, If we want to output an approximation of the terms of the minimal decomposition, with a relative error of 2 ?? , we can use Pan's algorithm (Pan, 2002.

, of degree D and let ? be the maximum bitsize of the coefficients a i . We study the bit complexity of computing suitable approximations of the ? j 's, ? j 's, and ? j 's of Equation (3), say ? j , ? j and ? j respectively, that induce an approximate decomposition correct up to bits. That is a decomposition such that f ? ? j ? j, complexity Let f ? Z[x, y] be a binary form as in Equation (1)

, The first step of the algorithm is to compute P v and P w , via the computation of three rows of the Extended GCD of two polynomials of degree D and D + 1 with coefficients of maximal sized ?. This can be achieved in O B (D 2 ?) bit operations (Gathen and Gerhard, 2013, Corollary 11.14.B), and the maximal bit size of P v and P w is O(D?), 2013.

P. If and Q. , If P v is not square-free, then we can compute that Q is not square-free (Corollary 23), thus at least one of the first 2D + 3 integer fits our requirements. We test them all. Each test corresponds to a GCD computation

, be the maximal bit size of Q. By Remark 37, we can assume that y does not divide Q, consider y = 1 and treat Q as an univariate polynomial

M. Ansola, A. Daz-cano, and M. A. Zurro, Semialgebraic decomposition of real binary forms of a given degree's space, 2017.

C. Bajaj, The algebraic degree of geometric optimization problems, Discrete & Computational Geometry, vol.3, issue.1, pp.177-191, 1988.

M. R. Bender, J. Faugère, L. Perret, and E. Tsigaridas, A superfast randomized algorithm to decompose binary forms, Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation, pp.79-86, 2016.

A. Bernardi, N. S. Daleo, J. D. Hauenstein, and B. Mourrain, Tensor decomposition and homotopy continuation. Differential Geometry and its Applications, vol.55, pp.78-105, 2017.

A. Bernardi, A. Gimigliano, and M. Ida, Computing symmetric rank for symmetric tensors, Journal of Symbolic Computation, vol.46, issue.1, pp.34-53, 2011.

G. Blekherman, Typical real ranks of binary forms, Foundations of Computational Mathematics, vol.15, issue.3, pp.793-798, 2015.

M. Boij, E. Carlini, and A. Geramita, Monomials as sums of powers: the real binary case, Proceedings of the American Mathematical Society, vol.139, issue.9, pp.3039-3043, 2011.

A. Bostan, F. Chyzak, M. Giusti, R. Lebreton, G. Lecerf et al., Algorithmes efficaces en calcul formel, 2017.

J. Brachat, P. Comon, B. Mourrain, and E. Tsigaridas, Symmetric tensor decomposition, Linear Algebra and its Applications, vol.433, issue.11, pp.1851-1872, 2010.

S. Cabay and D. Choi, Algebraic computations of scaled Padé fractions, SIAM Journal on Computing, vol.15, issue.1, pp.243-270, 1986.

E. Carlini, M. V. Catalisano, L. Chiantini, A. V. Geramita, and Y. Woo, Symmetric tensors: rank, Strassen's conjecture and e-computability. Annali della Scuola Normale Superiore di Pisa, Classe di scienze, vol.18, issue.1, pp.363-390, 2018.

E. Carlini, M. V. Catalisano, and A. Oneto, Waring loci and the Strassen conjecture, Advances in Mathematics, vol.314, pp.630-662, 2017.

J. W. Cassels, Local fields, vol.3, 1986.

G. Comas and M. Seiguer, On the rank of a binary form, Foundations of Computational Mathematics, vol.11, issue.1, pp.65-78, 2011.

P. Comon, Tensors: a brief introduction, IEEE Signal Processing Magazine, vol.31, issue.3, pp.44-53, 2014.

P. Comon, G. Golub, L. Lim, and B. Mourrain, Symmetric tensors and symmetric tensor rank, SIAM Journal on Matrix Analysis and Applications, vol.30, issue.3, pp.1254-1279, 2008.

P. Comon and B. Mourrain, Decomposition of quantics in sums of powers of linear forms, Signal Processing, vol.53, issue.2, pp.93-107, 1996.

J. Draisma, E. Horobe?, G. Ottaviani, B. Sturmfels, and R. R. Thomas, The euclidean distance degree of an algebraic variety, Foundations of computational mathematics, vol.16, issue.1, pp.99-149, 2016.

