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

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 e1, e2,…, eiϵE with #{e1,…, ei} =i and View the MathML source for all j = 1,…, i and eϵEβ{e1,…,ej−1}, one has f({e1,…,ei})+Σk=1iη(ek)≥f(I)+ΣeϵIη(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 aϵJβI, there exists some bϵJβ(Iυ {a}) with f(Iυ {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