An Elegant Analytical Resolution of the Sprott-Zeraoulia Conjecture for Three-Dimensional Quadratic Differential Systems with Symmetric Jacobian Matrices
The Sprott--Zeraoulia conjecture, formulated on the basis of numerical experiments, states that three-dimensional quadratic continuous-time systems with symmetric Jacobian matrices cannot exhibit chaotic behavior. We give an analytical resolution of this conjecture in terms of the principal precise notions of chaos used for continuous dynamical systems. The starting point is the classical, but decisive, fact that symmetry of the Jacobian on $\mathbb{R}^3$ is equivalent to a global gradient representation. Hence every system in the conjectured class can be written as $\dot{x}~=~\nabla V(x)$, where $V$ is a polynomial of degree at most three, and the conjecture becomes a problem about cubic polynomial gradient flows. For bounded trajectories, the monotonicity property of gradient systems first confines every omega-limit set to a critical level of $V$. When the equilibria on that level are isolated, convergence follows from the connectedness of the omega-limit set. In the general case, which may include curves or surfaces of equilibria, the \L{}ojasiewicz gradient inequality implies finite length and convergence of every bounded forward trajectory to a single equilibrium. Consequently, compact invariant sets support no nonstationary recurrence, positive-entropy measure, topological transitivity, Smale horseshoe, or Devaney, Auslander--Yorke, Li--Yorke, mean Li--Yorke, and distributional chaos. Solutions escaping to infinity are treated through the Poincar\'e compactification. Finally, we connect the gradient formulation with earlier Darboux-theoretic criteria: invariant algebraic surfaces satisfy $\langle\nabla V,\nabla f\rangle=Kf$, and an invariant affine plane with constant cofactor produces an exact tangential--normal decomposition of the potential and the flow.
We study one-dimensional Jacobi operators of divergence-gradient type on $\ell^2(\mathbb{Z}_{\geq 0})$, with coefficients generated by the doubling map, $a_n(x)=a(2^n x \mathrm{mod} 1)$. For continuous positive sampling functions, we show that the almost-sure essential spectrum is an interval containing the bottom of the spectrum. Under the assumption $a\in C^1(\mathbb{T})$, we prove square-root asymptotics for the integrated density of states and obtain a small-energy expansion for the Lyapunov exponent as $E\to0^+$. The leading term of the Lyapunov exponent is linear and positive precisely under a nondegeneracy condition on the sampling function. In this case, we prove a large-deviation estimate for the transfer matrices with an explicit rate depending on $E$. As consequences, we obtain local H\"older continuity of the Lyapunov exponent and the integrated density of states near the bottom of the spectrum, as well as the Anderson localization. Some of these results extend the small-coupling results of Chulaevsky--Spencer and Bourgain--Schlag for Schr\"odinger operators generated by the doubling map to divergence-gradient type operators. The arguments adapt methods from those works, but require uniform control in the small-energy parameter and refined correlation estimates reflecting the divergence-gradient structure.
We consider the classical analytic linearization problem for vector fields on the torus $\mathbb{T}^d$ close to a constant vector field $\omega$. Our goals are twofold. First, we provide a geometric framework in which the arithmetic condition governing analytic linearization arises naturally from the orbit of a unimodular lattice associated with $\omega$ under a diagonal flow on $\operatorname{SL}(d,\mathbb{Z})\backslash \operatorname{SL}(d,\mathbb{R})$. Within this framework, a summability condition emerges as the natural criterion for convergence. We prove that it is equivalent to several classical formulations of the Brjuno condition for linear forms, including those involving best approximation vectors and switching times of the diagonal flow. As a byproduct, we obtain a new quantitative linearization theorem with fully explicit estimates. In particular, the loss of analyticity of the conjugacy is controlled by a Brjuno function.
