# 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.

## 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.

## 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*.

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

The tensors in section 5 come equipped with distinguished bases of A, B andC. 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 inlexicographical order. The files give each rank 1, one after the other. Arank 1 consists of 27 numbers. The first 9 give the linear combination ofai \otimes aj, the second 9 give the linear combination of the bi \otimes bj, andthe third 9 give the linear combination of the ci \otimes cj.