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Computing images of polynomial maps

Abstract

The image of a polynomial map is a constructible set. While computing its closure is standard in computer algebra systems, a procedure for computing the constructible set itself is not. We provide a new algorithm, based on algebro-geometric techniques, addressing this problem. We also apply these methods to answer a question of W. Hackbusch on the non-closedness of site-independent cyclic matrix product states for infinitely many parameters.

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Acknowledgments

Open access funding provided by Max Planck Society. We thank Wolfgang Hackbusch for posing the question which motivated this work and for the stimulating discussions. We are grateful to Bernd Sturmfels and Michael Joswig for many suggestions and encouraging remarks.

Funding

MM was supported by Polish National Science Center project 2013/08/A/ST1/00804 affiliated at the University of Warsaw.

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Correspondence to Mateusz Michałek.

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Communicated by: Ivan Oseledets

Appendix: Matrix product states

Appendix: Matrix product states

We recall here two representations of tensors that are inspired from physics [23]. The corresponding notions have found applications in deep learning where they are called tensor train or tensor ring decompositions [22, 28, 29, 32].

The symbol \(\mathbb {K}\) denotes the field \(\mathbb {C}\) or \(\mathbb {R}\). For any \(a \in \mathbb {Z}_{>0}\), the vector space \(\mathbb {K}^{a}\) comes with the standard basis \(e_{1},\dots ,e_{a}\). Therefore, a tensor \(T\in \mathbb {K}^{a_{1}}\times \dots \times \mathbb {K}^{a_{q}}\) may be represented as

$$ T=\sum\limits_{1\leq i_{j}\leq a_{j}} \lambda_{i_{1},\dots,i_{q}} e_{i_{1}}\otimes\dots\otimes e_{i_{q}}, $$

which is also written

$$ T[i_{1},\dots,i_{q}]=\lambda_{i_{1},\dots,i_{q}}. $$

Definition A.1 (Site-independent (cyclic) matrix product state)

Fix integers r > 0, k > 0, q > 1, and matrices \(M_{i}\in \mathbb {K}^{r\times r}\) for \(i=1,\dots ,k\). Let \(T\in (\mathbb {K}^{k})^{\otimes q}\) be a tensor given by

$$ T[i_{1},\dots,i_{q}]:=\text{tr}(M_{i_{1}}M_{i_{2}}{\cdots} M_{i_{q}}). $$

The set of all tensors that allow such a representation will be denoted by \(\text {IMPS}(r,k,q)\subset (\mathbb {K}^{k})^{\otimes q}\).

Example A.1

Let us consider the case of matrices (q = 2). Here elements of IMPS(r, k, 2) can be viewed as matrices M such that \(M[i_{1},i_{2}]=\text {tr}(M_{i_{1}}M_{i_{2}})\). This is equivalent to a factorization of M = AAt for some matrix \(A\in \text {Hom}(\mathbb {K}^{r},\mathbb {K}^{k^{2}})\). In particular, M ∈IMPS(r, k, 2) if and only if M is symmetric and has ranked at most k2. It follows that IMPS(r, k, 2) is closed.

When q = 2, the tensor T corresponds to a symmetric matrix. However, for q > 2, the tensor T will not be a symmetric tensor in general, though the identity \(T[i_{1},\dots ,i_{q}]=T[i_{q},i_{1},\dots ,i_{q-1}]\) continues to hold. In other words, the tensor has cyclic symmetries with respect to the order of the product of the matrices.

Definition A.1 can be regarded as a symmetrization of the following definition of a cyclic matrix product state, where the underlying graph for the tensor network is a cycle.

Fix an integer q > 1 and tuples of positive integers \(\mathfrak {a} = (a_{1},\dots ,a_{q})\), \(\mathfrak {b}=(b_{1},\dots ,b_{q})\). We set aq+ 1 = a1. Then the locus \(\text {MPS}(\mathfrak {a},\mathfrak {b},q) \subset \mathbb {K}^{b_{1}}\otimes {\dots } \otimes \mathbb {K}^{b_{q}}\) is given by the following definition.

Definition A.3 (Cyclic matrix product state)

A tensor \(T \in \mathbb {K}^{b_{1}}\otimes {\dots } \otimes \mathbb {K}^{b_{q}}\) is in \(\text {MPS}(\mathfrak {a},\mathfrak {b},q)\) if there exist matrices

$$ M_{i,j}\in \text{Hom}_{\mathbb{K}}(\mathbb{K}^{a_{j}},\mathbb{K}^{a_{j+1}}),\quad j=1,\dots,q, i = 1,\dots,b_{j}, $$

such that

$$ T[i_{1},\dots,i_{q}]:=\text{tr}(M_{i_{1},1}M_{i_{2},2}{\cdots} M_{i_{q},q}). $$

Example A.4

The situation for q = 2 is analogous to Example A.2. In this case, we have M ∈MPS((a1, a2), (b1, b2), 2) if and only if M = AB where \(A\in \text {Hom}({\mathbb {K}^{b}_{1}},\mathbb {K}^{a_{1}a_{2}})\) and \(B\in \text {Hom}(\mathbb {K}^{a_{1}a_{2}},\mathbb {K}^{b_{2}})\). This can happen if and only if the rank of the matrix M is at most a1a2. Therefore, \(\text {MPS}(\mathfrak {a},\mathfrak {b},2)\) is always closed.

