Positivity of Mixed Multiplicities

Let R = ⊕(u,v)∈N2R(u,v) be a standard bigraded algebra over an artinian local ring K = R(0,0). Standard means R is generated over K by a finite number of elements of degree (1, 0) and (0, 1). Since the length l(R(u,v)) of R(u,v) is finite, we can consider l(R(u,v)) as a function in two variables u and v. This function was first studied by van der Waerden [W] and Bhattacharya [B] who proved that there is a polynomial PR(u, v) of degree ≤ dimR − 2 such that l(R(u,v)) = PR(u, v) for u and v large enough. Katz, Mandal and Verma [KMV] found out that the degree of PR(u, v) is equal to rdimR − 2, where rdimR is the relevant dimension of R defined as follows. Let R++ denote the ideal generated by the homogeneous elements of degree (u, v) with u ≥ 1, v ≥ 1. Let ProjR be the set of all homogeneous prime ideals ℘ 6⊇ R++ of R. Then rdimR := max{dimR/℘| ℘ ∈ ProjR} if ProjR 6= ∅ and rdimR can be any negative integer if ProjR = ∅. If we write