For Jacobi polynomials with parameters greater than $-1/2$, and with the relative extrema enumerated from the endpoint $x=1$, the normalised modulus at the $k$th extremum of degree $n+1$ is proved to be strictly smaller than that at the $k$th extremum of degree $n$, for $1\leq k\leq n$. This proves the Szeg\H{o} conjecture, recorded in the 1975 fourth edition of his classic monograph Orthogonal Polynomials, and strengthens it by removing the ordering assumption on the parameters. The proof transforms the Jacobi equation to angular form and compares Pr\"ufer amplitudes at equal phase. Combined with a reduction of de Oliveira Filho and a separate quadratic-transformation argument for the boundary case, the result also settles a question concerning the Lov\'asz theta number of spherical distance graphs in every dimension at least four.
In this paper, we address the following question: if a flat torus $\mathbb{T}^n$ is isometrically and minimally embedded into a sphere $\mathbb{S}^N$, must its translation group extend to the isometry group of the ambient sphere? As shown by Robert Bryant, for $n=2$ the answer is positive. Furthermore, while Ying Lu, Peng Wang, and Zhenxiao Xie recently demonstrated that the answer is negative for immersions when $n \geq 3$, the question for embeddings remained open. This problem is deeply tied to the work of Mikhail Gromov and Anton Petrunin concerning optimal curvature bounds. Petrunin proved that any immersion of a torus into a unit ball must have a maximum normal curvature of at least $\sqrt{\frac{3n}{n+2}}$. This bound is attained, for example, by families of tori constructed by Gromov. We call the tori that attain this optimal bound"Gromov tori". In this work, we first demonstrate that any Gromov torus is intrinsically flat, lies within a sphere, and is minimal inside it. We then establish the necessary and sufficient conditions for defining these tori. Finally, we present our main result: for dimensions $n \ge 3$, there exists a non-equivariant embedded Gromov torus, which provides a definitive negative answer to the question above.
We settle two conjectures of Gautschi on the degree dependence of the zeros of the Jacobi polynomials $P_n^{(\alpha,\beta)}$, $\alpha,\beta>-1$, and obtain results substantially stronger than those conjectured. The conjectures stem from a line of questions originating in spherical cubature and hyperinterpolation. A Liouville transformation and Sturm comparison yield affine comparison principles with exact thresholds for the pointwise monotonicity of the rescaled potential. We prove that an increasing affine ordering with a degree-independent shift exists if and only if $|\beta|\leq1/2$. For the spectral scale $n+(\alpha+\beta+1)/2$, we determine the exact parameter regions for the two opposite orderings and show that no uniform spectral ordering is possible outside them. We also characterise all equality cases and derive finite-degree bounds in terms of Bessel zeros. The resulting classifications are exact and cannot be enlarged: outside the stated parameter regions the corresponding uniform zero orderings necessarily fail.
For positive integers $n$ and $M$, Sundaram defined $ F_{n,M}=\sum_{\substack{d\mid n\\d\le M}} Lie_{n/d}[p_d] $ and conjectured that every coefficient in its Schur expansion is a nonnegative integer. We prove the conjecture. The power-sum expansion isolates the contribution of the identity class, and the proof divides Young diagrams at the intrinsic majority threshold. If the first row or first column contains more than half of the boxes, the Bernstein creation formula separates each rectangular-cycle character value into a hook constant and a remainder supported only on short cycles. The restrictions of the trivial and sign characters to the cyclic subgroup evaluate the constants exactly, whereas a uniform binomial contraction controls every distance from the boundary at once. If neither a row nor a column has a majority, Swanson's opposite-hook estimate gives a Catalan-scale lower bound for the dimension. The Fomin--Lulov rectangular-character estimate, together with uniform bounds for the truncated M\"obius weights, then shows that the identity class dominates. Sundaram's plethystic identity consequently implies that $\prod_{r=1}^M(1-p_r)^{-1}$ is Schur-positive for every fixed $M$. We also record a degree-$12$ counterexample to a separate congruence-class product conjecture attributed to Richard Stanley.
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.
In this paper, we study a Riesz-type weighted sum of the Liouville function, $f(x):=-\sum_{n\le x}\frac{\lambda(n)\log n}{\sqrt n}\log\frac{x}{n}$ for sufficiently large $x$. Motivated by a recent research on the sign criteria for the Riemann Hypothesis arising from weighted prime-counting functions, we investigate the sign behavior of $f(x)$ and its relation to the zeros of the Riemann zeta function. We first prove that if $f(x)$ is non-negative for all sufficiently large $x$, then the Riemann Hypothesis holds. The proof is based on the Mellin transform of $f$ and the analytic properties of $\frac{\zeta(2s)}{\zeta(s)}$. Conversely, assuming the Riemann Hypothesis, the Simple Zero Conjecture, and an absolute convergence condition involving the nontrivial zeta zeros, we derive an explicit formula for $f(x)$ in terms of these zeros. In particular, we show that $f(x)\sim \frac{(\log x)^3}{12|\zeta(\frac{1}{2})|}$ as $x\to \infty$. Consequently, we show that under these hypotheses, $f(x)$ is eventually positive.
We show that the weak Markus--Yamabe conjecture fails in every dimension $n\geq14$. We first prove a chain realization theorem: every polynomial Keller map $F=\I+H$ of $\R^n$ with component degrees $d_1,\ldots,d_n$ yields an explicit polynomial vector field on $\R^N$, with $N=\sum_i\max(d_i,2)-n$, whose Jacobian matrix has spectrum $\{-1\}$ at every point and whose singularities are in bijection with any prescribed fiber of $F$. Applied to the recent counterexample to the Jacobian conjecture, this gives a Hurwitz vector field of degree seven on $\R^{14}$ with three rational singularities. A rank-reduced dehomogenization of the associated cubic stabilization gives, independently, a Hurwitz field of degree three on $\R^{18}$. The dimensions $3\leq n\leq13$ remain open.
Álvaro Castañeda, Gerardo Honorato, Francisco Valenzuela-Henríquez· 0 citations
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