Proposition A.5

The sets IMPS and MPS may be represented as

  1. 1.

    \(\text {IMPS}(r,k,q)=\{f^{\otimes q}(\mathfrak {M}_{r,\dots ,r}) \mid f\in \text {Hom}(\mathbb {K}^{r\times r},\mathbb {K}^{k})\}\).

  2. 2.

    \(\text {MPS}(\mathfrak {a},\mathfrak {b},q)=\{(f_{1}\otimes {\dots } \otimes f_{q})(\mathfrak {M}_{a_{1},\dots ,a_{q}}) \mid f_{i} \in \text {Hom}(\mathbb {K}^{a_{i} \times a_{i+1}},\mathbb {K}^{b_{i}})\}\).

Proof

The proofs of both statements are similar. We prove the first one, as it is more important for this paper. We will be interpreting elements of \(\text {Hom}(\mathbb {K}^{r\times r},\mathbb {K}^{k})\) as r2 × k matrices. First, we note that there is a natural bijection φ between k-tuples of r × r matrices \(\mathcal {M}:=(A_{1},\dots ,A_{k})\) and matrices \(\varphi (\mathcal {M})\in \text {Hom}(\mathbb {K}^{r\times r},\mathbb {K}^{k})\). For 1 ≤ ik, the i th column of \(\varphi (\mathcal {M})\) is the representation of Ai as a vector of length r2.

Write \(M_{i}={\sum }_{p,q=1}^{r} a_{i,p,q} e_{p,q}\), where ep, q is the matrix with 1 in its (p, q)th entry and zeros everywhere else. Note that \(\varphi (\mathcal {M})(e_{p,q})=a_{i,p,q}\).

We prove the claim by showing that the tensor T ∈IMPS(r, k, q) associated to \(\mathcal {M}\) equals \(\varphi (\mathcal {M})(\mathfrak {M}_{r,\dots ,r})\). Indeed, we have

$$ \begin{array}{@{}rcl@{}} T &=&\sum\limits_{1\leq i_{j}\leq k} \text{tr}(M_{i_{1}}{\cdots} M_{i_{q}}) e_{i_{1}}\otimes\dots\otimes e_{i_{q}} \\ &=& \sum\limits_{1\leq i_{j}\leq k}\left( \sum\limits_{1\leq p_{j}\leq r} a_{i_{1},p_{1},p_{2}}a_{i_{2},p_{2},p_{3}}\cdots a_{i_{q-1},p_{q-1},p_{q}}a_{i_{q},p_{q},p_{1}}\right)e_{i_{1}}\otimes\dots\otimes e_{i_{q}}, \end{array} $$

where in all sums 1 ≤ jq. We can simplify further:

$$ \begin{array}{@{}rcl@{}} T&= &\sum\limits_{\begin{array}{llll}1\leq i_{j}\leq k \\ 1\leq p_{j}\leq r \end{array}}(a_{i_{1},p_{1},p_{2}}e_{i_{1}})\otimes\cdots\otimes (a_{i_{q},p_{q},p_{1}}e_{i_{q}}) \\ &=&\sum\limits_{1\leq p_{j}\leq r}\left( \sum\limits_{1\leq i_{1}\leq k}a_{i_{1},p_{1},p_{2}}e_{i_{1}}\right)\otimes\cdots\otimes \left( \sum\limits_{1\leq i_{q}\leq k}a_{i_{q},p_{q},p_{1}}e_{i_{q}}\right) \\ &=&\sum\limits_{1\leq p_{j}\leq r}\varphi(\mathcal{M})(e_{p_{1},p_{2}})\otimes\cdots\otimes\varphi(\mathcal{M})(e_{p_{q},p_{1}}) \\ &=&\varphi(\mathcal{M})^{\otimes q}(\mathfrak{M}_{r,\dots,r}). \end{array} $$

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Harris, C., Michałek, M. & Sertöz, E.C. Computing images of polynomial maps. Adv Comput Math 45, 2845–2865 (2019). https://doi.org/10.1007/s10444-019-09715-8

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Keywords

  • Polynomial maps
  • Constructible set
  • Matrix product states

Mathematics Subject Classification (2010)

  • Primary 14Q15
  • Secondary 68U05
  • 15A69