  Let P and Q be two polynomials in K[x, y] with degree at most d, where K is a
field. Denoting by R $\in$ K[x] the resultant of P and Q with respect to y, we
present an algorithm to compute R mod x^k in O~(kd) arithmetic operations in K,
where the O~ notation indicates that we omit polylogarithmic factors. This is
an improvement over state-of-the-art algorithms that require to compute R in
O~(d^3) operations before computing its first k coefficients.
