GAPJulia_Singular_blog.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Call Singular from GAP\n",
    "\n",
    "Run the examples from [a blog post by Bill Hart](https://wbhart.blogspot.de/2017/01/singular-and-julia.html).\n",
    "\n",
    "First load the GAP package."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "LoadPackage( \"JuliaExperimental\", false );;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A Nemo residue ring over Singular's ring of integers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: Residue Ring of Integer Ring modulo 23>"
      ]
     },
     "execution_count": 2,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R:= Julia.Nemo.ResidueRing( Julia.Singular.ZZ, 23 );"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: 19>"
      ]
     },
     "execution_count": 3,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R(12) + R(7);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\"Singular.n_Zn\""
      ]
     },
     "execution_count": 4,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "JuliaTypeInfo( R(12) );"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: Residue Ring of Integer Ring modulo 23>"
      ]
     },
     "execution_count": 5,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Julia.Base.parent( R(12) );"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Polynomial rings"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "indetnames:= Julia.Base.convert( JuliaEvalString( \"Array{String,1}\" ),\n",
    "ConvertedToJulia( [ \"x\", \"y\", \"z\", \"t\" ] ) );;\n",
    "Rinfo:= Julia.Singular.PolynomialRing( Julia.Singular.QQ, indetnames );;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: Singular Polynomial Ring (QQ),(x,y,z,t),(dp(4),C)>"
      ]
     },
     "execution_count": 8,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R:= Rinfo[1];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[ <Julia: x>, <Julia: y>, <Julia: z>, <Julia: t> ]"
      ]
     },
     "execution_count": 9,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "indets:= ConvertedFromJulia( Rinfo[2] );"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "x:= indets[1];; y:= indets[2];; z:= indets[3];; t:= indets[4];;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Polynomials"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: x^3*y+x^2*y^2+x^2*y*z+2*x^2+2*x*y+2*x*z>"
      ]
     },
     "execution_count": 14,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "<Julia: x^2+x*y+2*x*z+y*z+z^2+x*t+y*t+z*t>"
      ]
     },
     "execution_count": 15,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f:= (x+y+z)*(x^2*y + 2*x);\n",
    "g:= (x+y+z)*(x+z+t);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: x+y+z>"
      ]
     },
     "execution_count": 16,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Julia.Base.gcd( f, g );"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ideals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: Singular Ideal over Singular Polynomial Ring (QQ),(x,y,z,t),(dp(4),C) with generators ()>"
      ]
     },
     "execution_count": 17,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I1:= Julia.Singular.Ideal( R );"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: Singular Ideal over Singular Polynomial Ring (QQ),(x,y,z,t),(dp(4),C) with generators (x, z*t+x)>"
      ]
     },
     "execution_count": 18,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I2:= Julia.Singular.Ideal( R, x, t*z + x );"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: 2>"
      ]
     },
     "execution_count": 19,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Julia.Singular.ngens( I2 );"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: z*t+x>"
      ]
     },
     "execution_count": 20,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I2[2];"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Groebner basis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: Singular Ideal over Singular Polynomial Ring (QQ),(x,y,z,t),(dp(4),C) with generators (x*y+1, x+y, 2*x+y+z)>"
      ]
     },
     "execution_count": 21,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I:= Julia.Singular.Ideal( R, x*y + 1, x+y, 2*x+y+z );"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: Singular Ideal over Singular Polynomial Ring (QQ),(x,y,z,t),(dp(4),C) with generators (y-z, 2*x+y+z, z^2-1)>"
