Arnold’s Piecewise Linear Filtrations, Analogues of Stanley–Reisner Rings and Simplicial Newton Polyhedra

Abstract

In 1974, the author proved that the codimension of the ideal (g1,g2,…,gd) generated in the group algebra K[Zd] over a field K of characteristic 0 by generic Laurent polynomials having the same Newton polytope Γ is equal to d!×Volume(Γ). 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!×Volume(Γ), whose equivalence classes form a basis of the quotient algebra K[Zd]/(g1,g2,…,gd). The set Bsh is constructed inductively from any shelling sh of the polytope Γ. Using the Bsh structure, we prove that the associated graded K -algebra grΓ(K[Zd]) constructed from the Arnold–Newton filtration of K -algebra K[Zd] 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 (f1,f2,…,fd) with the same Newton polytope Γ, the set Bsh defines a monomial basis of the quotient algebra K[Zd]/(g1,g2,…,gd).