{
"cells": [
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## The locus $uMPS(D,d,N)$ of uniform matrix product states is defined as the constructible image of the polynomial map $$ (\\mathbb{C}^{D \\times D})^d \\to (\\mathbb{C}^d)^{\\otimes N}, (M_0, \\ldots, M_{d-1}) \\mapsto \\sum_{0 \\leq i_1, \\ldots, i_N \\leq d-1} tr(M_{i_1} \\cdots M_{i_N}) e_{i_1} \\otimes \\cdots \\otimes e_{i_N} \\mbox{.}$$\n",
"## Here we consider the case $(D,d,N) = (2,2,5)$.\n",
"\n",
"Our implementation $\\mathtt{ConstructibleImage}$ finishes in around 6 minutes and outperforms\n",
"* $\\mathtt{totalImage}$ (Macaulay2 package $\\mathtt{TotalImage}$ [HMS18,CMS])\n",
"* $\\mathtt{grobcov}$ (Singular package $\\mathtt{grobcov.lib}$ [MW10])\n",
"* $\\mathtt{PolynomialMapImage}$ ($\\mathsf{Maple}$ package $\\mathtt{RegularChains}$ [CGL+07])\n",
"* $\\mathtt{Comprehensive}$ ($\\mathsf{Maple}$ package $\\mathtt{AlgebraicThomas}$ [BGLHR12])\n",
"\n",
"which did not finish after a day.\n",
"\n",
"\n",
"[HMS18] Corey Harris, Mateusz Michałek, and Emre Can Sertöz, Computing images of polynomial maps, (arXiv:1801.00827), 2018.\n",
"\n",
"[CMS] Adam Czaplin ́ski, Mateusz Michałek, and Tim Seynnaeve, Uniform matrix product states from an algebraic geometer’s point of view.\n",
"\n",
"[MW10] Antonio Montes and Michael Wibmer, Gröbner bases for polynomial systems with parameters, J. Symbolic Comput. 45 (2010), no. 12, 1391–1425. MR 2733386\n",
"\n",
"[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.\n",
"\n",
"[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"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"CapAndHomalg v\u001b[32m1.0.0\u001b[39m\n",
"Imported OSCAR's components GAP and Singular_jll\n",
"Type: ?CapAndHomalg for more information\n"
]
}
],
"source": [
"using CapAndHomalgt"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"slideshow": {
"slide_type": "-"
}
},
"outputs": [],
"source": [
"LoadPackage( \"ZariskiFrames\" )"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"outputs": [
{
"data": {
"text/plain": [
"GAP: Q"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ℚ = HomalgFieldOfRationalsInSingular( )"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"GAP: Q[t00000,t00001,t00011,t00111,t01111,t11111,t00101,t11010]"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"B = ℚ[\"t00000,t00001,t00011,t00111,t01111,t11111,t00101,t11010\"]"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"GAP: Q[s_0,s_1,s_2,s_3,s_4]"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"T = ℚ[\"s_0..4\"]"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [],
"source": [
