Algorithmes de calcul de la réduction de Hermite d'une matrice à coefficients polynomiaux

In this paper, we study some algorithms for computing an Hermite reduction of a matrix with polynomial entries which avoid the swell-up of the size of intermediary objects and have a good sequential complexity. The second and the third algorithms generalize the subresultant method for computing the gcd of two polynomials. The last one is optimal in the sense that it computes an Hermite reduction with a minimal degree change of basis matrix. The Hermite reduction with polynomials entries amounts to a linear algebra problem over the coefficient field with a good control of the dimensions. Our problem of linear algebra is a progressive triangulation of matrices. So it is feasible exactly when there exists a polynomial time algorithm computing determinants of matrices with entries in the coefficient field of the polynomials given as input. Theoreticals results are better than for previously known algorithms, and practical results are interesting.

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