Authors: Enrica Barrilli, Oriol Reig

Extension Varieties of Tensors

Let $T\in S_d$ be a symmetric tensor of order $d$ in $n$ variables that we want to decompose, and let $r$ be its rank. An extension variety parametrises the lifts of the tensor to a larger space, while keeping the rank unchanged. The lift can increase the order, by considering the space of tensors of order $d^{\prime }>d$ (a degree extension), or the number of variables, by considering the space of tensors in $n^{\prime }>n$ variables (a dimension extension). The motivation is that a decomposition which is hard to find for $T$, may become accessible from a suitable lift.

Let $K$ be an algebraically closed field of characteristic $0$, $S_d$ the space of homogeneous polynomials of degree $d$ in $x_0,\dots ,x_n$, identified with the symmetric tensors of order $d$, and $R=K[x_1,\dots ,x_n]$ with $R_d$ the polynomials of degree at most $d$. We look for

$$T=\sum _{i=1}^r{\omega }_i{\ell }_i^d,{\omega }_i\in K\setminus \{0\},{\ell }_i\in S_1,$$

with $r$ minimal. After a generic change of coordinates and scaling we may assume that ${\xi }_{i,0}=1$, so that

$${\ell }_i=(\mathbf{x},{\xi }_i)=x_0+x_1{\xi }_{i,1}+\cdots +x_n{\xi }_{i,n},{\xi }_i\in K^n.$$

Thanks to apolarity, $T$ corresponds to a functional ${\Lambda }_T\in R_d^{\ast }$ with ${\Lambda }_T({\mathbf{x}}^{\alpha })=a_{\alpha }{\left(\genfrac{}{}{0ex}{}{d}{\alpha }\right)}^{-1}$, where $a_{\alpha }$ is the coefficient of ${\mathbf{x}}^{\alpha }$ in $T$, and finding a decomposition of $T$ is equivalent to finding a decomposition of the dual element

$${\Lambda }_T=\sum _{i=1}^r{\omega }_i{\mathrm{ev}}_{{\xi }_i}.$$

Let

$$\rho :(R_{d^{\prime }}^{\prime }{)}^{\ast }\to (R_d{)}^{\ast },$$

where

$$R^{\prime }=K[x_1,\dots ,x_{n^{\prime }}]\supseteq R,n^{\prime }\ge n,d^{\prime }\ge d.$$

An extension of ${\Lambda }_T$ corresponds to an element $\tilde{\Lambda }\in R_{d^{\prime }}^{\prime \ast }$ such that $\rho (\tilde{\Lambda })={\Lambda }_T$ and its Hankel operator has the same rank $r$. A degree extension corresponds to $n^{\prime }=n$, $d^{\prime }>d$, and a dimension extension to $d^{\prime }=d$, $n^{\prime }>n$.

Hankel operators and extensions

To any $\Lambda \in R^{\ast }$ we associate the Hankel operator $H_{\Lambda }:R\to R^{\ast }$, $p\mapsto p\star \Lambda$, where $(p\star \Lambda )(q)=\Lambda (pq)$. Its kernel $I_{\Lambda }$ is an ideal, $A_{\Lambda }=R/I_{\Lambda }$ is Gorenstein, and ${\dim }_K{A}_{\Lambda }=\mathrm{rank}{H}_{\Lambda }$. We know the first terms of $\Lambda$, namely the moments $h_{\alpha }={\Lambda }_T({\mathbf{x}}^{\alpha })$ for $|\alpha |\le d$, and we look for the remaining ones.

Fix a set $B=\{{\mathbf{x}}^{{\beta }_1},\dots ,{\mathbf{x}}^{{\beta }_r}\}$ of monomials connected to $1$, i.e. each $m\in B$ is $1$ or $m=x_j{m}^{\prime }$ with $m^{\prime }\in B$. Put $B^{+}=B\cup x_{1}B\cup \cdots \cup x_{n}B$.

We consider the formal Hankel matrix

$$H_{\Lambda }^B(\mathbf{h})=(h_{\alpha +\beta }{)}_{\alpha ,\beta \in B},$$

where $h_{\gamma }\text{ is the given value if }|\gamma |\le d$ and a new unknown otherwise, and, when $\det H_{\Lambda }^B(\mathbf{h})\ne 0$, the formal multiplication matrices

$$M_i(\mathbf{h})=(H_{\Lambda }^B(\mathbf{h}){)}^{-1}{H}_{x_i\star \Lambda }^B(\mathbf{h}),1\le i\le n.$$

The constraints on the unknown moments come from commutation [BCMT, Thm. 4.2]: there exists an extension $\tilde{\Lambda }\in R^{\ast }$ with $\mathrm{rank}{H}_{\tilde{\Lambda }}=r$ and $B$ a basis of $A_{\tilde{\Lambda }}$ if and only if $\det H_{\Lambda }^B(\mathbf{h})\ne 0$ and

$$M_i(\mathbf{h})M_j(\mathbf{h})-M_j(\mathbf{h})M_i(\mathbf{h})=0,1\le i<j\le n,$$

and then $\tilde{\Lambda }$ is unique. This is a polynomial system of degree at most $2$ in the unknown moments. An equivalent description asks that all the $(r+1)\times (r+1)$ minors of $H_{\Lambda }^{B^{+}}(\mathbf{h})$ vanish.

Since $H_{x_i\star \Lambda }=M_{x_i}^t\circ H_{\Lambda }$, the points of $\tilde{\Lambda }$ are given by the common eigenvectors of the multiplication-by- $x_i$ operators $M_{x_i}$, whose eigenvalues are the coordinates ${\xi }_{j,i}$ of the points. If the joint eigenvalues are $r$ distinct points, then $I_{\tilde{\Lambda }}$ is radical, and the weights ${\omega }_j$ are obtained from the linear system ${\Lambda }_T={\sum }_j{\omega }_j{\mathrm{ev}}_{{\xi }_j}$ on $R_d$. Otherwise, it gives a generalized additive decomposition of length $r$ [BMT].

A dimension extension lifts $T\in S_d$ to a tensor in $n^{\prime }>n$ variables, by the same principle: we look for $\tilde{\Lambda }$ on $R^{\prime }=K[x_1,\dots ,x_{n^{\prime }}]$ with $\rho (\tilde{\Lambda })={\Lambda }_T$ and $\mathrm{rank}{H}_{\tilde{\Lambda }}=r$, and the extension variety is the set of all such $\tilde{\Lambda }$.

References

[BCMT] J. Brachat, P. Comon, B. Mourrain, E. Tsigaridas, Symmetric tensor decomposition, arXiv:0901.3706, 2009.

[BMT] E. Barrilli, B. Mourrain, D. Taufer, Generalized additive decompositions of symmetric tensors, arXiv:2510.25681, 2025.