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Section: New Results

Algebraic representations for geometric modeling

Fitting ideals and multiple-points of surface parameterizations

Participants : Nicolàs Botbol, Laurent Busé.

Given a birational parameterization $\phi$ of an algebraic surface $𝒮$ in the projective space ${\mathbb{P}}^3$, the purpose of this ongoing work is to investigate the sets of points $D_k\left(\phi \right)$ on $𝒮$ whose preimage consists in $k$ or more points, counting multiplicity. Our main result is an explicit description of these algebraic sets $D_k\left(\phi \right)$ in terms of Fitting ideals of some graded parts of a symmetric algebra associated to the parameterization $\phi$.

This work is done in collaboration with Marc Chardin (University Pierre et Marie Curie).

Algebraic geometry tools for the study of entanglement: an application to spin squeezed states

Participant : Alessandra Bernardi.

In [18] a short review of Algebraic Geometry tools for the decomposition of tensors and polynomials is given from the point of view of applications to quantum and atomic physics. Examples of application to assemblies of indistinguishable two-level bosonic atoms are discussed using modern formulations of the classical Sylvester's algorithm for the decomposition of homogeneous polynomials in two variables. In particular, the symmetric rank and symmetric border rank of spin squeezed states is calculated as well as their Schrödinger-cat-like decomposition as the sum of macroscopically different coherent spin states; Fock states provide an example of states for which the symmetric rank and the symmetric border rank are different.

This is a joint work with I. Carusotto (University of Trento, Italy).

A partial stratification of secant varieties of Veronese varieties via curvilinear subschemes.

Participant : Alessandra Bernardi.

In [11] we give a partial quasi-stratification of the secant varieties of the order $d$ Veronese variety $X_{m,d}$ of ${\mathbb{P}}^m$. It covers the set ${\sigma }_t{\left(X_{m,d}\right)}^{†}$ of all points lying on the linear span of curvilinear subschemes of $X_{m,d}$, but two quasi-strata may overlap. For low border rank, two different quasi-strata are disjoint and we compute the symmetric rank of their elements. Our tool is the Hilbert schemes of curvilinear subschemes of Veronese varieties. To get a stratification we attach to each $P\in {\sigma }_t{\left(X_{m,d}\right)}^{†}$ the minimal label of a quasi-stratum containing it.

This is a joint work with E. Ballico (University of Trento, Italy).

Decomposition of homogeneous polynomials with low rank.

Participant : Alessandra Bernardi.

Let $F$ be a homogeneous polynomial of degree $d$ in $m+1$ variables defined over an algebraically closed field of characteristic 0 and suppose that $F$ belongs to the $s$-th secant variety of the $d$-uple Veronese embedding of ${\mathbb{P}}^m$ into ${\mathbb{P}}^{\left(\genfrac{}{}{0pt}{}{m+d}{d}\right)-1}$ but that its minimal decomposition as a sum of $d$-th powers of linear forms $M_1,...,M_r$ is $F=M_1^d+\cdots +M_r^d$ with $r>s$. In [12] , we show that if $s+r\le 2d+1$ then such a decomposition of $F$ can be split into two parts: one of them is made by linear forms that can be written using only two variables. The other part is uniquely determined once one has fixed the first part. We also obtain a uniqueness theorem for the minimal decomposition of $F$ if $r$ is at most $d$ and a mild condition is satisfied.

This is a joint work with E. Ballico (University of Trento, Italy).

Higher secant varieties of ${\mathbb{P}}^n\times {\mathbb{P}}^1$ embedded in bi-degree $(a,b)$

Participant : Alessandra Bernardi.

In [15] , we compute the dimension of all the higher secant varieties to the Segre-Veronese embedding of ${\mathbb{P}}^n\times {\mathbb{P}}^1$ via the section of the sheaf $𝒪(a,b)$ for any $n,a,b\in {\mathbb{Z}}^{+}$. We relate this result to the Grassmann Defectivity of Veronese varieties and we classify all the Grassmann $(1,s-1)$-defective Veronese varieties.

This is a joint work with E. Ballico, M. V. Catalisano (University of Trento, Italy).