"ϕ = RingMap( HomalgMatrix( \"-1/4*s_0^5+5/4*s_0*s_2^2,-1/4*s_0^4*s_1+1/4*s_1*s_2^2+s_0*s_2*s_3,-1/4*s_0^3*s_1^2-1/4*s_0*s_1^2*s_2+1/2*s_0^2*s_1*s_3-1/4*s_0^3*s_4+1/2*s_1*s_2*s_3+3/4*s_0*s_2*s_4,-1/4*s_0^2*s_1^3-1/4*s_1^3*s_2+1/2*s_0*s_1^2*s_3-1/4*s_0^2*s_1*s_4+3/4*s_1*s_2*s_4+1/2*s_0*s_3*s_4,-1/4*s_0*s_1^4+s_1*s_3*s_4+1/4*s_0*s_4^2,-1/4*s_1^5+5/4*s_1*s_4^2,-1/4*s_0^3*s_1^2+1/4*s_0*s_1^2*s_2-1/2*s_0^2*s_1*s_3+1/4*s_0^3*s_4+1/2*s_1*s_2*s_3+s_0*s_3^2-1/4*s_0*s_2*s_4,-1/4*s_0^2*s_1^3+1/4*s_1^3*s_2-1/2*s_0*s_1^2*s_3+1/4*s_0^2*s_1*s_4+s_1*s_3^2-1/4*s_1*s_2*s_4+1/2*s_0*s_3*s_4\", Length( Indeterminates( B ) ), 1, T ), B, T );"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Q[s_0,s_1,s_2,s_3,s_4]\n",
" ^\n",
" |\n",
"[ -1/4*s_0^5+5/4*s_0*s_2^2, -1/4*s_0^4*s_1+1/4*s_1*s_2^2+s_0*s_2*s_3, -1/4*s_0^3*s_1^2-1/4*s_0*s_1^2*s_2+1/2*s_0^2*s_1*s_3-1/4*s_0^3*s_4+1/2*s_1*s_2*s_3+3/4*s_0*s_2*s_4, -1/4*s_0^2*s_1^3-1/4*s_1^3*s_2+1/2*s_0*s_1^2*s_3-1/4*s_0^2*s_1*s_4+3/4*s_1*s_2*s_4+1/2*s_0*s_3*s_4, -1/4*s_0*s_1^4+s_1*s_3*s_4+1/4*s_0*s_4^2, -1/4*s_1^5+5/4*s_1*s_4^2, -1/4*s_0^3*s_1^2+1/4*s_0*s_1^2*s_2-1/2*s_0^2*s_1*s_3+1/4*s_0^3*s_4+1/2*s_1*s_2*s_3+s_0*s_3^2-1/4*s_0*s_2*s_4, -1/4*s_0^2*s_1^3+1/4*s_1^3*s_2-1/2*s_0*s_1^2*s_3+1/4*s_0^2*s_1*s_4+s_1*s_3^2-1/4*s_1*s_2*s_4+1/2*s_0*s_3*s_4 ]\n",
" |\n",
" |\n",
"Q[t00000,t00001,t00011,t00111,t01111,t11111,t00101,t11010]\n"
]
}
],
"source": [
"Display( ϕ )"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"GAP: ( V_{Q[t00000,t00001,t00011,t00111,t01111,t11111,t00101,t11010]}( I1 ) \\ V_{Q[t00000,t00001,t00011,t00111,t01111,t11111,t00101,t11010]}( J1_1 ) ) ∪ ( V_{Q[t00000,t00001,t00011,t00111,t01111,t11111,t00101,t11010]}( I2 ) \\ V_{Q[t00000,t00001,t00011,t00111,t01111,t11111,t00101,t11010]}( J2_1 ) )"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"im = ConstructibleImage( ϕ, rbhull_decomposition = false )"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n",
"\n",
"\n",
"\n"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"VisualizeInJulia( im )"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n",
"\n",
"\n",
"\n"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"VisualizeInJulia( im, all = true )"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"scrolled": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"V( )\\\n",
" \\ V( <2*t00011^3*t11111-9*t00001*t00011*t00111*t11111+2*t00000*t00111^2*t1111\\\n",
"1+12*t00001^2*t01111*t11111-t00000*t00011*t01111*t11111-t00000*t00001*t11111^2\\\n",
"+3*t00111^3*t00101-11*t00011*t00111*t01111*t00101+3*t00001*t01111^2*t00101+12*\\\n",
"t00011^2*t11111*t00101-8*t00001*t00111*t11111*t00101-4*t00000*t01111*t11111*t0\\\n",
"0101-2*t00111*t01111*t00101^2+9*t00011*t11111*t00101^2+t11111*t00101^3-t00011*\\\n",
"t00111^2*t11010+t00011^2*t01111*t11010+16*t00001*t00111*t01111*t11010-8*t00000\\\n",
