Let $X\subset{\mathbb P}^{n-1}$ be a hypersurface of degree $d\ge3$ with ordinary double points, where $n\ge3$. The roots of Bernstein-Sato polynomial of its defining polynomial $f$ are given up to sign by 1, $(n-1)/2$, and $j/d$ for $j\in{\mathbb Z}\cap[n,nd-n-p_f]$ with $p_f$ a positive integer. Here $p_f$ is bounded above by the minimal positive integer $q_s$ satisfying $\binom{q_s+n-1}{n-1}>s:=|{\rm Sing}\,X|$, and we can verify that $p_f$ coincides with $q_s$ in the case the singular points of $X$ are in ``general position". We show that this upper bound is sharp in the case $\binom{\lfloor d/2\rfloor+n-2}{n-1}\ge s$ or $\binom{d+n-3}{n-1}\ge sn$ by providing a homogeneous polynomial of degree $d$ such that the associated projective hypersurface has ordinary double points at given $s$ points in sufficiently general position and is nonsingular outside them (using a theorem of Alexander and Hirschowitz for the second case). It is conjectured that the above sharp bound under the first hypothesis can be extended naturally to the case where $X$ has only $A_2$-singularities instead of ordinary double points.
We study the cone $\mathcal{M}_{n,2d}$ of nonnegative mean polynomials---real $n$-variate forms of degree $2d$ that can be expressed as weighted power means $M_{q,p}(Y,w)$ with $q>p$. This cone simultaneously generalises the cone of sums of squares $\Sigma_{n,2d}$ and the cone of sums of nonnegative circuit polynomials $\mathcal{C}_{n,2d}$. We prove that every square of an arbitrary polynomial belongs to the mean polynomial preprime $T_{\mathrm{mean}}$, that $T_{\mathrm{mean}}$ is strongly generating, and consequently that every polynomial strictly positive on a compact semialgebraic set admits a representation with mean polynomial certificates. We exhibit the Robinson form $\hat{R}$ as a separating example that lies in $\mathcal{M}_{4,4}$ but outside $\mathrm{SOSONC}_{4,4}$. Finally, we outline a convergent hierarchy of lower bounds for polynomial optimization based on the mean polynomial cone and discuss tractable depth-truncated approximations via signomial programming.
Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb F_q$, and let $\mathcal F\subsetneq\genfrac{[}{]}{0pt}{}{V}{k}$. The \emph{projective Ore-degree} of $\mathcal F$ is the minimum, over all $k$-subspaces $S\notin\mathcal F$, of the sum of the $\mathcal F$-degrees of the projective points contained in $S$. We prove sharp projective Ore analogues of the vector-space Erd\H{o}s--Ko--Rado and Hilton--Milner theorems. The Ore--Erd\H{o}s--Ko--Rado theorem holds for $n\ge2k+1$, with equality only for a full point-star. For nontrivial intersecting families, we determine the sharp Ore--Hilton--Milner threshold, together with the complete equality classification, when $q\ge3$ and $n\ge2k+1$, or when $q\ge2$ and $n\ge2k+2$. We further determine a sharp projective Ore-degree threshold forcing a direct-sum matching of size $s$ when $s\ge3$ and $n\ge(2s-1)k-s+4$, and derive a multicolour Ramsey consequence.
Mengyue Cao, Mei Lu, Xuyang Yan et al.· 0 citations
Let $\mathcal{L}_n \subset \mathcal{M}_2$ be the locus of genus-two curves admitting a maximal degree-$n$ elliptic subcover, cut out in $\mathbb{P}(2,4,6,10)$ by an irreducible weighted-homogeneous polynomial $F_n \in \mathbb{Z}[J_2,J_4,J_6,J_{10}]$. Let $\nu(n)$ be the degree of $X_1(n) \to X(1)$, let $G_{n^2}$ be the Siegel modular form of level one with divisor the Humbert surface $H_{n^2}$, and let $k(H_{n^2})$ be its weight. We prove that the meromorphic Siegel modular form $F_n(\tau)$ obtained from $F_n$ has a pole of order exactly $\nu(n)$ along the product locus, that $\chi_{10}^{\nu(n)} F_n(\tau)$ is a constant multiple of $G_{n^2}$, and that $\deg_w F_n = k(H_{n^2}) - 10\nu(n)$ for every $n \geq 2$, even or odd. We determine the restriction of $G_{n^2}$ to the product locus as an explicit product of modular polynomials and, for $n \geq 3$, its leading Fourier-Jacobi coefficient as a product of theta functions over the torsion points of exact order $n$, and we characterize $G_{n^2}$, up to scalar, as the unique form of its weight vanishing on a single torsion divisor. These data convert the computation of $F_n$ from elimination into a linear problem of the size the formula prescribes, which we carry out for $n=5$.
