On the effective reduction of an ideal

It is well known that the reduction of an $\mathfrak m$-primary ideal $\mathfrak q$ in the Noetherian local ring $(R,\mathfrak m)$ with infinite residue field $R/\mathfrak m$ may be given in the form of a sufficiently general linear combination of its generators. We give a condition for the existence of such a reduction in terms of the sum of the degrees of the prime divisors of the ideal fiber cone $\mathcal F_{\mathfrak q}(R)$ in the case of any Noetherian local ring.