Conjecture~2 of Hinrichs and Vyb\'iral asserts that every continuous, nonnegative, positive definite function $f$ on $\R^d$, normalized by $f(0)=1$, satisfies $[f(x_j-x_k)]_{j,k=1}^n\succeq \one\one^T/n$. We give an explicit trigonometric polynomial on $\R^7$ and eight points for which this inequality fails. All hypotheses are verified directly: positive definiteness follows from nonnegative Fourier coefficients, and pointwise nonnegativity follows from interpolation on a cube. For an explicit vector of signs, the quadratic form is $22/5$, whereas the proposed lower bound is $9/2$. The resulting gap is exactly $-1/10$. The example is compatible with the established bound for functions of the form $|g|^2$ with $g$ positive definite.\\ The solution was found in a single prompt try by a colleague using his private licence of ChatGPT 6. Our aim is to make the solution public, as well as to collect some ideas that emerged during the analysis of the solution.
For $m\geq 2$, let $c_p(m)$ be the all-dimensional best constant in $$ \left\|\sum_{k=1}^m A_k\right\|_p \leq c_p(m)\left\|\sum_{k=1}^m |A_k|\right\|_p. $$ Tang and Zhang conjectured an explicit formula for every finite $p>1$. We disprove the conjecture with two explicit real $2\times 2$ rank-one matrices at $p=3/2$. T...
Escobar (J Funct Anal 165(1):101-116, 1999) conjectured that for every $n\ge 3$, an $n$-dimensional compact Riemannian manifold with nonnegative Ricci curvature and all boundary principal curvatures bounded below by $\kappa>0$ must satisfy $\sigma_1\geq \kappa$. We disprove this conjecture for every $n\geq 3$ by constr...
We construct a symmetric numerical semigroup $\Gamma$ of multiplicity $56$, embedding dimension $26$, and Frobenius number $181$ such that, over every field $k$, the one-dimensional Gorenstein local domain $R=k[t^\Gamma]_{\mathfrak{m}}$ has a nonprincipal ideal $I=(t^{56},t^{70})R$ for which $I\otimes_R\operatorname{Ho...
Harwit and Sloane conjectured that every nonsingular entrywise-nonnegative matrix $A\in\mathbb R^{n\times n}$ satisfies $\|A^{-1}\|_F\ge 2n(n+1)^{-1}\|A\|_{\max}^{-1}$, with equality precisely for positive multiples of $S$-matrices. Cheng proved the conjecture in odd dimensions, while Frankel and Urschel proved the eve...
For every $n\ge2$, we construct $4n-2$ nondegenerate Gaussian measures on $\mathbb{R}^{6n-3}$ that cannot be simultaneously equipartitioned by two affine hyperplanes. This disproves the Ramos conjecture for two hyperplanes. Combined with known upper bounds, the construction shows that $3\cdot2^{s-1}-2$ is the least dim...
In this paper, we settle a conjecture of Polujan and Pott by constructing an explicit, infinite family of Maiorana--McFarland bent functions $f_t$ in $2(2^t-1)$ variables with algebraic degree $\deg(f_t) = t + 1$ for any integer $t \ge 2$. Our construction builds upon a minimal commutative algebra $I_t$, which naturall...
Yan-Sheng Wu, Jia-Xin Wang, Jong Yoon Hyun· 0 citations
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