Let $(X,Y)$ be independent standard real Gaussian random variables, and let $(P\in\mathbb C[X,Y])$ satisfy $\mathbb E!\left(P(X,Y)^m\right)=0$ for every $(m\ge1)$. Using the complex coordinates $Z=\frac{X+iY}{\sqrt2}, \qquad W=\frac{X-iY}{\sqrt2},$ we prove that the monomial support of $(P)$ is strictly one-sided with respect to the weight $\operatorname{wt}(Z^aW^b)=a-b$. Thus either every monomial occurring in $(P)$ has positive weight, or every monomial has negative weight. It follows that, for every $(Q\in\mathbb C[X,Y])$, $\mathbb E!\left(Q(X,Y)P(X,Y)^m\right)=0$ whenever $(m>\deg Q)$. This proves the two-variable Gaussian Moments Conjecture with the explicit threshold $m\ge\deg Q+1$. The main ingredient is a prime-isolation theorem for exposed faces of the Newton polygon of $(P)$. A $(p)$-adic valuation argument at moment indices of the form $(m=qp)$ separates the contribution of a chosen face from all remaining multinomial strata. Frobenius reduction then forces the constant terms of all positive powers of the associated one-variable Laurent polynomial to vanish. The theorem of Duistermaat and van der Kallen implies that the face has weights of one strict sign, while a planar convex-geometric argument rules out support containing weights of both signs. Combined with the known one-variable case and counterexamples in dimensions $(n\ge3)$, this determines the dimensions in which the Gaussian Moments Conjecture holds.
Let $X=\{X_j\}_{j=1}^\infty$ be a sequence of independent random variables whose densities and moments of order $2d$ are uniformly bounded. For a random vector $f(X)=(f_1(X),f_2(X))$ whose components are polynomial functionals of degree at most $d$, we prove that \[ [[f]]_{\mu,\infty}^{\frac1{2d-1}}\mu(f\in A) \le C\bigl(\lambda_2(A)\bigr)^{\frac1{2d-1}} \] for every Borel set $A\subset\mathbb R^2$, where $C$ depends only on $d$ and the uniform density and moment bounds, and $\lambda_2$ denotes the Lebesgue measure on $\mathbb R^2$. Here $[[f]]_{\mu,\infty}$ measures the failure of proportionality of the highest-order orthogonal-chaos components of $f_1$ and $f_2$ with respect to the law $\mu$ of $X$. Consequently, whenever these components are not proportional, the law of $f$ admits a density in the weak Lorentz space $L^{\frac{2d-1}{2d-2},\infty}(\mathbb R^2)$. This recovers the dichotomy established by Nualart and Tudor for two-dimensional Wiener chaos vectors and extends it beyond the Gaussian setting. We also obtain the lower bound \[ \int_{\mathbb R^\infty}\Delta_f\,d\mu \ge C[[f]]_{\mu,\infty}^2, \] where $\Delta_f$ is the determinant of the Gram matrix of $\nabla f_1$ and $\nabla f_2$. In the special case of Gaussian measures, this gives a relaxed version of the estimate conjectured by Nourdin, Nualart, and Poly.
We give explicit complex polynomials $P,Q$ in three independent standard real Gaussian variables such that \[ {\mathbb E}(P^m)=0,\qquad {\mathbb E}(QP^m)=m!\neq0 \] for every $m\geq1$. In natural complex linear coordinates, $P$ has five terms and total degree $4$. Hence the Gaussian Moments Conjecture is false in every dimension $n\geq3$. We also give a six-term cubic example in four variables, which was found first and already proves failure for every $n\geq4$. Both examples follow from the same coefficient identity. The search was prompted by Levent Alp\"oge's public announcement of an explicit three-dimensional counterexample to the Jacobian Conjecture. Although the main theorem of Derksen, van den Essen, and Zhao is stated globally in dimension, its proof has fixed-dimensional content: a noninvertible cubic-homogeneous Keller map in $r$ variables forces the failure of ${\mathrm GMC}(2r)$. Tracking a standard Bass--Connell--Wright reduction of the announced map gives a conservative cubic-homogeneous counterexample in $79$ variables, and hence a route-based failure of ${\mathrm GMC}(158)$. That route is nonconstructive at the final Gaussian step and does not furnish explicit polynomials $P,Q$. The much smaller explicit failures in dimensions $4$ and $3$ below were not derived from the announced Jacobian map.
