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| Main Author: | |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2606.01531 |
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| _version_ | 1866910279196999680 |
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| author | Pang, Zhekai |
| author_facet | Pang, Zhekai |
| contents | 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 \(n^{-1}J_n\). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_01531 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Proof of Dittert's conjecture for dimensions \texorpdfstring{\(n\ge 17\)}{n >= 17} Pang, Zhekai Rings and Algebras 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 \(n^{-1}J_n\). |
| title | Proof of Dittert's conjecture for dimensions \texorpdfstring{\(n\ge 17\)}{n >= 17} |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2606.01531 |