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 Γ is equal to d ! × V o l u m e ( 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 B s h of monomials of cardinality d ! × V o l u m e ( Γ ) , whose equivalence classes form a basis of the quotient algebra K [ Z d ] / ( g 1 , g 2 , … , g d ) . The set B s h is constructed inductively from any shelling s h of the polytope Γ . Using the B s h structure, we prove that the associated graded K -algebra g r Γ ( 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 Γ , the set B s h defines a monomial basis of the quotient algebra K [ Z d ] / ( g 1 , g 2 , … , g d ) .