"*t01111^2*t11010-17*t00001*t00011*t11111*t11010+5*t00000*t00111*t11111*t11010-\\\n",
"t00111^2*t00101*t11010-2*t00011*t01111*t00101*t11010-9*t00001*t11111*t00101*t1\\\n",
"1010+t00001*t01111*t11010^2+7*t00000*t11111*t11010^2,5*t00001*t00111*t01111^2-\\\n",
"t00000*t01111^3-2*t00001*t00111^2*t11111-2*t00001*t00011*t01111*t11111+t00000*\\\n",
"t00111*t01111*t11111+t00001^2*t11111^2-3*t00111^2*t01111*t00101-t00011*t01111^\\\n",
"2*t00101+4*t00011*t00111*t11111*t00101-t00000*t11111^2*t00101+2*t01111^2*t0010\\\n",
"1^2+t00111*t11111*t00101^2+t00011*t00111*t01111*t11010-4*t00001*t01111^2*t1101\\\n",
"0-3*t00001*t00111*t11111*t11010+t00000*t01111*t11111*t11010+t00111*t01111*t001\\\n",
"01*t11010-2*t00011*t11111*t00101*t11010-2*t11111*t00101^2*t11010+4*t00001*t111\\\n",
"11*t11010^2,t00000*t00111*t01111^2-t00001*t00011*t00111*t11111+t00001^2*t01111\\\n",
"*t11111-t00111^3*t00101+2*t00011*t00111*t01111*t00101-t00001*t01111^2*t00101+t\\\n",
"00011^2*t11111*t00101-t00000*t01111*t11111*t00101+4*t00111*t01111*t00101^2-t11\\\n",
"111*t00101^3+t00011*t00111^2*t11010-t00011^2*t01111*t11010-t00001*t00111*t0111\\\n",
"1*t11010-t00000*t01111^2*t11010-t00001*t00011*t11111*t11010-3*t00111^2*t00101*\\\n",
"t11010-3*t00011*t01111*t00101*t11010+t00001*t11111*t00101*t11010+2*t00011*t001\\\n",
"11*t11010^2+t00001*t01111*t11010^2+t00000*t11111*t11010^2,5*t00011^2*t01111^2+\\\n",
"3*t00000*t01111^3-2*t00011^2*t00111*t11111+5*t00001*t00111^2*t11111-8*t00001*t\\\n",
"00011*t01111*t11111-4*t00000*t00111*t01111*t11111-t00001^2*t11111^2+2*t00000*t\\\n",
"00011*t11111^2-9*t00111^2*t01111*t00101+17*t00011*t01111^2*t00101-2*t00011*t00\\\n",
"111*t11111*t00101+5*t00001*t01111*t11111*t00101-9*t01111^2*t00101^2+6*t00111^3\\\n",
"*t11010-24*t00011*t00111*t01111*t11010-6*t00001*t01111^2*t11010+7*t00011^2*t11\\\n",
"111*t11010+7*t00001*t00111*t11111*t11010-t00000*t01111*t11111*t11010+9*t00111*\\\n",
"t01111*t00101*t11010-11*t00011*t11111*t00101*t11010+3*t11111*t00101^2*t11010+6\\\n",
"*t00111^2*t11010^2+6*t00011*t01111*t11010^2-t00001*t11111*t11010^2+3*t01111*t0\\\n",
"0101*t11010^2-6*t00111*t11010^3,10*t00001*t00011*t01111^2-3*t00001*t00011*t001\\\n",
"11*t11111-8*t00001^2*t01111*t11111+t00000*t00011*t01111*t11111+t00000*t00001*t\\\n",
"11111^2-3*t00111^3*t00101+3*t00011*t00111*t01111*t00101-11*t00001*t01111^2*t00\\\n",
"101+14*t00001*t00111*t11111*t00101+12*t00111*t01111*t00101^2-15*t00011*t11111*\\\n",
"t00101^2-3*t11111*t00101^3+5*t00011*t00111^2*t11010-11*t00011^2*t01111*t11010-\\\n",
"12*t00001*t00111*t01111*t11010+6*t00000*t01111^2*t11010+7*t00001*t00011*t11111\\\n",
"*t11010-5*t00000*t00111*t11111*t11010-7*t00111^2*t00101*t11010+6*t00011*t01111\\\n",
