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

On the Complexity of the F5 Gr\"obner basis Algorithm