Morneau-Guérin, Frédéric 
ORCID: https://orcid.org/0000-0001-7610-4648 et Singh, Sarishti ORCID: https://orcid.org/0000-0001-6536-6327
(sous presse).
Arithmetic complexity of Erdős matrices. Linear Algebra and its Applications. 10.1016/j.laa.2026.08.031
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- Revised_Arithmetic complexity of Erdos matrices.pdf
Contenu du fichier : Manuscrit accepté (révisé après évaluation) Accès restreint jusqu'à fin- 1 janvier 2027. |
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| Catégorie de document : | Articles de revues |
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| Évaluation par un comité de lecture : | Oui |
| Étape de publication : | Accepté (sous presse) |
| Résumé : | Erdős matrices are the doubly stochastic matrices for which equality holds in the Marcus--Ree inequality (which states that the maximal trace of M is greater or equal to the square of the Frobenius norm of M). Every such matrix has rational entries, but little is known about the growth of the denominators that occur. We initiate the study of this arithmetic complexity. Using a block-diagonal construction involving the matrices T_{p+1} (zero diagonal, constant off-diagonal entries 1/p, where p ranges over the primes, we prove that the largest denominator D_n of an n by n Erdős matrix satisfies D_n is greater or equal to exp(c (n log n)^(1/2)) for some constant strictly positive c and all n greater or equal to 1. On the other hand, by relating an Erdős matrix to the Laplacian of its bipartite support graph and applying the Matrix--Tree Theorem, we establish the upper bound D_n is less than or equal to n^(2n). As a corollary, we obtain that the number of distinct denominators up to dimension N is at least c(N/log N)^(1/2). We also observe that the Gram matrix associated with a Birkhoff decomposition admits a natural representation-theoretic decomposition into trivial and standard |
| Adresse de la version officielle : | https://www.sciencedirect.com/science/article/abs/... |
| Déposant: | Morneau-Guérin, Frédéric |
| Responsable : | Frédéric Morneau-Guérin |
| Dépôt : | 25 aout 2026 18:48 |
| Dernière modification : | 26 aout 2026 18:21 |
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