Proof of Dittert's conjecture for dimensions \texorpdfstring{\(n\ge 17\)}{n >= 17}

Abstract

Dittert's conjecture gives a sharp upper bound for the Dittert functional on nonnegative matrices whose entries sum to \(n\). It extends the van der Waerden permanent problem from the doubly stochastic polytope to a larger simplex in which row and column sums are allowed to vary. We prove the conjecture for every dimension \(n\ge 17\). The proof combines the Knopp--Sinkhorn lower bound for boundary points of the doubly stochastic polytope with a refined scaling step in the Cheon--Wanless method. The main improvement is a sharper subset-sum estimate for the row and column sums of a near maximizer, which reduces the scalar dilation needed to obtain a doubly superstochastic matrix. This strengthened comparison is sufficient to exclude boundary maximizers in all dimensions \(n\ge 17\), and the known positive-support characterization then identifies the unique maximizer as \(n^{-1}J_n\).