"*t00101*t11010+13*t00001*t11111*t00101*t11010+2*t01111*t00101^2*t11010+8*t0001\\\n",
"1*t00111*t11010^2-3*t00001*t01111*t11010^2-3*t00000*t11111*t11010^2-4*t00111*t\\\n",
"00101*t11010^2,t00000*t00011*t01111^2-t00001^2*t00111*t11111-t00011*t00111^2*t\\\n",
"00101+t00011^2*t01111*t00101+t00001*t00111*t01111*t00101-t00000*t01111^2*t0010\\\n",
"1+t00001*t00011*t11111*t00101+t00000*t00111*t11111*t00101+2*t00111^2*t00101^2-\\\n",
"3*t00011*t01111*t00101^2-t00001*t11111*t00101^2+2*t01111*t00101^3+t00011^2*t00\\\n",
"111*t11010-t00001*t00111^2*t11010-t00001*t00011*t01111*t11010-t00000*t00111*t0\\\n",
"1111*t11010+t00001^2*t11111*t11010-t00000*t00011*t11111*t11010-4*t00111*t00101\\\n",
"^2*t11010-2*t00011^2*t11010^2+3*t00001*t00111*t11010^2+t00000*t01111*t11010^2+\\\n",
"4*t00011*t00101*t11010^2-2*t00001*t11010^3,15*t00001^2*t01111^2+12*t00001*t000\\\n",
"11^2*t11111-51*t00001^2*t00111*t11111+3*t00000*t00011*t00111*t11111+27*t00000*\\\n",
"t00001*t01111*t11111-3*t00000^2*t11111^2-7*t00011*t00111^2*t00101-21*t00011^2*\\\n",
"t01111*t00101+26*t00001*t00111*t01111*t00101-52*t00000*t01111^2*t00101+57*t000\\\n",
"01*t00011*t11111*t00101+25*t00000*t00111*t11111*t00101+59*t00111^2*t00101^2-13\\\n",
"0*t00011*t01111*t00101^2-16*t00001*t11111*t00101^2+59*t01111*t00101^3+20*t0001\\\n",
"1^2*t00111*t11010-48*t00001*t00111^2*t11010+59*t00001*t00011*t01111*t11010+6*t\\\n",
"00000*t00111*t01111*t11010-25*t00001^2*t11111*t11010-38*t00000*t00011*t11111*t\\\n",
"11010+34*t00011*t00111*t00101*t11010+25*t00001*t01111*t00101*t11010+12*t00000*\\\n",
"t11111*t00101*t11010-118*t00111*t00101^2*t11010-67*t00011^2*t11010^2+47*t00001\\\n",
"*t00111*t11010^2+23*t00000*t01111*t11010^2+134*t00011*t00101*t11010^2-67*t0000\\\n",
"1*t11010^3,3*t00111^3*t01111-11*t00011*t00111*t01111^2+3*t00001*t01111^3+4*t00\\\n",
"011^2*t01111*t11111+3*t00001*t00111*t01111*t11111-t00000*t01111^2*t11111-2*t00\\\n",
"001*t00011*t11111^2-2*t00111*t01111^2*t00101-3*t00111^2*t11111*t00101+2*t00011\\\n",
"*t01111*t11111*t00101+t00001*t11111^2*t00101-2*t01111*t11111*t00101^2+3*t00111\\\n",
"^2*t01111*t11010+8*t00011*t01111^2*t11010-t00011*t00111*t11111*t11010-5*t00001\\\n",
"*t01111*t11111*t11010+t00000*t11111^2*t11010-t01111^2*t00101*t11010+5*t00111*t\\\n",
"11111*t00101*t11010-3*t00111*t01111*t11010^2-4*t00011*t11111*t11010^2+2*t11111\\\n",
"*t00101*t11010^2,5*t00011*t00111^2*t01111+27*t00000*t01111^3-7*t00011^2*t00111\\\n",
"*t11111+8*t00001*t00111^2*t11111-5*t00001*t00011*t01111*t11111-18*t00000*t0011\\\n",
"1*t01111*t11111-5*t00001^2*t11111^2+7*t00000*t00011*t11111^2-37*t00111^2*t0111\\\n",
"1*t00101+51*t00011*t01111^2*t00101+7*t00011*t00111*t11111*t00101-5*t00001*t011\\\n",
"11*t11111*t00101+4*t00000*t11111^2*t00101+18*t01111^2*t00101^2+11*t00111*t1111\\\n",
