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Register a new userFlat families by strongly stable ideals and a generalization of Groebner bases
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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.
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We describe the universal Groebner basis of the ideal of maximal minors and the ideal of $2$-minors of a multigraded matrix of linear forms. Our results imply that the ideals are radical and provide bounds on the regularity. In particular, the ideals of maximal minors have linear resolutions. Our main theoretical contribution consists of introducing two new classes of ideals named after Cartwright and Sturmfels, and proving that they are closed under multigraded hyperplane sections. The gins of the ideals that we study enjoy special properties.
We present an algorithm for computing Groebner bases of vanishing ideals of points that is optimized for the case when the number of points in the associated variety is less than the number of indeterminates. The algorithm first identifies a set of essential variables, which reduces the time complexity with respect to the number of indeterminates, and then uses PLU decompositions to reduce the time complexity with respect to the number of points. This gives a theoretical upper bound for its time complexity that is an order of magnitude lower than the known one for the standard Buchberger-Moeller algorithm if the number of indeterminates is much larger than the number of points. Comparison of implementations of our algorithm and the standard Buchberger-Moeller algorithm in Macaulay 2 confirm the theoretically predicted speedup. This work is motivated by recent applications of Groebner bases to the problem of network reconstruction in molecular biology.
A contemporary and exciting application of Groebner bases is their use in computational biology, particularly in the reverse engineering of gene regulatory networks from experimental data. In this setting, the data are typically limited to tens of points, while the number of genes or variables is potentially in the thousands. As such data sets vastly underdetermine the biological network, many models may fit the same data and reverse engineering programs often require the use of methods for choosing parsimonious models. Grobner bases have recently been employed as a selection tool for polynomial dynamical systems that are characterized by maps in a vector space over a finite field. While there are numerous existing algorithms to compute Groebner bases, to date none has been specifically designed to cope with large numbers of variables and few distinct data points. In this paper, we present an algorithm for computing Grobner bases of zero-dimensional ideals that is optimized for the case when the number m of points is much smaller than the number n of indeterminates. The algorithm identifies those variables that are essential, that is, in the support of the standard monomials associated to a polynomial ideal, and computes the relations in the Groebner bases in terms of these variables. When n is much larger than m, the complexity is dominated by nm^3. The algorithm has been implemented and tested in the computer algebra system Macaulay 2. We provide a comparison of its performance to the Buchberger-Moeller algorithm, as built into the system.
Let $X$ be a set of points whose coordinates are known with limited accuracy; our aim is to give a characterization of the vanishing ideal $I(X)$ independent of the data uncertainty. We present a method to compute a polynomial basis $B$ of $I(X)$ which exhibits structural stability, that is, if $widetilde X$ is any set of points differing only slightly from $X$, there exists a polynomial set $widetilde B$ structurally similar to $B$, which is a basis of the perturbed ideal $ I(widetilde X)$.
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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