from polynomial to real-analytic $p$ without explicitly specifying this extension or the global analytic conventions.

D. Grundmeier and J. Lebl ask: for every $n\geq2$ and nonnegative integer $k$, is there a finite constant $c(n,k)$ such that every real-analytic $p:{\mathbb{C}}^n\to{\mathbb{R}}$ satisfying \[ \operatorname{rank}(p|_H)\leq k\quad\text{for every affine complex hyperplane }H\subset{\mathbb{C}}^n \] satisfies $\operatorname{rank}(p)\leq c(n,k)$? An affine hyperplane has the form $H=\{z:a\cdot z=b\}$ with $a\ne0$. The word "affine" is the intended interpretation reported by the aid, and is essential. The original remark emphasizes that the question is worthwhile even when $p$ is a polynomial and a sum of squares, meaning all signs can be positive.

The aid attributes an affirmative result, without a positivity hypothesis, to Grundmeier--Lebl--Vivas: the rank in this intended affine real-analytic Hermitian setting is bounded by a function denoted $R_{n-1,n}(k)$. Its exact formula, theorem domains and local-to-global conventions are not supplied; this notation records the reported bound rather than an independently specified numerical function.

Its quantitative benchmark is fully explicit. For $d\geq0$ and a multi-index $\alpha\in\{0,1,\ldots\}^n$, put $z^\alpha=\prod_jz_j^{\alpha_j}$ and $|\alpha|=\sum_j\alpha_j$, and set \[ P_{n,d}(z,\overline z)=\sum_{|\alpha|\leq d}|z^\alpha|^2. \] The stated ranks are \[ \operatorname{rank}P_{n,d}=\binom{n+d}{d},\qquad \operatorname{rank}(P_{n,d}|_H)=\binom{n-1+d}{d} \quad\text{for every affine complex hyperplane }H. \] Consequently any admissible bound satisfies \[ c\left(n,\binom{n-1+d}{d}\right)\geq\binom{n+d}{d}, \qquad c(2,k)\geq\frac{k(k+1)}2\quad(k\geq1). \]

The aid also reports that replacing affine hyperplanes by only central hyperplanes makes the assertion false already for $n=2$, $k=1$. A concrete homogeneous formulation of that obstruction is \[ Q_d(z_1,z_2)=\sum_{j=0}^d|z_1^jz_2^{d-j}|^2,\qquad\operatorname{rank}Q_d=d+1. \] On each complex line through the origin, parametrized by $z=tv$ with $v\ne0$, its restriction is $Q_d(v)|t|^{2d}$, of rank $1$. This explains why the affine and central formulations cannot be interchanged.

Short English statement

Bound the Hermitian rank of a real-analytic function from uniform rank bounds on every affine complex hyperplane. The supplied aid reports an affirmative theorem without positivity and gives a polynomial lower-bound family; restricting only to hyperplanes through the origin changes the problem and permits unbounded rank.

Sources

• Local UnsolvedMath registry, permanent ID $20002867$, source code catalogue entry; canonical aim-several-complex-variables-notes.json, notes[24].
• AIM workshop Emerging applications of complexity for CR mappings, Problem 1, attributed to D. Grundmeier and J. Lebl, including its polynomial sum-of-squares remark: https://aimath.org/WWN/crmappings/crmappings.pdf. The original literature field is empty. The supplied verification reports normalization of OCR spacing and notation without mathematical change.
• The supplied research aid attributes the affine rank-bound theorem to Grundmeier, Lebl and Vivas, but provides no title, publication date or complete authoritative reference, and no formula for $R_{n-1,n}(k)$.
• Research attempt 1: partial_result, status solved, classification SOLVED-IN-LITERATURE, rigor proved, novelty confidence low. This machine-generated aid is not peer reviewed and requires independent verification before citation.