Let a_1 <= ... <= a_n be positive integers. A monomial ideal L in a polynomial ring R=k[x_1,...,x_n] is called an (a_1,...a_n)-lex-plus-powers (LPP) ideal if it satisfies two conditions:
(1) L is minimally generated by x_1^{a_1}, ..., x_n^{a_n} and monomials m_1, ..., m_t. (2) Suppose that r is a monomial such that deg r = deg m_i, and r > m_i 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.
LPP ideals arise in conjectures of Eisenbud-Green-Harris and Charalambous-Evans in algebraic geometry, Hilbert functions, and graded free resolutions. They are conjectured to play a role analogous to that of lexicographic ideals in theorems of Macaulay on Hilbert functions and Bigatti, Hulett, and Pardue on resolutions.
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 |
The object LPP is a method function.