We determine the exact finite-dimensional threshold in the zero-subspace problem of Aron and Rueda for complex homogeneous polynomials. More precisely, for every $d$ and $k$ we determine the least $m$ such that every $d$-homogeneous polynomial on $\mathbb{C}^m$ vanishes on a $k$-dimensional linear subspace. We also determine the exact threshold for arbitrary polynomials of degree at most $d$ to be constant on a $k$-dimensional linear subspace. The two thresholds are different. In the homogeneous case the exact threshold follows from Tevelev's theorem on isotropic subspaces and closedness of the incidence locus. In the bounded-degree case we first eliminate the linear homogeneous component by passing to its kernel; the remaining components, of degrees $2,\ldots,d$, form the system to which the Debarre--Manivel theorem is applied. For $k=2$ we give a separate proof using top Chern classes and Newton's inequalities.
Makeev [2] stated that every finite Borel measure in $\mathbb R^d$ assigning zero mass to hyperplanes can be cut by $d$ mutually orthogonal hyperplanes so that every pair divides the measure into four equal parts, and outlined a proof strategy, but the key steps were left incomplete. We give a direct and elementary pro...
Let $q=p^k$, and let $A_1,\ldots,A_{r-1}$ be invertible $n\times n$ matrices over ${\mathbb F}_q$. We prove that, if $k\ge r$, there is a vector $x$ for which \[ x,A_1x,\ldots,A_{r-1}x \] are all nowhere zero. For $r=2$ this recovers the theorem of Alon and Tarsi over nonprime finite fields. The proof tracks one monomi...
Suppose we have $r$ linearly independent hypersurfaces of degree $d$ in $\mathbb{P}^m$ (or $\mathbb{A}^m$) defined over a finite field $\mathbb{F}_q$, whose intersection is at most $k$-dimensional. What is the largest possible number of $\mathbb{F}_q$-rational points in the intersection? We conjecture an exact formula...
Let $I\subseteq\mathbb C[x_1,\ldots,x_n]$ be an ideal generated by homogeneous polynomials of a common degree $d$. We give a polynomial partial differential equation criterion guaranteeing that $-n/d$ is a root of the Bernstein-Sato polynomial $b_I(s)$. We apply this criterion to the ideal of maximal minors of a generi...
For a fixed prime $p$, let $\mu_p$ denote the $p$-adic Plancherel measure. Then the first main goal of this paper is to prove effective (and moreover explicit) $\mu_p$-equidistribution of the $p$-th Hecke eigenvalues over the Atkin--Lehner subspaces $S_k^\sigma(N) \subseteq S_k(N)$ and $S_k^{\operatorname{new}, \sigma}...
Aarya J. Kumar, Sargam Mondal, Erick Ross et al.· 2 citations
Let $m\geq 2$ and $d\geq 7$. We consider homogeneous polynomials in $2m+2$ variables of the form $f=F_0(u_0,v_0)+\cdots+F_m(u_m,v_m)$, where the $F_i$ are independently very general squarefree binary forms of degree $d$. We prove the rational noncommutative Hodge conjecture for the dg category $\mathrm{MF}^{\mathrm{gr}...
Xun Lin, Shi-Zhuo Zhang· 0 citations
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