      ]
     },
     "execution_count": 22,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gbasis:= Julia.Base.std( I );"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: 3>"
      ]
     },
     "execution_count": 23,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Julia.Singular.ngens( gbasis );"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Syzygy matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: Singular Module over Singular Polynomial Ring (QQ),(x,y,z,t),(dp(4),C), with Generators:x*gen(3)-2*x*gen(2)+y*gen(3)-y*gen(2)-z*gen(2)y^2*gen(3)-y^2*gen(2)-y*z*gen(2)-y*gen(1)+z*gen(1)-gen(3)+2*gen(2)x*y*gen(2)-x*gen(1)-y*gen(1)+gen(2)>"
      ]
     },
     "execution_count": 24,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M:= Julia.Singular.syz( I );"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: [0, -y+z, -x-y-2*x-y-z, -y^2-y*z+2, x*y+1x+y, y^2-1, 0]>"
      ]
     },
     "execution_count": 25,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Currently there is no 'Julia.Singular.Matrix'.\n",
    "JuliaFunction( \"Matrix\", \"Singular\" )( M );"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ideal arithmetic"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: Singular Ideal over Singular Polynomial Ring (QQ),(x,y,z,t),(dp(4),C) with generators (x*y+1, x+y, 2*x+y+z)>"
      ]
     },
     "execution_count": 26,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I:= Julia.Singular.Ideal( R, x*y + 1, x+y, 2*x+y+z );"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: Singular Ideal over Singular Polynomial Ring (QQ),(x,y,z,t),(dp(4),C) with generators (2*x-y+1, x+2*z)>"
      ]
     },
     "execution_count": 27,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "J:= Julia.Singular.Ideal( R, 2*x-y + 1, 2*z+x);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: Singular Ideal over Singular Polynomial Ring (QQ),(x,y,z,t),(dp(4),C) with generators (x^2*y^2+2*x*y+1, x^2*y+x*y^2+x+y, 2*x^2*y+x*y^2+x*y*z+2*x+y+z, x^2+2*x*y+y^2, 2*x^2+3*x*y+y^2+x*z+y*z, 4*x^2+4*x*y+y^2+4*x*z+2*y*z+z^2)>"
      ]
     },
     "execution_count": 28,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I^2;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: Singular Ideal over Singular Polynomial Ring (QQ),(x,y,z,t),(dp(4),C) with generators (2*x^2*y-x*y^2+x*y+2*x-y+1, x^2*y+2*x*y*z+x+2*z, 2*x^2+x*y-y^2+x+y, x^2+x*y+2*x*z+2*y*z, 4*x^2-y^2+2*x*z-y*z+2*x+y+z, 2*x^2+x*y+5*x*z+2*y*z+2*z^2)>"
      ]
     },
     "execution_count": 29,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I * J;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: Singular Ideal over Singular Polynomial Ring (QQ),(x,y,z,t),(dp(4),C) with generators (x*y, x, 2*x)>"
      ]
     },
     "execution_count": 30,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Julia.Singular.lead( I );"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Resolutions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: Singular Ideal over Singular Polynomial Ring (QQ),(x,y,z,t),(dp(4),C) with generators (x, y, z, t)>"
      ]
     },
     "execution_count": 31,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I:= Julia.Singular.MaximalIdeal( R, 1 );"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: Singular Ideal over Singular Polynomial Ring (QQ),(x,y,z,t),(dp(4),C) with generators (t, z, y, x)>"
      ]
     },
     "execution_count": 32,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I:= Julia.Base.std( I );"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: Singular Resolution:R^1 <- R^4 <- R^6 <- R^4 <- R^1>"
      ]
     },
     "execution_count": 33,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r:= Julia.Singular.sres( I, 5 );"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Access to modules in the resolution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: Singular Module over Singular Polynomial Ring (QQ),(x,y,z,t),(dp(4),C), with Generators:-z*gen(1)+t*gen(2)-y*gen(1)+t*gen(3)-y*gen(2)+z*gen(3)-x*gen(1)+t*gen(4)-x*gen(2)+z*gen(4)-x*gen(3)+y*gen(4)>"
      ]
     },
