The main result of the paper is a flat extension theorem for positive linear functionals on *-algebras. The theorem is applied to truncated moment problems on cylinder sets, on matrices of polynomials and on enveloping algebras of Lie algebras.
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 approac
h 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.
We introduce various notions of rank for a symmetric tensor, namely: rank, border rank, catalecticant rank, generalized rank, scheme length, border scheme length, extension rank and smoothable rank. We analyze the stratification induced by these rank
s. The mutual relations between these stratifications, allow us to describe the hierarchy among all the ranks. We show that strict inequalities are possible between rank, border rank, extension rank and catalecticant rank. Moreover we show that scheme length, generalized rank and extension rank coincide.
The spline space $C_k^r(Delta)$ attached to a subdivided domain $Delta$ of $R^{d} $ is the vector space of functions of class $C^{r}$ which are polynomials of degree $le k$ on each piece of this subdivision. Classical splines on planar rectangular gr
ids play an important role in Computer Aided Geometric Design, and spline spaces over arbitrary subdivisions of planar domains are now considered for isogeometric analysis applications. We address the problem of determining the dimension of the space of bivariate splines $C_k^r(Delta)$ for a triangulated region $Delta$ in the plane. Using the homological introduced by Billera (1988), we number the vertices and establish a formula for an upper bound on the dimension. There is no restriction on the ordering and we obtain more accurate approximations to the dimension than previous methods and furthermore, in certain cases even an exact value can be found. The construction makes also possible to get a short proof for the dimension formula when $kge 4r+1$, and the same method we use in this proof yields the dimension straightaway for many other cases.
The tensor decomposition addressed in this paper may be seen as a generalisation of Singular Value Decomposition of matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix
problem and give a new criterion for flat extension of Quasi-Hankel matrices. We connect this criterion to the commutation characterisation of border bases. A new algorithm is described. It applies for general multihomogeneous tensors, extending the approach of J.J. Sylvester to binary forms. An example illustrates the algebraic operations involved in this approach and how the decomposition can be recovered from eigenvector computation.
In this paper, we give new explicit representations of the Hilbert scheme of $mu$ points in $PP^{r}$ as a projective subvariety of a Grassmanniann variety. This new explicit description of the Hilbert scheme is simpler than the previous ones and glob
al. It involves equations of degree $2$. We show how these equations are deduced from the commutation relations characterizing border bases. Next, we consider infinitesimal perturbations of an input system of equations on this Hilbert scheme and describe its tangent space. We propose an effective criterion to test if it is a flat deformation, that is if the perturbed system remains on the Hilbert scheme of the initial equations. This criterion involves in particular formal reduction with respect to border bases.
Let $f_1,...,f_s in mathbb{K}[x_1,...,x_m]$ be a system of polynomials generating a zero-dimensional ideal $I$, where $mathbb{K}$ is an arbitrary algebraically closed field. We study the computation of matrices of traces for the factor algebra $A :=
CC[x_1, ..., x_m]/ I$, i.e. matrices with entries which are trace functions of the roots of $I$. Such matrices of traces in turn allow us to compute a system of multiplication matrices ${M_{x_i}|i=1,...,m}$ of the radical $sqrt{I}$. We first propose a method using Macaulay type resultant matrices of $f_1,...,f_s$ and a polynomial $J$ to compute moment matrices, and in particular matrices of traces for $A$. Here $J$ is a polynomial generalizing the Jacobian. We prove bounds on the degrees needed for the Macaulay matrix in the case when $I$ has finitely many projective roots in $mathbb{P}^m_CC$. We also extend previous results which work only for the case where $A$ is Gorenstein to the non-Gorenstein case. The second proposed method uses Bezoutian matrices to compute matrices of traces of $A$. Here we need the assumption that $s=m$ and $f_1,...,f_m$ define an affine complete intersection. This second method also works if we have higher dimensional components at infinity. A new explicit description of the generators of $sqrt{I}$ are given in terms of Bezoutians.
In this note we prove a generalization of the flat extension theorem of Curto and Fialkow for truncated moment matrices. It applies to moment matrices indexed by an arbitrary set of monomials and its border, assuming that this set is connected to 1.
When formulated in a basis-free setting, this gives an equivalent result for truncated Hankel operators.
This paper describes and analyzes a method for computing border bases of a zero-dimensional ideal $I$. The criterion used in the computation involves specific commutation polynomials and leads to an algorithm and an implementation extending the one p
rovided in [MT05]. This general border basis algorithm weakens the monomial ordering requirement for grob bases computations. It is up to date the most general setting for representing quotient algebras, embedding into a single formalism Grobner bases, Macaulay bases and new representation that do not fit into the previous categories. With this formalism we show how the syzygies of the border basis are generated by commutation relations. We also show that our construction of normal form is stable under small perturbations of the ideal, if the number of solutions remains constant. This new feature for a symbolic algorithm has a huge impact on the practical efficiency as it is illustrated by the experiments on classical benchmark polynomial systems, at the end of the paper.
