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Register a new userStrongly stable ideals and Hilbert polynomials
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Added by
Paolo Lella
Publication date
2014
fields
Informatics Engineering
and research's language is
English
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The texttt{StronglyStableIdeals} package for textit{Macaulay2} provides a method to compute all saturated strongly stable ideals in a given polynomial ring with a fixed Hilbert polynomial. A description of the main method and auxiliary tools is given.
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Let $(R,mathfrak m)$ be an analytically unramified local ring of positive prime characteristic $p.$ For an ideal $I$, let $I^*$ denote its tight closure. We introduce the tight Hilbert function $H^*_I(n)=ell(R/(I^n)^*)$ and the corresponding tight Hilbert polynomial $P_I^*(n)$ where $I$ is an $mathfrak m$-primary ideal. It is proved that $F$-rationality can be detected by the vanishing of the first coefficient of $P_I^*(n).$ We find the tight Hilbert polynomial of certain parameter ideals in hypersurface rings and Stanley-Reisner rings of simplicial complexes.
We will define the Alexander duality for strongly stable ideals. More precisely, for a strongly stable ideal $I subset Bbbk[x_1, ldots, x_n]$ with ${rm deg}(mathsf{m}) le d$ for all $mathsf{m} in G(I)$, its dual $I^* subset Bbbk[y_1, ldots, y_d]$ is a strongly stable ideal with ${rm deg}(mathsf{m}) le n$ for all $mathsf{m} in G(I^*)$. This duality has been constructed by Fl$o$ystad et al. in a different manner, so we emphasis applications here. For example, we will describe the Hilbert serieses of the local cohomologies $H_mathfrak{m}^i(S/I)$ using the irreducible decomposition of $I$ (through the Betti numbers of $I^*$).
Let J be a strongly stable monomial ideal in S=K[x_1,...,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 Groebner basis and its constructive capabilities. Indeed, although not every J-marked basis is a Groebner basis with respect to some term order, a sort of normal form modulo I (with the ideal I in 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 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 Groebner bases about syzygies.
In this paper we find the tight closure of powers of parameter ideals of certain diagonal hypersurface rings. In many cases the associated graded ring with respect to tight closure filtration turns out to be Cohen-Macaulay. This helps us find the tight Hilbert polynomial in these diagonal hypersurfaces. We determine the tight Hilbert polynomial in the following cases: (1) F-pure diagonal hypersurfaces where number of variables is equal to the degree of defining equation, (2) diagonal hypersurface rings where characteristic of the ring is one less than the degree of defining equation and (3) quartic diagonal hypersurface in four variables.
We propose a new algorithm for multiplying dense polynomials with integer coefficients in a parallel fashion, targeting multi-core processor architectures. Complexity estimates and experimental comparisons demonstrate the advantages of this new approach.
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