Multijoints and Factorisation

We solve the dual multijoint problem and prove the existence of so-called factorisations for arbitrary fields and multijoints of kⱼ 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 k₁ + … + kd = n k 1 + … + k d = n . There is a constant $$C=C(n)$$ C = C ( n ) so that for any field F F and for any finitely supported function S : Fⁿ → R≥ 0 S : F n → R ≥ 0 , there are factorising functions skⱼ : Fⁿ× \,Gr\,(kⱼ, Fⁿ)→ R≥ 0 s k j : F n × Gr ( k j , F n ) → R ≥ 0 such that aligned ( V₁ ⋯ Vd) S( p) ᵈ ≤ ∏ ⱼ₌₁ᵈ skⱼ( p, Vⱼ) , aligned V 1 ∧ ⋯ ∧ V d S p d ≤ ∏ j = 1 d s k j p , V j , for every p∈ Fⁿ p ∈ F n and every tuple of planes Vⱼ∈ \,Gr\,(kⱼ, Fⁿ) V j ∈ Gr ( k j , F n ) , and aligned ∑ p∈ π ⱼ s(p, e(π ⱼ)) =C | | S| | d, aligned ∑ p ∈ π j s ( p , e ( π j ) ) = C S d , for every kⱼ k j -plane π ⱼ⊂ Fⁿ π j ⊂ F n , where e(π ⱼ)∈ \,Gr\,(kⱼ,Fⁿ) e ( π j ) ∈ Gr ( k j , F n ) , is the translate of π ⱼ π j that contains the origin and ∧ denotes the discrete wedge product.