Symmetric tensor rank with a tangent vector: a generic uniqueness theorem

Participant : Alessandra Bernardi.

Let $X_{m,d}\subset {\mathbb{P}}^N$, $N:=\left(\genfrac{}{}{0pt}{}{m+d}{m}\right)-1$, be the order $d$ Veronese embedding of ${\mathbb{P}}^m$. Let $\tau \left(X_{m,d}\right)\subset {\mathbb{P}}^N$, be the tangent developable of $X_{m,d}$. For each integer $t\ge 2$ let $\tau (X_{m,d},t)\subseteq {\mathbb{P}}^N$, be the join of $\tau \left(X_{m,d}\right)$ and $t-2$ copies of $X_{m,d}$. In [13] , we prove that if $m\ge 2$, $d\ge 7$ and $t\le 1+\lfloor \left(\genfrac{}{}{0pt}{}{m+d-2}{m}\right)/(m+1)\rfloor$, then for a general $P\in \tau (X_{m,d},t)$ there are uniquely determined $P_1,\cdots ,P_{t-2}\in X_{m,d}$ and a unique tangent vector $\nu$ of $X_{m,d}$ such that $P$ is in the linear span of $\nu \cup \{P_1,\cdots ,P_{t-2}\}$. In other words, a degree $d$ linear form $f$ (a symmetric tensor $T$ of order $d$) associated to $P$ may be written as

with $L_i$ linear forms on ${\mathbb{P}}^m$ ($v_i$ vectors over a vector field of dimension $m+1$ respectively), $1\le i\le t$, that are uniquely determined (up to a constant).

This is a joint work with E. Ballico (University of Trento, Italy).

General tensor decomposition, moment matrices and applications.

Participants : Alessandra Bernardi, Bernard Mourrain.

In [17] the tensor decomposition addressed 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.

This is a joint work with J. Brachat and P. Comon (i3S, CNRS).

On the cactus rank of cubic forms

Participant : Alessandra Bernardi.

In this work, we prove that the smallest degree of an apolar 0-dimensional scheme of a general cubic form in $n+1$ variables is at most $2n+2$, when $n\ge 8$, and therefore smaller than the rank of the form. For the general reducible cubic form the smallest degree of an apolar subscheme is $n+2$, while the rank is at least $2n$.

This is a joint work with K. Ranestad (University of Oslo, Norway) that will be published in 2013 in the Journal of Symbolic Computation. The preprint is available at http://hal.inria.fr/inria-00630456 .

Tensor ranks on tangent developable of Segre varieties

Participant : Alessandra Bernardi.

In [14] we describe the stratification by tensor rank of the points belonging to the tangent developable of any Segre variety. We give algorithms to compute the rank and a decomposition of a tensor belonging to the secant variety of lines of any Segre variety. We prove Comon's conjecture on the rank of symmetric tensors for those tensors belonging to tangential varieties to Veronese varieties.

This is a joint work with E. Ballico (University of Trento, Italy).

On the dimension of spline spaces on planar T-meshes

Participant : Bernard Mourrain.

In [33] , we analyze the space of bivariate functions that are piecewise polynomial of bi-degree $\le (m,m^{\text{'}})$ and of smoothness $r$ along the interior edges of a planar T-mesh. We give new combinatorial lower and upper bounds for the dimension of this space by exploiting homological techniques. We relate this dimension to the weight of the maximal interior segments of the T-mesh, defined for an ordering of these maximal interior segments. We show that the lower and upper bounds coincide, for high enough degrees or for hierarchical T-meshes which are enough regular. We give a rule of subdivision to construct hierarchical T-meshes for which these lower and upper bounds coincide. Finally, we illustrate these results by analyzing spline spaces of small degrees and smoothness.

On the problem of instability in the dimension of a spline space over a T-mesh

Participant : Bernard Mourrain.

In [23] , we discuss the problem of instability in the dimension of a spline space over a T-mesh. For bivariate spline spaces $S(5,5,3,3)$ and $S(4,4,2,2)$, the instability in the dimension is shown over certain types of T-meshes. This result could be considered as an attempt to answer the question of how large the polynomial degree $(m,m\prime )$ should be relative to the smoothness $(r,r\prime )$ to make the dimension of a spline space stable. We show in particular that the bound $m\ge 2r+1$ and $m\prime \ge 2r\prime +1$ are optimal.

