A Proof of the Dittert Conjecture via Boundary Rigidity
Abstract
Let 𝒦n be the simplex of nonnegative n × n matrices whose entries sum to n. For A in 𝒦n, with row sums r1,…,rn and column sums c1,…,cn, define Φ(A)=∏i=1n ri+∏j=1n cj−per(A). Dittert conjectured that Φ is uniquely maximized by the uniform matrix Un=Jn/n, with maximum 2−n!/nn. We prove the conjecture for every n ≥ 2. Hwang's theorem already determines every positive maximizer, so the essential problem is to exclude maximizers on the boundary of 𝒦n. 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 ≥ 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.
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BibTeX
@misc{dai2026proof,
title = {{A Proof of the Dittert Conjecture via Boundary Rigidity}},
author = {Xinan Dai and Wenhao Deng and Yingdong Shi and Tailin Wu and Yuchen Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10545},
eprint = {2610.10545},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10545},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Proof of the Dittert Conjecture via Boundary Rigidity
AU - Xinan Dai
AU - Wenhao Deng
AU - Yingdong Shi
AU - Tailin Wu
AU - Yuchen Yang
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.10545
UR - https://arxiv.org/abs/2610.10545
AB - Let 𝒦_n be the simplex of nonnegative n × n matrices whose entries sum to n. For A in 𝒦_n, with row sums r_1,…,r_n and column sums c_1,…,c_n, define Φ(A)=∏_i=1^n r_i+∏_j=1^n c_j−per(A). Dittert conjectured that Φ 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 ≥ 2. Hwang's theorem already determines every positive maximizer, so the essential problem is to exclude maximizers on the boundary of 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 ≥ 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.
ER -