The locus $uMPS(D,d,N)$ of uniform matrix product states is defined as the constructible image of the polynomial map $$ (\mathbb{C}^{D \times D})^d \to (\mathbb{C}^d)^{\otimes N}, (M_0, \ldots, M_{d-1}) \mapsto \sum_{0 \leq i_1, \ldots, i_N \leq d-1} tr(M_{i_1} \cdots M_{i_N}) e_{i_1} \otimes \cdots \otimes e_{i_N} \mbox{.}$$¶

Here we consider the case $(D,d,N) = (2,2,3)$.¶

Our implementation $\mathtt{ConstructibleImage}$ finishes in less than 15 seconds and outperforms

$\mathtt{totalImage}$ (Macaulay2 package $\mathtt{TotalImage}$ [HMS18,CMS])
$\mathtt{grobcov}$ (Singular package $\mathtt{grobcov.lib}$ [MW10])
$\mathtt{PolynomialMapImage}$ ($\mathsf{Maple}$ package $\mathtt{RegularChains}$ [CGL+07])
$\mathtt{Comprehensive}$ ($\mathsf{Maple}$ package $\mathtt{AlgebraicThomas}$ [BGLHR12])

[HMS18] Corey Harris, Mateusz Michałek, and Emre Can Sertöz, Computing images of polynomial maps, (arXiv:1801.00827), 2018.

[CMS] Adam Czaplin ́ski, Mateusz Michałek, and Tim Seynnaeve, Uniform matrix product states from an algebraic geometer’s point of view.

[MW10] Antonio Montes and Michael Wibmer, Gröbner bases for polynomial systems with parameters, J. Symbolic Comput. 45 (2010), no. 12, 1391–1425. MR 2733386

[CGL+07] Changbo Chen, Oleg Golubitsky, François Lemaire, Marc Moreno Maza, and Wei Pan, Com- prehensive triangular decomposition, Computer Algebra in Scientific Computing (Berlin, Heidelberg) (Victor G. Ganzha, Ernst W. Mayr, and Evgenii V. Vorozhtsov, eds.), Springer Berlin Heidelberg, 2007, pp. 73–101.

[BGLHR12] Thomas Bächler, Vladimir Gerdt, Markus Lange-Hegermann, and Daniel Robertz, Algorithmic Thomas decomposition of algebraic and differential systems, J. Symbolic Comput. 47 (2012), no. 10, 1233–1266, (arXiv:1108.0817). MR 2926124

In [1]:

using CapAndHomalg

CapAndHomalg v1.0.2
Imported OSCAR's components GAP and Singular_jll
Type: ?CapAndHomalg for more information

In [2]:

LoadPackage( "ZariskiFrames" )

In [3]:

UseSystemSingular(true)

Out[3]:

true

In [4]:

ℚ = HomalgFieldOfRationalsInSingular( )

Out[4]:

GAP: Q

In [5]:

B = ℚ["t000,t001,t011,t111"]

Out[5]:

GAP: Q[t000,t001,t011,t111]

In [6]:

T = ℚ["s_0..4"]

Out[6]:

GAP: Q[s_0,s_1,s_2,s_3,s_4]

In [7]:

ϕ = RingMap( HomalgMatrix( "-(1/2)*s_0^3+(3/2)*s_0*s_2,-(1/2)*s_0^2*s_1+(1/2)*s_1*s_2+s_0*s_3,-(1/2)*s_0*s_1^2+s_1*s_3+(1/2)*s_0*s_4,-(1/2)*s_1^3+(3/2)*s_1*s_4", Length( Indeterminates( B ) ), 1, T ),  B, T );

In [8]:

Display( ϕ )

Q[s_0,s_1,s_2,s_3,s_4]
  ^
  |
[ -1/2*s_0^3+3/2*s_0*s_2, -1/2*s_0^2*s_1+1/2*s_1*s_2+s_0*s_3, -1/2*s_0*s_1^2+s_1*s_3+1/2*s_0*s_4, -1/2*s_1^3+3/2*s_1*s_4 ]
  |
  |
Q[t000,t001,t011,t111]

In [9]:

im = ConstructibleImage( ϕ )

Out[9]:

GAP: ( V_{Q[t000,t001,t011,t111]}( I1 ) \ V_{Q[t000,t001,t011,t111]}( J1_1 ) )

In [10]:

Display( im )

V( <> ) \ ∅

In [11]:

Visualize( im )

In [12]:

Visualize( im, all = true )

In [13]:

imf = ConstructibleImage( ϕ, freeness = true )

Out[13]:

GAP: ( V_{Q[t000,t001,t011,t111]}( I1 ) \ V_{Q[t000,t001,t011,t111]}( J1_1 ) )

In [14]:

imf == im

Out[14]:

true

In [15]:

Display( imf )

V( <> ) \ ∅

In [16]:

Visualize( imf )

In [17]:

Visualize( imf, all = true )

In [ ]: