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TitleFlat families by strongly stable ideals and a generalization of Gröbner bases
Author(s) Francesca Cioffi, margherita Roggero
TypeArticle in Journal
AbstractLet J be a strongly stable monomial ideal in S = K[x_0, , x_n] and let Mf(J) be the family of all homogeneous ideals I in S such that the set of all terms outside J is a K -vector basis of the quotient S/I. We show that an ideal I belongs to Mf(J) if and only if it is generated by a special set of polynomials, the J -marked basis of  I , that in some sense generalizes the notion of reduced Gröbner basis and its constructive capabilities. Indeed, although not every J -marked basis is a Gröbner basis with respect to some term order, a sort of reduced form modulo I ∈ Mf(J) can be computed for every homogeneous polynomial, so that a J -marked basis can be characterized by a Buchberger-like criterion. Using J -marked bases, we prove that the family Mf(J) can be endowed, in a very natural way, with a structure of an affine scheme that turns out to be homogeneous with respect to a non-standard grading and flat in the origin (the point corresponding to J ), thanks to properties of J -marked bases analogous to those of Gröbner bases about syzygies.
KeywordsFamily of schemes, Strongly stable ideal, Gröbner basis, Flatness
URL http://www.sciencedirect.com/science/article/pii/S0747717111000800
JournalJournal of Symbolic Computation
Pages1070 - 1084
Translation No
Refereed No