Well-layered maps and the maximum-degree k × k-subdeterminant of a matrix of rational functions

Abstract

Given a finite set E and a map ƒ: P(E) → IR ∪ {−∞}, we define ƒ to be well-layered, if and only if for every map η: E → IR and every finite sequence e 1, e 2,…, e i ϵ E with #{ e 1,…, e i } = i and ƒ({e 1,…, e j}) + Σ k=1 jη(e k)≥ƒ({e 1,…, e j−1,e}) + Σ k=1 j−1η(e k) + η(e) for all j = 1,…, i and e ϵ Eβ{ e 1,…, e j−1 }, one has f({ e 1,…, e i })+ Σ k=1 i η( e k )≥ f( I)+ Σ eϵI η( e) for every I⊆ E with # I= i. In this note, we show that a map f is well-layered if and only if for every I ⊆ J ⊆ E with #( JβI) ≥ 3 and with f( I) ≠ −∞ or I = φ and for every a ϵ JβI, there exists some b ϵ Jβ( I υ { a}) with f( I υ { a}) + f( Jβ{ a}) ≤ f( Iυ { b}) + f ( Jβ{ b}), and if in addition f( I) = −∞ for all subsets I of some fixed cardinality i with 0< i<# E implies f( I′) = −∞ for all subsets I′ with i < # I′ < # E. In addition, we provide some “generic” examples of well-layered maps related to p-adic geometry, and we indicate some interesting applications related to control theory.