math.AG — Algebraic Geometry

The Generalized Vanishing Conjecture: The Two-Variable Theorem and the First Failing Dimension

Contributed by Roy van Rijn

\documentclass[11pt]{article} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools,microtype} \usepackage[T1]{fontenc} \usepackage[hidelinks]{hyperref} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \newcommand{\N}{\mathbb N} \newcommand{\Q}{\mathbb Q} \newcommand{\Z}{\mathbb Z} \newcommand{\Newt}{\operatorname{Newt}} \newcommand{\Supp}{\operatorname{Supp}} \newcommand{\CT}{\operatorname{CT}} \newcommand{\wt}{\operatorname{wt}} \title{The Generalized Vanishing Conjecture:\\ The Two-Variable Theorem and the First Failing Dimension} \author{Roy van Rijn\\{\small Independent researcher}} \date{4 August 2026} \hypersetup{ pdftitle={The Generalized Vanishing Conjecture: The Two-Variable Theorem and the First Failing Dimension}, pdfauthor={Roy van Rijn}, pdfsubject={The exact dimensional boundary of the Generalized Vanishing Conjecture}, pdfkeywords={Generalized Vanishing Conjecture, constant-coefficient differential operators, Newton polygons, Hall's theorem, Gaussian moments, counterexample} } \begin{document} \maketitle \begin{abstract} We determine the exact dimensional range of the Generalized Vanishing Conjecture for constant-coefficient differential operators over a field of characteristic zero. In two variables we prove that \[ \Lambda^m(P^m)=0\quad(m\geq1) \qquad\Longrightarrow\qquad \Lambda^m(QP^m)=0\quad(m\gg0) \] for every fixed polynomial $Q$. The proof follows two moving faces. Hall's marriage theorem puts the lowest operator face strictly to the right of the highest polynomial face near ordinary degree. A prime-dilation argument shows that, while the two faces have different weights, their horizontal Newton intervals cannot overlap. Since the supports are finite, the two piecewise-linear weight envelopes must eventually meet. Their first meeting is an unequal-weight common threshold, where a fixed multiplier cannot repair the resulting linear derivative deficit. The only deep external theorem is the constant-term theorem of Duistermaat and van der Kallen. We then give a homogeneous three-variable counterexample. With $\rho=t^2+xy$, $A=\rho+x^2$, $C=y\rho^2-2xt^2\rho-x^3t^2$, and $\Delta=4\partial_x\partial_y+\partial_t^2$, the pair \[ \Lambda=\Delta^6,\qquad P=AC^2 \] satisfies $\Lambda^m(P^m)=0$ while $\Lambda^m(x^2P^m)\ne0$ for every $m\geq1$. The construction is a homogeneous endpoint-contact lift of Long's three-dimensional Gaussian counterexample. It is the smallest member of an explicit cusp-profile family with arbitrary winding, profile, and radial suspension; in particular every $\Delta^k$, $k\geq6$, has such a failure. Consequently unrestricted, and already homogeneous, GVC holds in dimension $n$ exactly when $n\leq2$. \end{abstract} \section{Introduction} For a polynomial $\lambda(X,Y)$, write \[ \Lambda=\lambda(\partial_x,\partial_y). \] The Generalized Vanishing Conjecture asks whether, for all polynomials $P,Q$ over a characteristic-zero field, \begin{equation}\label{eq:gvc} \Lambda^m(P^m)=0\quad\text{for every }m\geq1 \quad\Longrightarrow\quad \Lambda^m(QP^m)=0\quad\text{for all sufficiently large }m. \end{equation} It grew out of the Vanishing Conjecture and its relation to the Jacobian Conjecture; see Zhao~\cite{Zhao} and van den Essen, Willems, and Zhao~\cite{EWZ}. There were several particularly important precursors. De Bondt proved GVC for every product of linear differential forms, and therefore for every homogeneous operator in two variables~\cite{deBondt}. Feng and Sun proved the nonhomogeneous binary case \[ \Lambda=(\partial_x-\Phi(\partial_y))\partial_y \] for an arbitrary one-variable polynomial $\Phi$~\cite{FengSun}. Wilson's independent July 2026 preprint proves the two-variable Gaussian Moments Conjecture by a related Newton-face and prime-isolation method ~\cite{WilsonGMC}. Wilson treats the Gaussian functional rather than an arbitrary constant-coefficient operator. In the present argument, Hall localization and separation on shifted output rays are the additional steps that pass from that philosophy to unrestricted binary GVC. Also in 2026, Long found a five-term quartic in three real Gaussian variables for which all pure Gaussian moments vanish while one fixed mixed moment equals $m!