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 for this problem in both the projective and affine settings, assuming $q\geq d+1$. The case $k=m-1$ recovers the Beelen-Datta-Ghorpade conjecture [2] and the case $k=0$ recovers the zero-dimensional conjecture in [15]. Another interesting special case of the conjecture is $k=m-r$, which corresponds to the complete intersection of $r$ degree $d$ polynomials. The case $r=1$, $k=m-1$ was proven by Serre in [16] who showed that if $F$ is a degree $d$ homogeneous polynomial, then $|V(F)(\mathbb{F}_q)|\leq dq^{m-1}+\pi_{m-2}(q)$. We prove the case $r=2$ and $k=m-2$, that is, if $F_1, F_2$ are coprime, degree $d$ homogeneous polynomials, then $|V(F_1,F_2)(\mathbb{F}_q)|\leq d^2q^{m-2}+\pi_{m-3}(q)$.
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
A classical geometric result says that every nonzero cycle of the mod-$2$ incidence map from $d$-subsets to $(d-1)$-subsets of $[n]$ has support at least $d+1$, with equality attained by the boundary of a simplex on $d+1$ vertices. We prove an analogous result for the subspace lattice of $\mathbb{F}_q^n$, determining the minimum support size of a nonzero $d$-cycle over a field $K$ of characteristic $p \mid q+1$. Surprisingly, the boundary of a $(d+1)$-space is not always optimal. Shorter cycles occur for $d=1$, and for $d=2$ when $n\ge4$, and otherwise, the boundary of a $(d+1)$-space is shortest. For $d\ge4$, we prove a gap-stability result: every cycle with support less than $(2-10/q)$ times the minimum is a multiple of the boundary of a $d+1$ space. We also construct support-controlled cones, yielding a direct geometric analysis of the dimensions in which the homology groups of the subspace incidence complex vanish and explicit lower bounds on its coboundary expansion. The degree-$1$ expansion estimate is an ingredient in the stability theorem.
Let $K$ be a number field with ring of integers $\mathcal{O}_K$, and let $A,B,C \in \mathcal{O}_K \setminus\{0\}$. Denote by $S_K'$ the set of prime ideals of $\mathcal{O}_K$ dividing $2ABC$. Assuming two standard conjectures concerning the modularity of mod-$p$ Galois representations and the Eichler-Shimura correspondence over number fields, we study the asymptotic behavior of the generalized Fermat equation $Ax^p+By^p+Cz^p=0$ over $K$. Using the modular method, we establish an asymptotic criterion in terms of the solutions of the associated $S_K'$-unit equation. As an application, we obtain asymptotic results for certain imaginary quadratic fields $K=\mathbb{Q}(\sqrt{-d})$. In particular, for a family of squarefree integers $d$, we determine the relevant $S_K'$-unit solutions explicitly and deduce that the generalized Fermat equation has no asymptotic solutions. Finally, we show that this family of squarefree integers has relative density $5/6$ among all squarefree positive integers.
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.
Let $\cX\subseteq\Pj^n_{\Z}$ be a fixed integral quasiprojective subscheme, smooth over $\Z$ of relative dimension $r$. For each fixed $m\ge1$, we bound the probability that the $m$th principal-parts jet of the restriction of a uniform degree-$d$ form to $\cX_p$ has a positive-dimensional zero scheme. The bound is $C(d+1)^{N_m}p^{-\lambda_m(d)}$, where $N_m=\binom{r+m}{m}$ and $\lambda_m(d)=\floor{m(d+1)/(m+1)}$. For $m=1$, this gives the Bertini singular-locus estimate $C(d+1)^{r+1}p^{-\ceil{d/2}}$. It settles Poonen's arithmetic Bertini Conjecture~5.2 and, after increasing the degree threshold, yields $p^{-A}$ for every fixed $A>0$. For $c\le r$ independent hypersurfaces, the probability of a positive-dimensional Jacobian rank-degeneracy locus is bounded both by $C\sum_i(d_i+1)^{r+1}p^{-\ceil{d_i/2}}$ and by $C'(d_{\min}+1)^{r+1}p^{-\ceil{d_{\min}/2}}$. The proof uses filtered $Q$-adic decompositions, triangular normal Taylor blocks, and Jacobian-pivot charts of uniformly controlled complexity.
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 $\mathrm{MF}^{\mathrm{gr}}(f)$ of graded matrix factorizations. Its Hochschild homology is the direct sum of the scalar-invariant Jacobian sector and $d-1$ one-dimensional point sectors. Boundary--bulk images of explicit rank-one factorizations generate a lattice of rank $(d-1)^{m+1}$ in the identity sector, while grading shifts of the stabilized residue field generate all point sectors. A reduced-Burau calculation shows that the identity-sector lattice exhausts the rational Hodge classes at a very general parameter. Consequently, $\dim_{\mathbb{Q}}\operatorname{Hdg}\bigl(\mathrm{MF}^{\mathrm{gr}}(f),\mathbb{Q}\bigr)=(d-1)^{m+1}+d-1$, and the rational topological $K$-rank is $\bigl((d-1)^{2m+2}+d-1\bigr)/d+d-1$. Finally, applying the additivity of the noncommutative Hodge conjecture for semiorthogonal decompositions together with Orlov's decompositions for Fano, Calabi--Yau, and general-type hypersurfaces proves the rational Hodge conjecture for the associated smooth projective hypersurface.
Xun Lin, Shi-Zhuo Zhang· 0 citations
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