"1*t00101^2+6*t00111^3*t11010-8*t00011*t00111*t01111*t11010-75*t00001*t01111^2*\\\n",
"t11010+2*t00011^2*t11111*t11010+29*t00001*t00111*t11111*t11010-20*t00000*t0111\\\n",
"1*t11111*t11010+10*t00111*t01111*t00101*t11010-53*t00011*t11111*t00101*t11010-\\\n",
"19*t11111*t00101^2*t11010+6*t00111^2*t11010^2+6*t00011*t01111*t11010^2+58*t000\\\n",
"01*t11111*t11010^2+3*t01111*t00101*t11010^2-6*t00111*t11010^3,10*t00001*t00111\\\n",
"^2*t01111-9*t00001*t00011*t00111*t11111-4*t00001^2*t01111*t11111+3*t00000*t000\\\n",
"11*t01111*t11111+t00000*t00001*t11111^2-11*t00111^3*t00101+13*t00011*t00111*t0\\\n",
"1111*t00101-5*t00001*t01111^2*t00101+2*t00011^2*t11111*t00101+16*t00001*t00111\\\n",
"*t11111*t00101-6*t00000*t01111*t11111*t00101+14*t00111*t01111*t00101^2-15*t000\\\n",
"11*t11111*t00101^2-5*t11111*t00101^3+5*t00011*t00111^2*t11010-3*t00011^2*t0111\\\n",
"1*t11010-26*t00001*t00111*t01111*t11010+6*t00000*t01111^2*t11010+5*t00001*t000\\\n",
"11*t11111*t11010-3*t00000*t00111*t11111*t11010-3*t00111^2*t00101*t11010-4*t000\\\n",
"11*t01111*t00101*t11010+15*t00001*t11111*t00101*t11010+4*t00011*t00111*t11010^\\\n",
"2+t00001*t01111*t11010^2-t00000*t11111*t11010^2,t00000*t00111^2*t01111-t00001*\\\n",
"t00011^2*t11111-2*t00011*t00111^2*t00101+2*t00011^2*t01111*t00101+7*t00001*t00\\\n",
"111*t01111*t00101-2*t00000*t01111^2*t00101-t00000*t00111*t11111*t00101-2*t0011\\\n",
"1^2*t00101^2+2*t00011*t01111*t00101^2-t00001*t11111*t00101^2-2*t01111*t00101^3\\\n",
"+2*t00011^2*t00111*t11010-2*t00001*t00111^2*t11010-7*t00001*t00011*t01111*t110\\\n",
"10+2*t00001^2*t11111*t11010+t00000*t00011*t11111*t11010+4*t00111*t00101^2*t110\\\n",
"10+2*t00011^2*t11010^2-2*t00001*t00111*t11010^2+t00000*t01111*t11010^2-4*t0001\\\n",
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"t00001^2*t11010^2+41*t00000*t00011*t11010^2-66*t00000*t00101*t11010^2,3*t00001\\\n",
"*t00011^3-11*t00001^2*t00011*t00111+4*t00000*t00001*t00111^2+3*t00001^3*t01111\\\n",
"+3*t00000*t00001*t00011*t01111-2*t00000^2*t00111*t01111-t00000*t00001^2*t11111\\\n",
"+3*t00001*t00011^2*t00101+8*t00001^2*t00111*t00101-t00000*t00011*t00111*t00101\\\n",
"-5*t00000*t00001*t01111*t00101+t00000^2*t11111*t00101-3*t00001*t00011*t00101^2\\\n",
"-4*t00000*t00111*t00101^2-2*t00001^2*t00011*t11010-3*t00000*t00011^2*t11010+2*\\\n",
"t00000*t00001*t00111*t11010+t00000^2*t01111*t11010-t00001^2*t00101*t11010+5*t0\\\n",
"0000*t00011*t00101*t11010+2*t00000*t00101^2*t11010-2*t00000*t00001*t11010^2,5*\\\n",
"t00001*t00011^2*t11111^2-15*t00001^2*t00111*t11111^2+10*t00000*t00001*t01111*t\\\n",
"11111^2-t00000^2*t11111^3-5*t00011*t00111*t01111^2*t00101-5*t00001*t01111^3*t0\\\n",
"0101-5*t00011^2*t01111*t11111*t00101+15*t00001*t00111*t01111*t11111*t00101-19*\\\n",
