Mixed monomial bases

Abstract

Given a system of n generic Laurent polynomials

$$f_i (x) = \sum\limits_{q \in A_i } {c_{iq} x^q ;} \,\,\,\,\,q = (q{}_1,...,q_n );\,\,\,\,x^q = x_1^{q_1 } x_2^{q_2 } ...x_n^{q_n }$$

(1.1)

with support sets A _i ⊂ℤⁿ, we consider the ring

$$A{ : = }K\left[ {x_{1} ,x_1^{ - 1} ,...,x_{n,} x_n^{ - 1} } \right]/\left( {f_1 ,....f_n } \right),$$

where K is the field ℚ({c _iq }). The K-dimension of A equals the number of toric roots {x ∈ (ℂ*)ⁿ : f _i (x) = 0, 1 ≤ i ≤ n}. By Bernstein’s theorem [Ber], this number equals the mixed volume Mν(P ₁ ,..., P _n ) of the Newton polytopes P ⁱ := conv(A ⁱ ) . The objective of this note is to construct explicit K-bases for A, using the combinatorial technique of mixed subdivisions of the Minkowski sum P := P ₁ + ... + P ⁿ.

Partially supported by NSF grant CCR-9258533 (NYI).

Partially supported by a David and Lucile Packard Fellowship and NSF grants DMS-9201453 and DMS-9258547 (NYI).