This is a joint work with Berdinsky Dmitry, Oh Min-Jae and Kim Taewan (Department of Naval Architecture and Ocean Engineering Seoul National University, South Korea).

Homological techniques for the analysis of the dimension of triangular spline spaces

Participant : Bernard Mourrain.

The spline space $C_k^r\left(\Delta \right)$ 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 grids 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. In [34] , we address the problem of determining the dimension of the space of bivariate splines $C_k^r\left(\Delta \right)$ 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. Furthermore, in certain cases even an exact value can be found. The construction makes it also possible to get a short proof for the dimension formula when $k\ge 4r+1$, and the same method we use in this proof yields the dimension straightaway for many other cases.

This is a joint work with Nelly Villamizar (CMA, University of Oslo, Norway).

Analysis-suitable volume parameterization of multi-block computational domain in isogeometric applications

Participants : Bernard Mourrain, André Galligo.

Parameterization of computational domain is a key step in isogeometric analysis just as mesh generation is in finite element analysis. In [36] , we study the volume parameterization problem of multi-block computational domain in isogeometric version, i.e., how to generate analysis-suitable parameterization of the multi-block computational domain bounded by B-spline surfaces. Firstly, we show how to find good volume parameterization of single-block computational domain by solving a constraint optimization problem, in which the constraint condition is the injectivity sufficient conditions of B-spline volume parametrization, and the optimization term is the minimization of quadratic energy functions related to the first and second derivatives of B-spline volume parameterization. By using this method, the resulted volume parameterization has no self-intersections, and the isoparametric structure has good uniformity and orthogonality. Then we extend this method to the multi-block case, in which the continuity condition between the neighbor B-spline volume should be added to the constraint term. The effectiveness of the proposed method is illustrated by several examples based on three-dimensional heat conduction problem.

This is a joint work with Régis Duvigneau (Inria, EPI OPALE) and Xu Gang (College of computer - Hangzhou Dianzi University, China).

A new error assessment method in isogeometric analysis of 2D heat conduction problems

Participants : Bernard Mourrain, André Galligo.

In [35] , we propose a new error assessment method for isogeometric analysis of 2D heat conduction problems. A posteriori error estimation is obtained by resolving the isogeometric analysis problem with several $k$-refinement steps. The main feature of the proposed method is that the resulted error estimation surface has a B-spline form, according to the main idea of isogeometric analysis. Though the error estimation method is expensive, it can be used as an error assessment method for isogeometric analysis. Two comparison examples are presented to show the efficiency of the proposed method.

This is a joint work with Régis Duvigneau (Inria, EPI OPALE) and Xu Gang (College of computer - Hangzhou Dianzi University, China).

On the cut-off phenomenon for the transitivity of randomly generated subgroups

Participant : André Galligo.

Consider $K\ge 2$ independent copies of the random walk on the symmetric group $S_N$ starting from the identity and generated by the products of either independent uniform transpositions or independent uniform neighbor transpositions. At any time $n\in \mathbb{N}$, let $G_n$ be the subgroup of $S_N$ generated by the $K$ positions of the chains. In the uniform transposition model, we prove in [28] that there is a cut-off phenomenon at time $Nln\left(N\right)/\left(2K\right)$ for the non-existence of fixed point of $G_n$ and for the transitivity of $G_n$, thus showing that these properties occur before the chains have reached equilibrium. In the uniform neighbor transposition model, a transition for the non-existence of a fixed point of $G_n$ appears at time of order $N^{1+\frac{2}{K}}$ (at least for $K\ge 3$), but there is no cut-off phenomenon. In the latter model, we recover a cut-off phenomenon for the non-existence of a fixed point at a time proportional to $N$ by allowing the number $K$ to be proportional to $ln\left(N\right)$. The main tools of the proofs are spectral analysis and coupling techniques.

This is a joint work with Laurent Miclo (University of Toulouse).