$ at every power~\cite{LongGMC}. Long's example disproves the Gaussian Moments Conjecture in every dimension at least three, but it is not itself a constant-coefficient GVC counterexample. This paper closes the gap on both sides. First we prove \eqref{eq:gvc} for arbitrary, possibly nonhomogeneous, operators in two variables. We then lift the endpoint-coefficient mechanism behind Long's example to a homogeneous polynomial and a homogeneous differential operator in three variables. The two results meet at the exact boundary. \begin{theorem}[Binary Generalized Vanishing Conjecture]\label{thm:main} Let $k$ be a field of characteristic zero. Every constant-coefficient differential operator $\Lambda$ on $k[x,y]$ satisfies \eqref{eq:gvc}. \end{theorem} The global proof is organized as follows. Give both the operator variables and the polynomial variables weight $(s,1)$. Let $L(s)$ be the least operator weight and $U(s)$ the greatest polynomial weight. Hall's theorem orders the corresponding faces just to the right of $s=1$. Prime-dilated endpoint separation prevents the faces from passing one another while $U(s)-L(s)>0$. Finite support then forces this difference to reach zero. At the first zero all operator terms lie above one threshold and all polynomial terms lie below it, which gives eventual mixed vanishing. For orientation, the load-bearing dependencies in the binary argument are \[ \begin{array}{c} \text{Duistermaat--van der Kallen}+\text{split-symbol polarization} \longrightarrow \text{Proposition~\ref{prop:cutoff}},\\[2pt] \text{Duistermaat--van der Kallen}+\text{Hall localization} \longrightarrow \text{Lemma~\ref{lem:hall}},\\[2pt] \text{rational-point exposure}+\text{prime isolation} \longrightarrow \boxed{\text{Proposition~\ref{prop:shifted}}} \longrightarrow \text{Corollary~\ref{cor:horizontal}},\\[2pt] \text{Lemma~\ref{lem:hall}}+\text{Corollary~\ref{cor:horizontal}} \longrightarrow \text{first common threshold} \longrightarrow \text{Proposition~\ref{prop:threshold}} \longrightarrow \text{Theorem~\ref{thm:main}}. \end{array} \] Thus Proposition~\ref{prop:shifted} is the single arithmetic bridge from the local Hall gap to the global envelope argument. \section{Two standard reductions} We use the Newton polytope of a Laurent polynomial $F$, \[ \Newt(F)=\operatorname{conv}(\Supp(F)). \] The only substantial external input is the following theorem. \begin{theorem}[Duistermaat--van der Kallen~\cite{DK}]\label{thm:dk} Let $F$ be a Laurent polynomial over a characteristic-zero field. If $\CT(F^m)=0$ for every $m\geq1$, then $0\notin\Newt(F)$. \end{theorem} We will also use the following immediate consequence. \begin{lemma}[Exposing a rational point]\label{lem:expose} Let $F$ be a nonzero Laurent polynomial and let $\rho\in\Newt(F)\cap\Q^n$. Then there is an integer $a\geq1$ such that \[ [z^{a\rho}]F^a\ne0. \] For finitely many polynomials and rational points, the same $a$ may be chosen for all of them. \end{lemma} \begin{proof} Choose $N$ with $N\rho\in\Z^n$ and apply Theorem~\ref{thm:dk} contrapositively to $z^{-N\rho}F^N$. To use one exponent for several polynomials, place them in disjoint sets of Laurent variables, multiply them, and apply the same argument to the product. \end{proof} We next dispose of the range in which the degree of $P$ is no larger than the first positive order of $\Lambda$. This familiar split-symbol argument is included so that the main theorem has no hidden degree case. \begin{proposition}[Binary degree cutoff]\label{prop:cut