## Abstract

Let K _{n} denote the convex set consisting of all real nonnegative n×n matrices whose entries have sum n. For A ∈ K _{n} with row sums r _{1},⋯,r _{n} and column sums c _{1}, ⋯,c _{n}, define φ(A)=∏ ^{n} _{i=1} ^{r} _{i}+∏ ^{n} _{j=1}c _{j}-per(A). Dittert's conjecture asserts that the maximum of φ on K _{n} occurs uniquely at J _{n}=[1/n] _{n×n}. In this paper, we prove:(i)if A ∈ K ^{n} is partly decomposable then φ(A) < φ(J _{n});(ii)if the zeroes in A ∈ K _{n} form a block then A is not a φ-maximising matrix;(iii)φ(A) < φ(J _{n}) unless δ:=per(J _{n})-per(A) ≤ (n ^{4} _{e} ^{-2n}) and |k-∑i∈αr ^{i}|< √2δk,|k-∑i∈βc _{i}|<√2δk and ∑i∈α,j∈βaij <k+ √δk for all sets α,β of k integers chosen from {1,2,⋯,n}.

Original language | English |
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Pages (from-to) | 791-801 |

Number of pages | 11 |

Journal | Linear Algebra and Its Applications |

Volume | 436 |

Issue number | 4 |

DOIs | |

Publication status | Published - 15 Feb 2012 |

Externally published | Yes |