, 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.
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
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