A reducible obstruction and the quadratic irreducible case

Definitions and mathematical statement

Question (B. Stensones). Let $p(z,\bar z,w,\bar w)$ be a real-valued plurisubharmonic homogeneous polynomial of degree $2k$ on ${\mathbb{C}}^2$. If an algebraic curve $\Sigma$ avoids zero and $p|_\Sigma=0$, must $\Sigma$ be one level set of a homogeneous holomorphic polynomial? The source does not specify irreducibility, ordinary versus bihomogeneity, positivity beyond plurisubharmonicity, or exclusion of pluriharmonic terms. The aid uses ordinary real homogeneity $p(tz,tw)=t^{2k}p(z,w)$, $t>0$, and affine algebraic curves. A nonempty level avoiding the origin must use nonzero level value and positive polynomial degree.

Reported reducible obstruction. Under those literal conventions take \[ p(z,w)=\operatorname{Im}(zw),\qquad \Sigma=\{zw=1\}\cup\{zw=2\}. \] The polynomial is nonzero, homogeneous quadratic and pluriharmonic, hence plurisubharmonic. The pure one-dimensional algebraic curve avoids the origin and lies in its zero set. The aid reports that it is not a single level $\{H=c\}$ for any positive-degree homogeneous holomorphic $H$. Constancy of $H$ at a nonzero value on $zw=1$ forces $H$ to be a scalar power of $zw$, whose nonzero values at $zw=1$ and $zw=2$ differ. This counterexample relies on allowing reducibility and pluriharmonic polynomials; it does not refute a strengthened version excluding them.

Reported irreducible quadratic theorem. If $p$ is any nonzero plurisubharmonic homogeneous quadratic and $\Sigma$ is an irreducible algebraic curve avoiding zero with $p|_\Sigma=0$, the aid reports \[ \Sigma=\{H=c\},\qquad c\ne0, \] for a homogeneous holomorphic polynomial $H$ of degree one or two. Nonzero $p$ and irreducibility are hypotheses of this special theorem. No analogous all-degree conclusion is supplied.

Reported higher-degree reduction. At every smooth point of $\Sigma$, its complex tangent is Levi-null for $p$, since the restriction is constant and the Levi form is nonnegative. If $\Sigma=\{F=0\}$ with irreducible $F\in{\mathbb{C}}[z,w]$, then $F(0)\ne0$. For the Euler field $E=z\partial_z+w\partial_w$, the aid gives the criterion \[ \Sigma\text{ is a homogeneous holomorphic level} \quad\Longleftrightarrow\quad EF\text{ is constant modulo }(F). \] Equivalently $EF-c\in(F)$ for some scalar. For $d=\deg F$, this condition forces $F=F_d+F(0)$ with homogeneous $F_d$, hence the nonzero level $F_d=-F(0)$. The aid does not prove that tangential Levi-nullity forces this Euler criterion for every higher-degree irreducible curve. Thus the missing general step, rather than the reducible counterexample, is the target of a clarified irreducible version. All supplied claims require independent verification.

Short English statement

The literal wording admits a reducible quadratic counterexample using a pluriharmonic polynomial. The aid reports a positive theorem for nonzero quadratic polynomials and irreducible curves. In higher degree it reduces the desired conclusion to an Euler-field condition, without proving that condition from plurisubharmonicity.

Sources

• Local UnsolvedMath registry, permanent ID $20002870$, catalogue entry, aim-several-complex-variables-notes.json, notes[27], workshop Emerging applications of complexity for CR mappings, "Plurisubharmonic polynomials", Problem 1, B. Stensones: https://aimath.org/WWN/crmappings/crmappings.pdf.
• Supplied research summary on the reducible obstruction, irreducible quadratic theorem, tangential Levi-nullity and Euler criterion. No independent authoritative theorem references are supplied.
• Attempt 1: partial_result, status at run "unknown", rigor "proved", low novelty confidence; registry status partially_solved. The machine-generated aid is not peer reviewed and requires independent verification before citation; no PDF visit is claimed.