Mixed monomial bases
Paul S. Pedersen, Bernd Sturmfels · Birkhäuser Basel eBooks · 1996
Given a system of n generic Laurent polynomials 1.1 $$f_i (x) = \sum\limits_{q \in A_i } {c_{iq} x^q ;} \,\,\,\,\,q = (q{}_1,...,q_n );\,\,\,\,x^q = x_1^{q_1 } x_2^{q_2 } ...x_n^{q_n }$$ 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 .