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Article Dans Une Revue International Mathematics Research Notices Année : 2004

A quadratic bound for the determinant and permanent problem.

Résumé

The determinantal complexity of a polynomial f is defined here as the minimal size of a matrix M with affine entries such that f = det M. This function gives a minoration of the more traditional size of an arithmetical formula. Consider the polynomial "permanent" permd of a d × d matrix with entries Xi,j. A conjecture in complexity theory says that the determinantal complexity (dc) of permd should not be polynomial in d. In this article we prove that dc(permd)> d^2/2, improving the previously known linear minoration. We also begin a systematic study of the function dc, and compute it for the homogeneous polynomials of degree 2.

Dates et versions

hal-00818200 , version 1 (26-04-2013)

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Thierry Mignon, Ressayre Nicolas. A quadratic bound for the determinant and permanent problem.. International Mathematics Research Notices, 2004, 79, pp.4241-4253. ⟨10.1155/S1073792804142566⟩. ⟨hal-00818200⟩
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