By Madhu Sudan

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The preprocessing stage we describe is randomized and has a small probability of failing. 2. BLACK-BOX FACTORIZATION OF MULTIVARIATE POLYNOMIALS 47 factor (provided that we manage to factor some polynomial with 3 variables). In the rest of this section we 9 assume that we are working over a eld that is big enough, f is monic in x, and @f @x 6= 0. 1 Preprocessing Stage Our rst task is to determine the number of factors of f. Assume that f has n + 1 variables x; y1; : : :; yn, it is monic in x, the derivative @f @x is non-zero, and the total degree of f is at most d.

1 there exists g g (mod I) and h h (mod I) such that f g h (mod I 2 ). We know g g is a multiple of yk and g is monic in x. Therefore we can apply the division algorithm to divide the polynomial adef = (g g)=yk by g and obtain polynomials q and r such that a = qg + r where degx r < degxg. Let g0 = g + yk r. Clearly, g0 is monic and degx g0 = degx g. Moreover, g0 = g (1 yk qg ) + yk q(g g) I g (1 + u) where u = yk qg 2 I. 1 that f g0h0 (mod I 2 ) for h0 = h (1 u), ie. g0 divides f modulo I 2 . This proves that there exists a solution g0 such that g0 is monic and of degree degx g.

The rest of the argument will also be analogous to the bivariate case, for instance instead of computing resultants over R(y) (as was done in the case of R x; y]) we will now compute resultants over R(y1 ; y2 ; : : :; yn ), etc. We omit the details which can be easily lled in. The second approach is more e cient than the rst one we suggested, but still both the methods perform poorly for one non-circumventable (at least apparently) reason. Even though the original multivariate polynomial may be sparse and hence have small description, it may very well be the case that one of its factors in not sparse.