Supplementary Files for
Dimension of Tensor Network Varieties
by A. Bernardi, C. De Lazzari, F. Gesmundo
We collect the supplementary material required to complete the computer calculations described in Section 5 of the paper.
The calculations are performed in Macaulay2.
Lower bounds for the dimension of Tensor Network Varieties
The lower bound on the dimension is obtained by computing the rank of the differential
of the parametrization map Phi (technically, of the map bar(Phi) defined on the sum of homomorphism spaces)
defined in Section 2 at a random point of its domain.
The scripts compute the dimension of the image of the differential.
We provide two files, one for the case of three factors (cycle C3) and one for the case of four factors (cycle C4).
The user only needs to change the input nn at line 7 specifying the local dimensions they desire
Additional documentation is included in the files.
Proof of Thm. 5.2: Containment of TNS in Z
The inclusion is proved symbolically, showing that for a generic choice of local maps (X1,X2,X3),
the tensor T = (X1,X2,X3) ( T(C3,(2,2,2)) is contained in the variety Z
The script imposes the normalization described in the proof
then it computes the intersection L \cap sigma2
Additional documentation is included in the file.
Proof of Thm. 5.3: Evaluation of the Degree 6 Invariant
The invariant is evaluated symbolically, showing that for a generic choice of local maps (X1,X2,X3,X4),
it vanishes identically on the tensor T = (X1,X2,X3,X4) ( T(C4,(2,2,2,2))
The script imposes the normalization described in the proof
then it evaluates the invariant
Additional documentation is included in the file.