Let $X_1,X_2,\ldots$ be independent $\mathrm{Bernoulli}(\theta)$ random variables, and let $\bar X_n = n^{-1}(X_1 + \cdots + X_n)$. We prove that, for every real $p \geq 1$, the sequence $\{\mathsf{E}(\bar X_n^p)\}_{n \geq 1}$ is log-convex. This proves the Bernoulli case of a conjecture of Lamkin and Tkocz [Canad. Math. Bull., 65(2):271-278, 2022]. The proof conditions on the total number of successes among $2n$ trials and reduces the desired inequality to a convex-order comparison between two normalized quadratic functions of hypergeometric random variables. All the log-convexity inequalities are strict for $p>1$ and $0<\theta<1$.
We prove the following theorem: let $\mathcal{M}_d$ denote the set of $d \times d$ matrices over an infinite field $K$, and let ${(K^d)^*}$ be the set of row vectors. Define an action of $\mathrm{SL}_d(K)$ on $X:= \mathcal{M}_d \oplus (K^d)^*$ by \[ g \cdot (A,\phi) = (gAg^{-1}, \phi g^{-1}).\] Then $K[X]^{\mathrm{SL}_d}$ is a polynomial ring, generated by the coefficients of the characteristic polynomial of $A$ and one further invariant, namely $\Delta(A,\phi):= \det(\phi,\phi A,\phi A^2,\ldots, \phi A^{d-1})^t.$ Our proof is entirely classical in nature, but we give an interpretation of the result and its proof in terms of quiver representation theory.
Let $R$ be an $m\times m$ correlation matrix satisfying $R-\mathbf{1}\mathbf{1}^{\mathsf T}/m\succeq0$, let $X\sim\mathcal{N}(0,R)$, and let $Z_1,\ldots,Z_m$ be independent standard Gaussian random variables. We prove $\max_i X_i\leq_{\mathrm{st}}\max_i Z_i$, with equality in distribution if and only if $R=I_m$. We use this comparison to resolve the Weak Simplex Conjecture: among $d+1$ equiprobable equal-energy signals in $\mathbb{R}^d$ transmitted over an additive white Gaussian noise channel, the regular simplex is the unique maximizer of the average probability of correct maximum-likelihood decoding at every signal-to-noise ratio. The same comparison proves the Simplex Mean Width Conjecture and gives the exact finite-energy performance of deterministic no-feedback AWGN codes with equiprobable messages, no restriction on the number of channel uses, and a maximal per-codeword energy constraint. The proof uses a Gaussian product inequality for log-concave functions whose first moments with respect to standard Gaussian measure vanish. A variational argument chooses one exponential tilt and one truncation endpoint in each coordinate so that this product inequality applies and a Gaussian change of measure returns all coordinates to the prescribed common threshold. A strict form of the product inequality also shows that, unless $R=I_m$, $\mathbb{P}\{X\leq c\mathbf{1}\}>\Phi(c)^m$ for every finite $c$, and hence gives the distributional equality statement. A Lean formalization is available at https://github.com/abhmul/weak-simplex-conjecture-lean.
Let $A=(a_{ij})$ be an $n\times n$ real-valued random matrix with independent, mean-zero, variance-one entries whose fourth moments are uniformly at most $K$. Suppose that there exists $\kappa \in (0, 1)$ such that the entries of $A$ satisfy $$ \max_{i,j}\sup_{u \in \mathbb{R}} \mathbb{P}(\lvert a_{ij} - u\rvert<1) \le \kappa. $$ We prove that there are constants $c,C>0$, depending only on $K$ and $\kappa$, such that for every fixed invertible $n\times n$ matrix $M$ and every $\varepsilon\ge0$, $$ \mathbb{P}!\left(s_{\min}(MA) \le \frac{\varepsilon}{\lVert M^{-1}\rVert_{\mathrm{HS}}}\right) \le C\varepsilon + e^{-cn}. $$ In the Gaussian case, we also show that the above estimate is sharp in the sense that $\mathbb{E}[s_{\min}(MA)]\asymp \lVert M^{-1}\rVert_{\mathrm{HS}}^{-1}.$