### Supplementary Files for

###

# Kronecker Powers of Tensors and Strassen's Laser Method

### by A. Conner, F. Gesmundo, J. M. Landsberg, E. Ventura

We collect the supplementary files required to complete the computer
calculations described in the paper. The softwares we use are Sage and Macaulay2.
The files are collected in different appendices:
## Appendix A: A randomly chosen tensor in C3 x C3 x C3

This data file addresses the Claim of Theorem 2.13*. We provide the
coordinates of a randomly chosen tensor in C3 \otimes C3 \otimes C3 and
an approximate decomposition of its Kronecker square.

We include one file, with enclosed documentation. It contains the following two pieces of data:
the coordinates of a tensor T in C3 \otimes C3 \otimes C3,
represented as three matrices of size 3x3, M_1,M_2,M_3: in coordinates
T_ijk = (M_i)_jk. The entries of the matrices were randomly chosen from
the uniform distribution on [-1,1];
the coordinates of 22
rank one tensors in C9 \otimes C9 \otimes C9, forming a rank 22
decomposition of T \boxtimes T. This is a list of 22 elements
Z_1,...,Z_22. Each element Z_j is a list of 3 elements a,b,c. The
elements a,b,c are matrices of size 3x3 representing the coordinates of
three vectors in C9 = C3 \otimes C3. This is the same representation as
the one of Section 5 of the paper.

Download the file here.

## Appendix B: Rank 18 Decomposition of 3x3 determinant

This is a Macaulay2 file which verifies the Claim in the proof of
Theorem 2.11 (Section 5.1). It is one file, with enclosed
documentation.

Download the file here.

## Appendix C: Border Rank 17 Decomposition of 3x3 determinant

This
is a set of files addressing Claims made in the proof of Theorem 2.12
(Section 5.2). The set consists of 7 files, which can be downloaded in
an archive tar.gz file.

We include seven files with enclosed documentation. The role of each file is briefly explained below:
checkingType1eqns.m2: A Macaulay2 file verifying the equations of type 1, as explained in the paper;
checkingType2eqns.m2: A Macaulay2 file verifying the equations of type 2, as explained in the paper;
eqnsType1.m2: The list of equations of type 1, for easy access: they can be computed directly as well;
eqnsType2.m2: The list of equations of type 2, for easy access: they can be computed directly as well;
monomialsType1.m2: The monomials appearing in equations of type 1, with the corresponding expression in terms of the root y*;
monomialsType2.m2:
The monomials appearing in equations of type 2, with the corresponding
expression in terms of the chosen root u in each equation;
yy_exps.m2: The expressions of the elements y_j in terms of the root y*.

Download the archive file here.

For the reader's convenience, we include the minimal polynomials of the

roots z_i with an approximation of their value in

det_br_17_entries.txt on the border rank 17 decomposition.
## Appendix D: Matrices of Koszul flattenings of Coppersmith-Winograd powers

This
set of files addresses the Claims on the matrices representing the
Koszul flattenings of the Kronecker square and the Kronecker cube of
the small Coppersmith-Winograd tensor, in the proof of Theorem 2.1 and
Theorem 2.2 (Sections 4.5 and Section 4.6). The set consists of 3
files, which can be downloaded in an archive tar.gz file.

We include three files with enclosed documentation. The role of each file is briefly explained below:
cwsquare.txt: A data file for easy access to the four
matrices Phi_1, ... , Phi_4 describing the map on the multiplicity
spaces in the proof of Theorem 2.1;
cwcube.txt: A data file
for easy access to the eight matrices Psi_1, ... , Psi_8 describing the
map on the multiplicity spaces in the proof of Theorem 2.2;
cwmatrices.sage: A sage file: it generates the relevant matrices using the method

described in Section 7 of the paper, and computes their ranks

Download the archive file here.

## Appendix E: Tight tensors in C3 x C3 x C3

This is a Macaulay2
file addressing the Claims made in the proofs of Section 6.
Documentation is enclosed and a description of the various parts of the
file is given in the paper.

Download the file here.

files T9 T6 T5 T4 T3 T2

The tensors in section 5 come equipped with distinguished bases of A, B and

C. for a basis a1 a2, a3, I take for the tensor square the basis a1\ot a1,

a1\otimes a2, a1\otimes a3, a2\otimes a1, ... a3\otimes a3, that is, the indices taken in

lexicographical order. The files give each rank 1, one after the other. A

rank 1 consists of 27 numbers. The first 9 give the linear combination of

ai \otimes aj, the second 9 give the linear combination of the bi \otimes bj, and

the third 9 give the linear combination of the ci \otimes cj.