"t00000*t01111^2*t11111*t00101+13*t00001*t00011*t11111^2*t00101+8*t00000*t00111\\\n",
"*t11111^2*t00101+20*t00111*t01111^2*t00101^2+13*t00111^2*t11111*t00101^2-54*t0\\\n",
"0011*t01111*t11111*t00101^2+9*t01111*t11111*t00101^3-26*t00000*t01111^3*t11010\\\n",
"+6*t00011^2*t00111*t11111*t11010-20*t00001*t00111^2*t11111*t11010+33*t00001*t0\\\n",
"0011*t01111*t11111*t11010+19*t00000*t00111*t01111*t11111*t11010-7*t00001^2*t11\\\n",
"111^2*t11010-18*t00000*t00011*t11111^2*t11010+15*t00111^2*t01111*t00101*t11010\\\n",
"-40*t00011*t01111^2*t00101*t11010+14*t00011*t00111*t11111*t00101*t11010+31*t00\\\n",
"001*t01111*t11111*t00101*t11010-35*t01111^2*t00101^2*t11010-40*t00111*t11111*t\\\n",
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"+28*t00000*t01111*t11111*t11010^2-3*t00111*t01111*t00101*t11010^2+87*t00011*t1\\\n",
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"00000*t01111^3*t00101+8*t00011^2*t00111*t11111*t00101-30*t00001*t00111^2*t1111\\\n",
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"11010^3+30*t00001*t11010^4,25*t00001^3*t01111*t11111-15*t00000*t00001*t00011*t\\\n",
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"1^2+9*t00001*t00111*t01111*t00101^2-34*t00000*t01111^2*t00101^2+108*t00001*t00\\\n",
"011*t11111*t00101^2+19*t00000*t00111*t11111*t00101^2+71*t00111^2*t00101^3-156*\\\n",
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"00001*t01111^2*t11010-25*t00001^2*t00011*t11111*t11010+8*t00000*t00011^2*t1111\\\n",
"1*t11010+11*t00000*t00001*t00111*t11111*t11010-3*t00000^2*t01111*t11111*t11010\\\n",
"-10*t00001*t00111^2*t00101*t11010+140*t00001*t00011*t01111*t00101*t11010-31*t0\\\n",
"0000*t00111*t01111*t00101*t11010-92*t00001^2*t11111*t00101*t11010-36*t00000*t0\\\n",
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"0011^3*t11010^2-60*t00001*t00011*t00111*t11010^2+14*t00000*t00111^2*t11010^2+7\\\n",
"5*t00001^2*t01111*t11010^2-32*t00000*t00011*t01111*t11010^2+10*t00000*t00001*t\\\n",
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",3*t00000*t00011*t00111^2*t11111-6*t00000*t00001*t00111*t01111*t11111-3*t00001\\\n",
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"1*t01111*t11010^2+26*t00001^2*t11111*t11010^2-23*t00000*t00011*t11111*t11010^2\\\n",
"-14*t00011*t00111*t00101*t11010^2+t00001*t01111*t00101*t11010^2+33*t00000*t111\\\n",
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"114*t00000^2*t00101^2*t11010^2-1586*t00000^2*t00001*t11010^3> )\n",
"\n",
"∪\n",
"\n",
"V( ) \\ ∅\n"
]
}
],
"source": [
"Display( im )"
]
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{
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}
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