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

arXiv:2608.28274v1 Announce Type: new Abstract: Let nge2 be even, let lambda=(lambda_1,ldots,lambda_n)inR^n have pairwise distinct coordinates, and define the difference-power matrix [ A_d(lambda) := igl[(lambda_r-lambda_s)^digr]{r,s=1}^n, qquad dinN. ] In 1928, Colombo proved that et A{n-1}(lambda)ne0---and hence et A_{n-1}(lambda)>0---and that operatorname{rank} A_d(lambda)=d+1 for 0le d<n-1. He conjectured that [ et A_d(lambda)ne0 qquadext{for every } dge 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 dge n+1. We prove nonsingularity for all these odd exponents, thereby completing Colombo's conjecture. Consequently, [ operatorname{rank} A_d(lambda)=min{n,d+1} qquad(dinN). ] 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.

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Source: arXiv cs.LG | 2026-08-31