  We consider the Euler equation for an incompressible fluid on a three
dimensional torus, and the construction of its solution as a power series in
time. We point out some general facts on this subject, from convergence issues
for the power series to the role of symmetries of the initial datum. We then
turn the attention to a paper by Behr, Necas and Wu in ESAIM: M2AN 35 (2001)
229-238; here, the authors chose a very simple Fourier polynomial as an initial
datum for the Euler equation and analyzed the power series in time for the
solution, determining the first 35 terms by computer algebra. Their
calculations suggested for the series a finite convergence radius \tau_3 in the
H^3 Sobolev space, with 0.32 < \tau_3 < 0.35; they regarded this as an
indication that the solution of the Euler equation blows up. We have repeated
the calculations of Behr, Necas and Wu, using again computer algebra; the order
has been increased from 35 to 52, using the symmetries of the initial datum to
speed up computations. As for \tau_3, our results agree with the original
computations of Behr, Necas and Wu (yielding in fact to conjecture that 0.32 <
\tau_3 < 0.33). Moreover, our analysis supports the following conclusions: (a)
The finiteness of \tau_3 is not at all an indication of a possible blow-up. (b)
There is a strong indication that the solution of the Euler equation does not
blow up at a time close to \tau_3. In fact, the solution is likely to exist, at
least, up to a time \theta_3 > 0.47. (c) Pade' analysis gives a rather weak
indication that the solution might blow up at a later time.