A. Dür, On computing the canonical form for a binary form of odd degree, Journal of Symbolic Computation, vol.8, issue.4, pp.327-333, 1989.

I. García-marco, P. Koiran, and T. Pecatte, Reconstruction algorithms for sums of affine powers, Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation. ISSAC '17, pp.317-324, 2017.

J. V. Gathen and J. Gerhard, Modern computer algebra, 2013.

M. Giesbrecht, E. Kaltofen, and W. Lee, Algorithms for computing sparsest shifts of polynomials in power, chebyshev, and pochhammer bases, Journal of Symbolic Computation, vol.36, issue.3-4, pp.401-424, 2003.

M. Giesbrecht and D. S. Roche, Interpolation of shifted-lacunary polynomials, Computational Complexity, vol.19, issue.3, pp.333-354, 2010.

S. Gundelfinger, Zur theorie der binären formen, Journal für die reine und angewandte Mathematik, vol.100, pp.413-424, 1887.

J. D. Hauenstein, L. Oeding, G. Ottaviani, A. J. Sommese, and K. Rost, Homotopy techniques for tensor decomposition and perfect identifiability, Journal fr die reine und angewandte Mathematik (Crelles Journal), issue.0, p.0, 1984.

U. Helmke, Waring's problem for binary forms, Journal of pure and applied algebra, vol.80, issue.1, pp.29-45, 1992.

A. Iarrobino and V. Kanev, Power sums, Gorenstein algebras, and determinantal loci, 1999.

E. Kaltofen and L. Yagati, Improved sparse multivariate polynomial interpolation algorithms, Symbolic and Algebraic Computation, pp.467-474, 1989.

J. P. Kung, Canonical forms of binary forms: variations on a theme of Sylvester, Institute for Mathematics and Its Applications, vol.19, p.46, 1990.

J. P. Kung and G. Rota, The invariant theory of binary forms, Bulletin of the American Mathematical Society, vol.10, issue.1, pp.27-85, 1984.

J. M. Landsberg, Tensors: geometry and applications, 2012.

J. M. Mcnamee and V. Y. Pan, Numerical methods for roots of polynomials (II), 2013.

J. Nie, K. Ranestad, and B. Sturmfels, The algebraic degree of semidefinite programming, Mathematical Programming, vol.122, issue.2, pp.379-405, 2010.

L. Oeding and G. Ottaviani, Eigenvectors of tensors and algorithms for waring decomposition, Journal of Symbolic Computation, vol.54, pp.9-35, 2013.

V. Pan, Structured matrices and polynomials: unified superfast algorithms, 2001.

V. Y. Pan, Univariate polynomials: Nearly optimal algorithms for numerical factorization and root-finding, Journal of Symbolic Computation, vol.33, issue.5, pp.701-733, 2002.

V. Y. Pan and E. Tsigaridas, Accelerated approximation of the complex roots and factors of a univariate polynomial, Theoretical Computer Science, vol.681, pp.138-145, 2017.

V. Y. Pan and E. P. Tsigaridas, Nearly optimal computations with structured matrices, Theoretical Computer Science, vol.681, pp.117-137, 2017.

B. Reznick, Homogeneous polynomial solutions to constant coefficient pde's, Advances in Mathematics, vol.117, issue.2, pp.179-192, 1996.

B. Reznick, On the length of binary forms, Quadratic and Higher Degree Forms, pp.207-232, 2013.

B. Reznick, Some new canonical forms for polynomials, Pacific Journal of Mathematics, vol.266, issue.1, pp.185-220, 2013.

B. Reznick and N. Tokcan, Binary forms with three different relative ranks, Proceedings of the American Mathematical Society, vol.145, issue.12, pp.5169-5177, 2017.

B. A. Reznick, Sums of even powers of real linear forms, vol.463, 1992.

J. J. Sylvester, On a remarkable discovery in the theory of canonical forms and of hyperdeterminants. The London, Edinburgh, and Dublin Philosophical Magazine, Journal of Science, vol.2, issue.12, pp.391-410, 1851.

J. J. Sylvester, An essay on canonical forms, supplement to a sketch of a memoir on elimination, transformation and canonical forms, The collected papers of James Joseph Sylvester, vol.1, pp.203-216, 1904.