GAPJulia_Singular_car.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Call Singular from GAP\n",
    "\n",
    "Show the example from the [Computeralgebra-Rundrief Nr. 55, Oktober 2014](http://www.fachgruppe-computeralgebra.de/rundbrief/).\n",
    "\n",
    "First load the GAP package."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "LoadPackage( \"JuliaExperimental\", false );;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Part 1:  Just call the appropriate Julia functions from GAP.\n",
    "\n",
    "Define the ring with the default monomial ordering.\n",
    "The syntax looks complicated; the reason is that the Julia syntax\n",
    "for specifying the monomial ordering ('named arguments')\n",
    "is not available in the low level interface.\n",
    "(As a workaround, we can provide some Julia code, see Part 2 below.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: Any[\"x0\", \"x1\", \"x2\", \"x3\"]>"
      ]
     },
     "execution_count": 2,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "<Julia: String[\"x0\", \"x1\", \"x2\", \"x3\"]>"
      ]
     },
     "execution_count": 3,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "<Julia: (Singular Polynomial Ring (QQ),(x0,x1,x2,x3),(dp(4),C), Singular.spoly{Singular.n_Q}[x0, x1, x2, x3])>"
      ]
     },
     "execution_count": 4,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "<Julia: Singular Polynomial Ring (QQ),(x0,x1,x2,x3),(dp(4),C)>"
      ]
     },
     "execution_count": 5,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[ <Julia: x0>, <Julia: x1>, <Julia: x2>, <Julia: x3> ]"
      ]
     },
     "execution_count": 6,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "indetnames:= ConvertedToJulia( [ \"x0\", \"x1\", \"x2\", \"x3\" ] );\n",
    "indetnames:= Julia.Base.convert( JuliaEvalString( \"Array{String,1}\" ),\n",
    "                 indetnames );\n",
    "Rinfo:= Julia.Singular.PolynomialRing( Julia.Singular.QQ, indetnames );\n",
    "R:= Rinfo[1];\n",
    "indets:= ConvertedFromJulia( Rinfo[2] );\n",
    "x0:= indets[1];;  x1:= indets[2];;  x2:= indets[3];;  x3:= indets[4];;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define the polynomial $( x_1 + x_3 )^2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: x1^2+2*x1*x3+x3^2>"
      ]
     },
     "execution_count": 11,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p:= (x1+x3)^2;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define the ideal in R generated by $x_0^2 - x_1 x_3$ and $x_0 x_1 - x_2 x_3$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: Singular Ideal over Singular Polynomial Ring (QQ),(x0,x1,x2,x3),(dp(4),C) with generators (x0^2-x1*x3, x0*x1-x2*x3)>"
      ]
     },
     "execution_count": 12,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I:= Julia.Singular.Ideal( R, x0^2 - x1*x3, x0*x1 - x2*x3 );"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The corresponding matrix $i$ (of ideal generators)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: [x0^2-x1*x3, x0*x1-x2*x3]>"
      ]
     },
     "execution_count": 13,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i:= JuliaFunction( \"Matrix\", \"Singular\" )( I );"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The sum $I + I$ means the sum of ideals."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: Singular Ideal over Singular Polynomial Ring (QQ),(x0,x1,x2,x3),(dp(4),C) with generators (x0^2-x1*x3, x0*x1-x2*x3)>"
      ]
     },
     "execution_count": 14,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "J:= I + I;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Whereas $i + i$ means the sum of Singular matrices."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: [2*x0^2-2*x1*x3, 2*x0*x1-2*x2*x3]>"
      ]
     },
     "execution_count": 15,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i + i;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The squared ideal $I^2$ inside $R$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: Singular Ideal over Singular Polynomial Ring (QQ),(x0,x1,x2,x3),(dp(4),C) with generators (x0^4-2*x0^2*x1*x3+x1^2*x3^2, x0^3*x1-x0*x1^2*x3-x0^2*x2*x3+x1*x2*x3^2, x0^2*x1^2-2*x0*x1*x2*x3+x2^2*x3^2)>"
      ]
     },
     "execution_count": 16,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I2 := I^2;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Gröbner basis of the ideal $I$\n",
    "is returned as a new different (but mathematically equal) ideal $G$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: Singular Ideal over Singular Polynomial Ring (QQ),(x0,x1,x2,x3),(dp(4),C) with generators (x0*x1-x2*x3, x0^2-x1*x3, x1^2*x3-x0*x2*x3)>"
