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International Mathematics Research Notices (2004) 2004:4241-4253 , doi:10.1155/S1073792804142566
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Copyright © 2004 Hindawi Publishing Corporation. All rights reserved.

A quadratic bound for the determinant and permanent problem

Thierry Mignon and Nicolas Ressayre

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 x 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 Formula, improving the previously known minoration, Formula. We also begin a systematic study of the function dc, and compute it for the homogeneous polynomials of degree 2.


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