The primes that are "bad" in your sense will divide the number Resx(Resy(f,fracpartialfpartialx),Resy(f,fracpartialfpartialy)). (If I interpreted damiano's comment correctly).
All that is left is to bound this number. So:
Let M:=max(|aij|).
parallelResy(f,fracpartialfpartialx)parallelandparallelResy(f,fracpartialfpartialy)parallelare<(2d)!M2d
RightarrowparallelResx(Resy(f,fracpartialfpartialx),Resy(f,fracpartialfpartialy))parallel<(2d2)2d2((2d)!M2d)2d2ll(dM)4d3+O(d2)
So pick a random prime larger than this and then compute gcd(Resy(f,fracpartialfpartialx),Resy(f,fracpartialfpartialy)) in mathbbFp. The complexity is O(poly(d)timespoly(log(M)). Is this better than Groebner computations in mathbbQ? I have no idea...
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