Arnold's Piecewise Linear Filtrations, Analogues of Stanley-Reisner Rings and Simplicial Newton Polyhedra

In 1974, the author proved that the codimension of the ideal (g(1),g(2), ... ,g(d)) generated in the group algebra K[Z(d)] over a field K of characteristic 0 by generic Laurent polynomials having the same Newton polytope gamma is equal to d!xVolume(gamma). Assuming that Newtons polytope is simplicial and super-convenient (that is, containing some neighborhood of the origin), the author strengthens the 1974 result by explicitly specifying the set Bsh of monomials of cardinality d!xVolume(gamma), whose equivalence classes form a basis of the quotient algebra K[Z(d)]/(g(1),g(2), ... ,g(d)). The set B-sh is constructed inductively from any shelling sh of the polytope gamma. Using the B-sh structure, we prove that the associated graded K -algebra gr gamma(K[Z(d)]) constructed from the Arnold-Newton filtration of K -algebra K[Z(d)] has the Cohen-Macaulay property. This proof is a generalization of B. Kind and P. Kleinschmitt's 1979 proof that Stanley-Reisner rings of simplicial complexes admitting shelling are Cohen-Macaulay. Finally, we prove that for generic Laurent polynomials (f(1),f(2), ... ,f(d)) with the same Newton polytope gamma, the set Bsh defines a monomial basis of the quotient algebra K[Zd]/(g1,g2, ... ,gd).