Zero estimates for polynomials over commutative rings (Q801975)

The zero estimates in question are lower bounds for the degree of polynomials, over a commutative Noetherian ring R, which vanish on initial segments of a finitely generated subgroup \(\Gamma\) of the additive group \((R^+)^ n\) or the multiplicative semigroup \((R^{\times})^ n\). These estimates are generalizations of the zero estimates of \textit{D. W. Masser} and \textit{G. Wüstholz} [Invent. Math. 64, 489-516 (1981; Zbl 0467.10025)], for polynomials over the complex numbers or p-adic numbers, which are basic to transcendence theory. - These zero estimates are best possible in the sense that the exponent \(\chi\) (\(\Gamma)\) (Dirichlet exponent) of the estimates cannot be improved. For the case \(\chi (\Gamma)=0\), when the zero estimates themselves tell us nothing, this paper gives exact lower bounds for the degrees of polynomials vanishing on sufficiently large initial segments of \(\Gamma\). The method of proof of the estimates is in principle similar to that of Masser (loc. cit.) and uses techniques of commutative algebra. However, over an arbitrary commutative Noetherian ring R complications arise because of the nilpotent elements and associated prime ideals of R. Indeed, precisely the associated points of Spec R control the estimates in that the exponent \(\chi\) (\(\Gamma)\) is defined as a minimum of local exponents at the associated points.