M. Compute and . Res, = (H) 0 ? 2M ?1 (E) by means of Algorithm 2

, Compute D num = M ?1 (D ? ) + (D res ? M ?1 (D + ))

, G l of the vector space of polynomials in K

,. .. ,

G. Chèze and G. Lecerf, Lifting and recombination techniques for absolute factorization, J. Complexity, vol.23, issue.3, pp.380-420, 2007.

X. Dahan, M. Moreno-maza, É. Schost, and Y. Xie, On the complexity of the D5 principle, Proceedings of Transgressive Computing, pp.149-168, 2006.

C. Durvye and G. Lecerf, A concise proof of the Kronecker polynomial system solver from scratch, Expo. Math, vol.26, issue.2, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00682083

W. Fulton, Algebraic Curves -An Introduction to Algebraic Geometry, 1989.

J. Zur-gathen and J. Gerhard, Modern computer algebra, 2013.

B. Grenet, J. Van-der-hoeven, and G. Lecerf, Deterministic root finding over finite fields using Graeffe transforms, Appl. Algebra Engrg. Comm. Comput, vol.27, issue.3, pp.237-257, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01104251

G. Haché, Construction Effective des Codes Géométriques, 1996.

G. Haché and . Brill, Rapport de recherche n ? 1998-01, Laboratoire d'Arithmétique, de Calcul formel et d'Optimisation ESA -CNRS 6090, Noether appliqué aux courbes réduites, 1998.

F. Hess, Computing Riemann-Roch spaces in algebraic function fields and related topics, J. Symbolic Comput, vol.33, issue.4, pp.425-445, 2002.

J. Van-der-hoeven and G. Lecerf, Composition modulo powers of polynomials, Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation, ISSAC '17, pp.445-452, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01455722

J. Van-der-hoeven and G. Lecerf, Directed evaluation, HAL, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01966428

J. Van-der-hoeven and G. Lecerf, On the complexity exponent of polynomial system solving, HAL, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01848572

J. Van-der-hoeven and G. Lecerf, Fast computation of generic bivariate resultants, HAL, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02080426

J. Van-der-hoeven and G. Lecerf, Fast multivariate multi-point evaluation revisited, J. Complexity, vol.56, p.101405, 2020.
URL : https://hal.archives-ouvertes.fr/hal-01848571

X. Huang and V. Y. Pan, Fast rectangular matrix multiplication and applications, J. Complexity, vol.14, issue.2, pp.257-299, 1998.

K. Khuri-makdisi, Asymptotically fast group operations on Jacobians of general curves, Math. Comp, vol.76, issue.260, pp.2213-2239, 2007.

D. , L. Brigand, and J. Risler, Algorithme de Brill-Noether et codes de Goppa, vol.116, pp.231-253, 1988.

F. and L. Gall, Powers of tensors and fast matrix multiplication, 14: International Symposium on Symbolic and Algebraic Computation, pp.296-303, 2014.

A. , L. Gluher, and P. Spaenlehauer, A fast randomized geometric algorithm for computing Riemann-Roch spaces, Math. Comp, 2019.
URL : https://hal.archives-ouvertes.fr/hal-01930573

G. Lecerf, Fast separable factorization and applications, Appl. Algebra Engrg. Comm. Comput, vol.19, issue.2, pp.135-160, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00681420

G. Lecerf, On the complexity of the Lickteig-Roy subresultant algorithm, J. Symbolic Comput, vol.92, pp.243-268, 2019.
URL : https://hal.archives-ouvertes.fr/hal-01450869

T. Mulders and A. Storjohann, On lattice reduction for polynomial matrices, J. Symbolic Comput, vol.35, issue.4, pp.377-401, 2003.

V. Neiger, Fast computation of shifted Popov forms of polynomial matrices via systems of modular polynomial equations, Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation, ISSAC '16, pp.365-372, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01266014

V. Neiger, J. Rosenkilde, and G. Solomatov, Computing Popov and Hermite forms of rectangular polynomial matrices, Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation, ISSAC '18, pp.295-302, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01701867

M. Nüsken and M. Ziegler, Fast multipoint evaluation of bivariate polynomials, Algorithms -ESA 2004. 12th Annual European Symposium, vol.3221, pp.544-555, 2004.

G. Villard, On computing the resultant of generic bivariate polynomials, Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation, ISSAC '18, pp.391-398, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01921369