     "execution_count": 34,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r[2];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: Singular Module over Singular Polynomial Ring (QQ),(x,y,z,t),(dp(4),C), with Generators:y*gen(1)-z*gen(2)+t*gen(3)x*gen(1)-z*gen(4)+t*gen(5)x*gen(2)-y*gen(4)+t*gen(6)x*gen(3)-y*gen(5)+z*gen(6)>"
      ]
     },
     "execution_count": 35,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r[3];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: y*gen(1)-z*gen(2)+t*gen(3)>"
      ]
     },
     "execution_count": 36,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r[3][1];"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Arithmetic on module elements"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: -3*z*gen(1)+3*t*gen(2)>"
      ]
     },
     "execution_count": 37,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r[2][1] * 3;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: -y*gen(1)-z*gen(1)+t*gen(3)+t*gen(2)>"
      ]
     },
     "execution_count": 38,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r[2][1] + r[2][2];"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Standard bases and resolutions of modules"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: Singular Module over Singular Polynomial Ring (QQ),(x,y,z,t),(dp(4),C), with Generators:y*gen(1)-z*gen(2)+t*gen(3)x*gen(1)-z*gen(4)+t*gen(5)x*gen(2)-y*gen(4)+t*gen(6)x*gen(3)-y*gen(5)+z*gen(6)>"
      ]
     },
     "execution_count": 39,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "J:= Julia.Base.std( r[3] );"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: Singular Resolution:R^6 <- R^4 <- R^1>"
      ]
     },
     "execution_count": 40,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "J1:= Julia.Singular.sres( J, 4 );"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: Singular Module over Singular Polynomial Ring (QQ),(x,y,z,t),(dp(4),C), with Generators:-x*gen(1)+y*gen(2)-z*gen(3)+t*gen(4)>"
      ]
     },
     "execution_count": 41,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "J1[2];"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Generic multivariate polynomial code in Nemo, over a Singular coefficient ring"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "Rinfo:= Julia.Nemo.PolynomialRing( Julia.Singular.ZZ, indetnames );;\n",
    "R:= Rinfo[1];;\n",
    "indets:= ConvertedFromJulia( Rinfo[2] );;\n",
    "x:= indets[1];; y:= indets[2];; z:= indets[3];; t:= indets[4];;\n",
    "f:= (2*t^2078*z^15*y^53-28*t^1080*z^15*y^44+98*t^82*z^15*y^35)*x^9\n",
    "+(t^2003*y^40-14*t^1005*y^31+49*t^7*y^22)*x^7\n",
    "+((t^2100+t^2000)*y^40 +(-4*t^1079*z^16+156*t^1078*z^15)*y^33\n",
    "+(-14*t^1102-14*t^1002)*y^31+(28*t^81*z^16-1092*t^80*z^15)*y^24\n",
    "+(49*t^104+49*t^4)*y^22)*x^6\n",
    "+((-2*t^1004*z+78*t^1003)*y^20+(14*t^6*z-546*t^5)*y^11)*x^4\n",
    "+(((-2*t^1101-2*t^1001)*z+(78*t^1100+78*t^1000))*y^20\n",
    "+(2*t^80*z^17-156*t^79*z^16+3042*t^78*z^15)*y^13\n",
    "+((14*t^103+14*t^3)*z+(-546*t^102-546*t^2))*y^11)*x^3\n",
    "+(t^5*z^2-78*t^4*z+1521*t^3)*x\n",
    "+((t^102+t^2)*z^2+(-78*t^101-78*t)*z+(1521*t^100+1521));;\n",
    "g:= (4*t^1156*z^30*y^46-28*t^158*z^30*y^37)*x^9\n",
    "+(4*t^1081*z^15*y^33-28*t^83*z^15*y^24)*x^7\n",
    "+((4*t^1178+4*t^1078)*z^15*y^33+(-4*t^157*z^31+156*t^156*z^30)*y^26\n",
    "+(-28*t^180-28*t^80)*z^15*y^24)*x^6\n",
    "+(t^1006*y^20-7*t^8*y^11)*x^5\n",
    "+((2*t^1103+2*t^1003)*y^20+(-4*t^82*z^16+156*t^81*z^15)*y^13\n",
    "+(-14*t^105-14*t^5)*y^11)*x^4\n",
    "+((t^1200+2*t^1100+t^1000)*y^20+((-4*t^179-4*t^79)*z^16\n",
    "+(156*t^178+156*t^78)*z^15)*y^13+(-7*t^202-14*t^102-7*t^2)*y^11)*x^3\n",
    "+(-t^7*z+39*t^6)*x^2\n",
    "+((-2*t^104-2*t^4)*z+(78*t^103+78*t^3))*x\n",
    "+((-t^201-2*t^101-t)*z+(39*t^200+78*t^100+39));;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: -2*x^6*y^33*z^15*t^1078+14*x^6*y^24*z^15*t^80-x^4*y^20*t^1003+7*x^4*y^11*t^5-x^3*y^20*t^1100-x^3*y^20*t^1000+2*x^3*y^13*z^16*t^79-78*x^3*y^13*z^15*t^78+7*x^3*y^11*t^102+7*x^3*y^11*t^2+x*z*t^4-39*x*t^3+z*t^101+z*t-39*t^100-39>"
      ]
     },
     "execution_count": 51,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p:= Julia.Base.gcd( f, g );"
   ]
  }
 ],
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