  We study the complexity of Gr\"obner bases computation, in particular in the
generic situation where the variables are in simultaneous Noether position with
respect to the system.
  We give a bound on the number of polynomials of degree $d$ in a Gr\"obner
basis computed by Faug\`ere's $F_5$ algorithm~(Fau02) in this generic case for
the grevlex ordering (which is also a bound on the number of polynomials for a
reduced Gr\"obner basis, independently of the algorithm used). Next, we analyse
more precisely the structure of the polynomials in the Gr\"obner bases with
signatures that $F_5$ computes and use it to bound the complexity of the
algorithm.
  Our estimates show that the version of~$F_5$ we analyse, which uses only
standard Gaussian elimination techniques, outperforms row reduction of the
Macaulay matrix with the best known algorithms for moderate degrees, and even
for degrees up to the thousands if Strassen's multiplication is used. The
degree being fixed, the factor of improvement grows exponentially with the
number of variables.
