Weak Bezout inequality for D-modules

Original · EN

Let {wᵢ,ⱼ}₁≤ ᵢ≤ ₙ, ₁≤ ⱼ≤ ₛ ⊂ Lₘ=F(X₁,...,Xₘ)[∂ ∂ X₁,..., ∂ ∂ Xₘ] be linear partial differential operators of orders with respect to ∂ ∂ X₁,..., ∂ ∂ Xₘ at most d. We prove an upper bound n(4m²d{n,s})⁴ᵐ⁻ᵗ⁻¹⁽²⁽ᵐ⁻ᵗ⁾⁾ on the leading coefficient of the Hilbert-Kolchin polynomial of the left Lₘ-module <{w₁,ⱼ,..., wₙ,ⱼ}₁≤ ⱼ ≤ ₛ > ⊂ Lₘⁿ having the differential type t (also being equal to the degree of the Hilbert-Kolchin polynomial). The main technical tool is the complexity bound on solving systems of linear equations over algebras of fractions of the form Lₘ(F[X₁,..., Xₘ, ∂ ∂ X₁,..., ∂ ∂ Xₖ])⁻¹.