Combinatorial decompositions for monomial ideals

It is well known that Riquier introduced the notion of multiplicative variables applied to order ideals to represent initial conditions of partial differential equations as series. On the other hands, Janet introduced the concept of involutive division. Following Riquier and Janet, we focus on the following problem. Suppose one needs to compute a monomial ideal generated in some degree D and its escalier, knowing the Hilbert function and some monomials which must belong to the ideal/escalier. We give combinatorial tools to answer this question. First we define combinatorial decompositions of sets of terms, showing the criteria to decompose the set T≥D of the terms in degree greater or equal than D, into disjoint subsets called cones. This is done by assigning to each term t of degree D some multiplicative variables. The cone of t is then the set formed by t and all its multiples obtained multiplying it by all possible products of powers of multiplicative variables. Then, supposed one has selected a decomposition, we deal with the escalier/ideal partition problem, namely we study the rules which force a term to be in the ideal (or in the escalier) provided that another term does, so that both the ideal and the escalier turn out to be decomposed in disjoint cones by the decomposition set on the whole T≥D.