N. Chevallier, J. Dias, Jose Gaivao et al.· 0 citations
The oscillator (or metaplectic) semigroup is classically defined as the two-fold cover of the semigroup of positive complex symplectic matrices. Although geometrically precise, this definition does not provide an intrinsic operator-theoretic characterization of the bounded operators corresponding to positive symplectic matrices. This is in contrast with the real metaplectic group, whose relation with the symplectic group can be expressed directly in terms of the Schr\"odinger representation of the Heisenberg group through its intertwining property. In the complex setting, such a relation appears in the existing literature mainly in infinitesimal form or on suitable classes of Gaussian functions. In this work we prove that the (complexified) Schr\"odinger intertwining relation holds for every function in $L^2(\mathbb{R}^d)$. The main point is the converse statement: if an arbitrary complex symplectic matrix $S$ admits a nonzero bounded operator on $L^2(\mathbb{R}^d)$ satisfying this intertwining relation, then $S$ is necessarily positive and the operator coincides, up to a nonzero scalar, with the corresponding element of the oscillator semigroup. Consequently, positive complex symplectic matrices are exactly those admitting a nonzero bounded Schr\"odinger intertwiner, and the associated intertwining space is one-dimensional. This provides a Schr\"odinger-representation characterization of the oscillator semigroup, parallel to the classical one for the real metaplectic group.
We exhibit an explicit integer polynomial in five variables, of total degree $14$ and with constant Hessian determinant $128$, whose gradient is not injective. Consequently its formal Legendre transform is not a polynomial, and the Hessian conjecture $\HC_5$ is false. The counterexample is obtained from the six-variable doubling of Alp\"oge's 2026 Jacobian counterexample by a one-variable \emph{Schur descent}---a partial Legendre transform in a single variable. Combined with de~Bondt's theorem that $\HC_n$ holds for $n\le3$, with the elementary doubling and stabilization bridges relating the Jacobian conjectures $\JC_n$ to the Hessian conjectures $\HC_n$, and with Alp\"oge's refutation of $\JC_3$, this decides the Hessian conjecture in every dimension except $n=4$: $\HC_n$ is true for $n\le3$, false for $n\ge5$, and open only at $n=4$. Exactly two statements of the two families remain unsettled, $\JC_2$ and $\HC_4$, linked by $\HC_4 \Rightarrow \JC_2$. Along the way we record, as a warm-up, an explicit six-variable counterexample to $\HC_6$ with constant Hessian determinant $-4$ and non-injective gradient. This note adds the five-variable counterexample to, and updates the status recorded in, the first author's earlier educational preprint \cite{MengRG2026}.
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.
We show that, at a fake exponent of an $A$-hypergeometric system and for an ordered negative support family, intrinsic perturbation within $\ker_{\mathbb{Z}}(A)$ can produce a strictly smaller coefficient space than ambient perturbation, which answers a question of Okuyama--Saito. We measure the gap by an intrinsic-perturbation obstruction module, a quotient of two colon ideals whose graded dual is the ambient coefficient space modulo the intrinsic one, and we realize this module by two right-exact sequences involving $\operatorname{Tor}_1$ and present it finitely by two antichains. For a homogeneous system, we present as a finite-dimensional cokernel the quotient of the canonical formal solution space by the span of the canonical series obtained by intrinsic perturbation, taken over all exponents occurring in that space and all ordered negative support families, and we compute the codimension of that span; the presentation and the codimension transfer to the holomorphic solutions on a common nonsingular domain. We give configurations of lattice rank 1 and 2 with nonzero obstruction, one of which has an ordered distinguished collection and settles the case left open by Okuyama--Saito. We give, for every integer $q\geq 1$, a configuration of lattice rank 2 with a connected column matroid and with a normal affine semigroup generated by the reduced Gr\"obner basis vectors, whose obstruction module has dimension $q^2$ and for which the sums of the canonical series obtained by intrinsic perturbation span a subspace of the holomorphic solution space of codimension at least $\lceil 3q^2/4\rceil$, so this codimension is unbounded at fixed lattice rank.
