论文标题
多关节和分解
Multijoints and Factorisation
论文作者
论文摘要
我们解决了双重多期问题，并证明了$ k_j $ - 平面的任意字段和多关点的所谓“分解”。更普遍地，我们推断出应对Bourgain和Guth的定理的离散类似物。我们的结果是一个通用语句，它描述了离散楔形产品的属性，而无需任何明确引用多焦点，并表示如下：假设$ k_1 + \ ldots + k_d = n $。有一个常数$ c = c（n）$，因此对于任何字段$ \ mathbb {f} $，对于任何有限支持的函数$ s：\ mathbb {f}^n \ rightarrow \ rightarrow \ mathbb {r} _ {\ geq 0} $ \ Mathrm {gr}（k_j，\ Mathbb {f}^n）\ rightarrow \ Mathbb {r} _ {\ geq 0} $，因此$$（v_1 \ wedge \ wedge \ cdots \ cdge \ cdge \ wedge v_d） v_j），每$ p \ in \ mathbb {f}^n $以及平面的每一个元组$ v_j \ in \ mathrm {grm {gr}（k_j，k_j，\ mathbb {f}^n）$ $k_j$-plane $π_j\subset \mathbb{F}^n$, where $e(π_j)\in \mathrm{Gr}(k_j,\mathbb{F}^n)$ denotes the translate of $π_j$ that contains the origin and $\wedge$ denotes the discrete wedge product.
We solve the dual multijoint problem and prove the existence of so-called "factorisations" for arbitrary fields and multijoints of $k_j$-planes. More generally, we deduce a discrete analogue of a theorem due in essence to Bourgain and Guth. Our result is a universal statement which describes a property of the discrete wedge product without any explicit reference to multijoints and is stated as follows: Suppose that $k_1 + \ldots + k_d = n$. There is a constant $C=C(n)$ so that for any field $\mathbb{F}$ and for any finitely supported function $S : \mathbb{F}^n \rightarrow \mathbb{R}_{\geq 0}$, there are factorising functions $s_{k_j} : \mathbb{F}^n\times \mathrm{Gr}(k_j, \mathbb{F}^n)\rightarrow \mathbb{R}_{\geq 0}$ such that $$(V_1 \wedge\cdots\wedge V_d)S(p)^d \leq C\prod_{j=1}^d s_{k_j}(p, V_j),$$ for every $p\in \mathbb{F}^n$ and every tuple of planes $V_j\in \mathrm{Gr}(k_j, \mathbb{F}^n)$, and $$\sum_{p\in π_j} s(p, e(π_j)) =||S||_d$$ for every $k_j$-plane $π_j\subset \mathbb{F}^n$, where $e(π_j)\in \mathrm{Gr}(k_j,\mathbb{F}^n)$ denotes the translate of $π_j$ that contains the origin and $\wedge$ denotes the discrete wedge product.
