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Register a new userToric Border Bases
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Authors
Bernard Mourrain
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We extend the theory and the algorithms of Border Bases to systems of Laurent polynomial equations, defining toric roots. Instead of introducing new variables and new relations to saturate by the variable inverses, we propose a more efficient approach which works directly with the variables and their inverse. We show that the commutation relations and the inversion relations characterize toric border bases. We explicitly describe the first syzygy module associated to a toric border basis in terms of these relations. Finally, a new border basis algorithm for Laurent polynomials is described and a proof of its termination is given for zero-dimensional toric ideals.
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Given a multi-index sequence $$sigma$$, we present a new efficient algorithm to compute generators of the linear recurrence relations between the terms of $$sigma$$. We transform this problem into an algebraic one, by identifying multi-index sequences, multivariate formal power series and linear functionals on the ring of multivariate polynomials. In this setting, the recurrence relations are the elements of the kerne l$I$sigma$$ of the Hankel operator $H$sigma$$ associated to $$sigma$$. We describe the correspondence between multi-index sequences with a Hankel operator of finite rank and Artinian Gorenstein Algebras. We show how the algebraic structure of the Artinian Gorenstein algebra $A$sigma$$ associated to the sequence $$sigma$$ yields the structure of the terms $sigma$$alpha$ for all $$alpha$ $in$ N n$. This structure is explicitly given by a border basis of $A$sigma$$, which is presented as a quotient of the polynomial ring $K[x 1 ,. .. , xn$] by the kernel $I$sigma$$ of the Hankel operator $H$sigma$$. The algorithm provides generators of $I$sigma$$ constituting a border basis, pairwise orthogonal bases of $A$sigma$$ and the tables of multiplication by the variables in these bases. It is an extension of Berlekamp-Massey-Sakata (BMS) algorithm, with improved complexity bounds. We present applications of the method to different problems such as the decomposition of functions into weighted sums of exponential functions, sparse interpolation, fast decoding of algebraic codes, computing the vanishing ideal of points, and tensor decomposition. Some benchmarks illustrate the practical behavior of the algorithm.
In modeling physical systems it is sometimes useful to construct border bases of 0-dimensional polynomial ideals which are contained in the ideal generated by a given set of polynomials. We define and construct such subideal border bases, provide some basic properties and generalize a suitable variant of the Buchberger-Moeller algorithm as well as the AVI-algorithm to the subideal setting. The subideal version of the AVI-algorithm is then applied to an actual industrial problem.
Here we study the problem of generalizing one of the main tools of Groebner basis theory, namely the flat deformation to the leading term ideal, to the border basis setting. After showing that the straightforward approach based on the deformation to the degree form ideal works only under additional hypotheses, we introduce border basis schemes and universal border basis families. With their help the problem can be rephrased as the search for a certain rational curve on a border basis scheme. We construct the system of generators of the vanishing ideal of the border basis scheme in different ways and study the question of how to minimalize it. For homogeneous ideals, we also introduce a homogeneous border basis scheme and prove that it is an affine space in certain cases. In these cases it is then easy to write down the desired deformations explicitly.
The main topic of the paper is the construction of various explicit flat families of border bases. To begin with, we cover the punctual Hilbert scheme Hilb^mu(A^n) by border basis schemes and work out the base changes. This enables us to control flat families obtained by linear changes of coordinates. Next we provide an explicit construction of the principal component of the border basis scheme, and we use it to find flat families of maximal dimension at each radical point. Finally, we connect radical points to each other and to the monomial point via explicit flat families on the principal component.
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)$.
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