Given a field $\mathbb{F}$ and an integer $n \geq 2$, denote by $t_n(\mathbb{F})$ the greatest possible dimension for a vector space of $n$-by-$n$ matrices over $\mathbb{F}$ in which every element is triangularizable. It was recently proved that $t_n(\mathbb{F})=\frac{n(n+1)}{2}$ if and only if $\mathbb{F}$ is not quadratically closed, with the possible exception of finite fields with characteristic $2$ and less than $n-1$ elements. In this article, we prove that the equality $t_n(\mathbb{F})=\frac{n(n+1)}{2}$ holds for all perfect non-quadratically closed fields with characteristic $2$ -- with the possible exception of fields with cardinality $2$ -- and for these fields we obtain a key result for a future analysis of the spaces that have the critical dimension $t_n(\mathbb{F})$.
The factorizations over $\mathbb{F}_p$ of two supersingular polynomials $h_p(x)$ and $g_p(x)$ for genus $2$ curves, discussed by Ibukiyama, Katsura and Oort in their 1986 paper, are investigated. These polynomials are congruent modulo $p$ to the Jacobi polynomials $P_n^{(\alpha,0)}(1-2x)$, for $\alpha = \pm 1/4, \pm 1/6$, respectively. The number of their linear factors (mod $p$) is determined in terms of class numbers of the imaginary quadratic fields $\mathbb{Q}(\sqrt{-dp})$, where $d \in \{1,2,3\}$. The proofs use a quadratic transformation relating these polynomials to the Legendre polynomials $P_n(x)$; previous results on linear and binomial quadratic factors of $P_{(p-e)/4}(x)$ and $P_{(p-\bar e)/3}(x)$ proved by Brillhart and Morton; and properties of the irreducible quadratic factors of the class equations $H_{-3p}(X)$ and $H_{-12p}(X)$ (mod $p$). The quadratic transformation and linear factor results for $h_p(x) \equiv P_n^{(\pm1/4,0)}(1-2x)$ were first discovered using artificial intelligence.
For integers $n\ge 1$, consider assertions: $\mathbf{(P_n)}$: Let $\Omega \subset S^n$ be a spherical domain enclosed by totally geodesic $S^{n-1}$'s with non-obtuse dihedral angles. Then in $[0,2(n+1)]$, its Neumann spectrum can only take values among $\{0,n,2(n+1)\}$. Moreover, $n$ is a Neumann eigenvalue if and only if the corresponding eigenfunction is the restriction of a linear function in $\mathbb{R}^{n+1}$, while $2(n+2)$ is a Neumann eigenvalue if and only if the corresponding eigenfunction is restriction of a quadratic polynomial in $\mathbb{R}^{n+1}$. $\mathbf{(Q_n)}$: A weak solution $u$ to $\Delta u = f$ with the Neumann boundary condition, with $f$ H\"older continuous, in a conical polyhedral domain $\Omega$ in $\mathbb{R}^n$ with non-obtuse dihedral angles, is in $C^{2,\alpha}_{loc}(\overline{\Omega})$. We prove the implications \[\mathbf{(Q_n)} \Rightarrow \mathbf{(P_n)},\qquad \mathbf{(P_n)}\Rightarrow \mathbf{(Q_{n+1})}.\] Consequently, both assertions hold in all dimensions. These give the optimal Neumann eigenvalue lower bound and $C^2$ elliptic regularity in non-obtuse Riemannian polyhedral domains.
Nick Edelen, Chao Li· 0 citations
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