A Proof of the Dittert Conjecture via Boundary Rigidity

Abstract

Let ${\mathcal{K}}_n$ be the simplex of nonnegative $n\times n$ matrices whose entries sum to $n$. For $A$ in ${\mathcal{K}}_n$, with row sums $r_1,\dots ,r_n$ and column sums $c_1,\dots ,c_n$, define $\Phi (A)={\prod }_{i=1}^n{r}_i+{\prod }_{j=1}^n{c}_j-\mathrm{per}(A)$. Dittert conjectured that $\Phi$ is uniquely maximized by the uniform matrix $U_n=J_n/n$, with maximum $2-n!/n^n$. We prove the conjecture for every $n\ge 2$. Hwang's theorem already determines every positive maximizer, so the essential problem is to exclude maximizers on the boundary of ${\mathcal{K}}_n$. A reduction of Cheon and Wanless supplies two zero entries in distinct rows and columns. In order four, we classify all admissible support strata and exclude them by exact Karush-Kuhn-Tucker identities together with a Bernstein-form positivity certificate. For all $n\ge 5$, an attained-stratum averaging argument compresses a hypothetical boundary maximizer to a seven-parameter matrix with one multiplicity $m=n-2$. A symbolic odd-variable expansion, a cofactor identity at the two missing entries, and uniform coefficient inequalities then contradict the one-sided multiplier conditions. This argument is uniform in $m$ and uses no finite enumeration. We also give independent spectral proofs in dimensions 7 through 13 and prescribed-zero proofs in dimensions 14 and 15. Every dimension-dependent calculation is certified over the rationals and recorded explicitly in the appendices.