An algebraic proof of Colombo's difference-power determinant conjecture

Let n≥2 be even, let λ=(λ1,ldots,λn)∈ℝn have pairwise distinct coordinates, and define the difference-power matrix [ Ad(λ) := bigl[(λr-λs)dbigr]r,s=1n, qquad d∈mathbbN. ] In 1928, Colombo proved that det An-1(λ)ne0---and hence det An-1(λ)>0---and that operatornamerank Ad(λ)=d+1 for 0≤ d<n-1. He conjectured that [ det Ad(λ)ne0 qquadfor every d≥ n-1. ] For even d, the conjectured nonsingularity follows from previously published results on distance-power matrices. The remaining open cases were therefore the supercritical odd exponents d≥ n+1. We prove nonsingularity for all these odd exponents, thereby completing Colombo's conjecture. Consequently, [ operatornamerank Ad(λ)=minn,d+1 qquad(d∈mathbbN). ] Our proof converts a hypothetical kernel vector into a real binary form having more projective real linear factors, counted with multiplicity, than its real Waring length permits.