Without further information on the system, there is only one practical method, which consists of computing a Gröbner basis and deducing the degree of the denominator of the Hilbert series of the ideal generated by the equations.
The second step, which is usually the fastest, may be accelerated in the following way: Firstly, the Gröbner basis is replaced by the list of its leading monomials (this is already done for the computation of the Hilbert series).