Let a1 <= ... <= an be positive integers. A monomial ideal L in a polynomial ring R=k[x1,...,xn] is called an (a1,...an)-lex-plus-powers (LPP) ideal if it satisfies two conditions:
(1) L is minimally generated by x1a1, ..., xnan and monomials m1, ..., mt. (2) Suppose that r is a monomial such that deg r = deg mi, and r > mi in the lex order. Then r is in L.
LPP ideals are generalizations of Artinian lexicographic ideals. Condition (2) represents the lex portion of the LPP ideal; monomials that are not powers of a variable must satisfy a lexicographic condition similar to what generators of lex ideals satisfy. An LPP ideal also contains powers of all the variables in weakly increasing order.
i1 : R=ZZ/32003[a..c]; |
i2 : LPP(R,{1,3,6,5,3},{3,3,4}) 3 3 4 2 2 2 2 2 3 o2 = ideal (a , b , c , a b, a c, a*b , a*b*c , b c ) o2 : Ideal of R |
i3 : LPP(R,{1,3,4,2,1},{2,3,5}) --an Artinian lex ideal 2 3 5 2 2 3 o3 = ideal (a , b , c , a*b, a*c , b c, b*c ) o3 : Ideal of R |
i4 : LPP(R,{1,3,4,2,1},{2,4,3}) --exponents not in weakly increasing order |
i5 : LPP(R,{1,3,4,2,1},{2,2,3}) --no LPP ideal with this Hilbert function and power sequence |