  We study tight bounds and fast algorithms for LCLMs of several linear
differential operators with polynomial coefficients. We analyze the arithmetic
complexity of existing algorithms for LCLMs, as well as the size of their
outputs. We propose a new algorithm that recasts the LCLM computation in a
linear algebra problem on a polynomial matrix. This algorithm yields sharp
bounds on the coefficient degrees of the LCLM, improving by one order of
magnitude the best bounds obtained using previous algorithms. The complexity of
the new algorithm is almost optimal, in the sense that it nearly matches the
arithmetic size of the output.
