Multijoints and Factorisation

Abstract

AbstractWe solve the dual multijoint problem and prove the existence of so-called factorisations for arbitrary fields and multijoints of $$k_j$$ 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 $$k_1 + \ldots + k_d = n$$ k 1 + … + k d = n . There is a constant $$C=C(n)$$ C = C ( n ) so that for any field $$\mathbb {F}$$ F and for any finitely supported function $$S : \mathbb {F}^n \rightarrow \mathbb {R}_{\ge 0}$$ S : F n → R ≥ 0 , there are factorising functions $$s_{k_j} : \mathbb {F}^n\times {{\,\mathrm{{Gr}}\,}}(k_j, \mathbb {F}^n)\rightarrow \mathbb {R}_{\ge 0}$$ s k j : F n × Gr ( k j , F n ) → R ≥ 0 such that $$\begin{aligned} \left( V_1 \wedge \cdots \wedge V_d\right) S\left( p\right) ^d \le \prod _{j=1}^d s_{k_j}\left( p, V_j\right) , \end{aligned}$$ V 1 ∧ ⋯ ∧ V d S p d ≤ ∏ j = 1 d s k j p , V j , for every $$p\in \mathbb {F}^n$$ p ∈ F n and every tuple of planes $$V_j\in {{\,\mathrm{{Gr}}\,}}(k_j, \mathbb {F}^n)$$ V j ∈ Gr ( k j , F n ) , and $$\begin{aligned} \sum _{p\in \pi _j} s(p, e(\pi _j)) =C \left| \left| S\right| \right| _d, \end{aligned}$$ ∑ p ∈ π j s ( p , e ( π j ) ) = C S d , for every $$k_j$$ k j -plane $$\pi _j\subset \mathbb {F}^n$$ π j ⊂ F n , where $$e(\pi _j)\in {{\,\mathrm{{Gr}}\,}}(k_j,\mathbb {F}^n)$$ e ( π j ) ∈ Gr ( k j , F n ) , is the translate of $$\pi _j$$ π j that contains the origin and $$\wedge $$ ∧ denotes the discrete wedge product.