      ]
     },
     "execution_count": 17,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G:= Julia.Base.std( I );"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The syzygies of the generators of $G$ are the columns of a Singular module."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: Singular Module over Singular Polynomial Ring (QQ),(x0,x1,x2,x3),(dp(4),C), with Generators:x0*gen(1)-x1*gen(2)-gen(3)x1*x3*gen(1)-x2*x3*gen(2)-x0*gen(3)>"
      ]
     },
     "execution_count": 18,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S:= Julia.Singular.syz( G );"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To access the second column of $S$ use:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: x1*x3*gen(1)-x2*x3*gen(2)-x0*gen(3)>"
      ]
     },
     "execution_count": 19,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S[2];"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To create a matrix in `Singular.jl`, use the following.\n",
    "```\n",
    "Singular.matrix( R, 3, 2, [ x0, x3, x1, x2, x3, x0 ] )'\n",
    "```\n",
    "Via the GAP interface, the following works.\n",
    "(We have to adjust the Julia type of the array,\n",
    "in order to make it acceptable for `AbstractAlgebra.matrix`.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: [x0 x3][x1 x2][x3 x0]>"
      ]
     },
     "execution_count": 22,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "arr1:= ConvertedToJulia( [ x0, x3, x1, x2, x3, x0 ] );;\n",
    "arr2:= Julia.Base.convert(\n",
    "JuliaEvalString( \"Array{Singular.spoly{Singular.n_Q},1}\" ),\n",
    "arr1 );;\n",
    "m:= Julia.AbstractAlgebra.matrix( R, 3, 2, arr2 );"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To extract the $(2,1)$-entry from the matrix use the following.\n",
    "(Access to 'rows' or 'columns' of such a matrix is not supported.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Julia: x1>"
      ]
     },
     "execution_count": 23,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m[ 2, 1 ];"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Part 2:  Use some GAP structures on top of the Julia interface.\n",
    "\n",
    "Define the ring with the monomial ordering `\"degrevlex\"`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "#I  Global variable `x0' is already defined and will be overwritten\n",
      "#I  Global variable `x1' is already defined and will be overwritten\n",
      "#I  Global variable `x2' is already defined and will be overwritten\n",
      "#I  Global variable `x3' is already defined and will be overwritten\n",
      "#I  Assigned the global variables [ x0, x1, x2, x3 ]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Singular_QQ[x0,x1,x2,x3]"
      ]
     },
     "execution_count": 24,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R:= SingularPolynomialRing( Rationals, [ \"x0\", \"x1\", \"x2\", \"x3\" ] :\n",
    "        ordering:= \"dp\" );\n",
    "AssignGeneratorVariables( R );"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define the polynomial $( x_1 + x_3 )^2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<<Julia: x1^2+2*x1*x3+x3^2>>"
      ]
     },
     "execution_count": 26,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 27,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p:= (x1+x3)^2;\n",
    "IsSingularPolynomial( p );"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define the ideal in $R$ generated by $x_0^2 - x_1 x_3$ and $x_0 x_1 - x_2 x_3$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<semigroup>"
      ]
     },
     "execution_count": 28,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I:= Ideal( R, [ x0^2 - x1*x3, x0*x1 - x2*x3 ] );"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Do not mix up the ideal and its list of generators."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[ <<Julia: x0^2-x1*x3>>, <<Julia: x0*x1-x2*x3>> ]"
      ]
     },
     "execution_count": 29,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i:= GeneratorsOfIdeal( I );"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Gröbner basis of the ideal $I$\n",
    "is returned as a new different (but mathematically equal) ideal $G$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<semigroup>"
      ]
     },
     "execution_count": 30,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "[ <<Julia: x0*x1-x2*x3>>, <<Julia: x0^2-x1*x3>>, <<Julia: x1^2*x3-x0*x2*x3>> ]"
      ]
     },
     "execution_count": 31,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G:= GroebnerBasisIdeal( I );\n",
    "GeneratorsOfIdeal( G );"
   ]
  }
 ],
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