{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Basic use of the macaulay2 interface\n",
    "\n",
    "`m2(\"string\")` --> evaluate string in M2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "macaulay2(\"2 + 2\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Shorthand: let's type `m2` instead of `macaulay2`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "m2 = macaulay2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Let's construct a ring in M2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "m2.ring?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "R = m2.ring('QQ', '[x,y]')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "QQ[x..y, Degrees => {2:1}, Heft => {1}, MonomialOrder => {MonomialSize => 16}, DegreeRank => 1]\n",
       "                                                         {Lex => 2          }\n",
       "                                                         {Position => Up    }"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<class 'sage.interfaces.macaulay2.Macaulay2Element'>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(R)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Can we convert a M2 ring to a Sage ring? Let's try first for the rationals."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "QQ"
      ]
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m2(QQ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<class 'sage.interfaces.macaulay2.Macaulay2Element'>"
      ]
     },
     "execution_count": 102,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(m2(QQ))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<class 'sage.rings.rational_field.RationalField_with_category'>"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(QQ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "QQ"
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mQQ = m2(QQ)\n",
    "mQQ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "QQ"
      ]
     },
     "execution_count": 105,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mQQ.describe()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Ring"
      ]
     },
     "execution_count": 106,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m2('class %s' % mQQ._name)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Every objected created by the interface sets a new variable in the M2 session called `sage0`, `sage1`, `sage2`, ...\n",
    "\n",
    "We can use these variable names to access the objects in M2 via the command `m2(\"sage9\")`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 136,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'sage5'"
      ]
     },
     "execution_count": 136,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mQQ._name"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 137,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "QQ"
      ]
     },
     "execution_count": 137,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m2(\"sage5\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 138,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Ring"
      ]
     },
     "execution_count": 138,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m2(\"class sage5\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Can we convert M2 objects to Sage objects?\n",
    "\n",
    "Some objects, yes. Try the command `m2object.sage()`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "QQ"
      ]
     },
     "execution_count": 111,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mQQ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Macaulay2"
      ]
     },
     "execution_count": 112,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mQQ.parent()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 113,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mQQ.sage() is QQ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 139,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "QQ[x..y, Degrees => {2:1}, Heft => {1}, MonomialOrder => {MonomialSize => 16}, DegreeRank => 1]\n",
       "                                                         {Lex => 2          }\n",
       "                                                         {Position => Up    }"
      ]
     },
     "execution_count": 139,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 140,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Multivariate Polynomial Ring in x, y over Rational Field"
      ]
     },
     "execution_count": 140,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R.sage()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {},
   "outputs": [],
   "source": [
    "M = m2.matrix(\"{{1,2,3},{4,5,6/1}}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "| 1 2 3 |\n",
       "| 4 5 6 |"
      ]
     },
     "execution_count": 115,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1 2 3]\n",
       "[4 5 6]"
      ]
     },
     "execution_count": 116,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.sage()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Full MatrixSpace of 2 by 3 dense matrices over Rational Field"
      ]
     },
     "execution_count": 117,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_.parent()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### What version of Sage am I using?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'SageMath version 8.9.beta3, Release Date: 2019-07-19'"
      ]
     },
     "execution_count": 118,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "version()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Use a M2 package through the interface\n",
    "\n",
    "Here we will use the \"NormalToricVarieties\" to construct the smooth Fano toric variety, and compute its rays and cones."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "NormalToricVarieties"
      ]
     },
     "execution_count": 127,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m2.needsPackage('\"NormalToricVarieties\"')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = m2.smoothFanoToricVariety(3, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 129,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sage47"
      ]
     },
     "execution_count": 129,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Hmm...it would be nice to have a better string representation of this object."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 130,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sage47"
      ]
     },
     "execution_count": 130,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X.describe()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 131,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{{1, 0, 0}, {0, 1, 0}, {-1, -1, 0}, {0, 0, 1}, {0, 1, -1}}"
      ]
     },
     "execution_count": 131,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X.rays()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 132,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{{0, 1, 3}, {0, 1, 4}, {0, 2, 3}, {0, 2, 4}, {1, 2, 3}, {1, 2, 4}}"
      ]
     },
     "execution_count": 132,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X.max()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[0, 1, 3], [0, 1, 4], [0, 2, 3], [0, 2, 4], [1, 2, 3], [1, 2, 4]]"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X.max().sage()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Let's convert the smooth Fano Toric Variety computed with M2 to a Sage `Fan`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 133,
   "metadata": {},
   "outputs": [],
   "source": [
    "rays = X.rays().sage()\n",
    "cones = X.max().sage()\n",
    "P = Fan(rays=rays, cones=cones)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 134,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Rational polyhedral fan in 3-d lattice N"
      ]
     },
     "execution_count": 134,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 135,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
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       "  'color pmesh translucent 0.6 [255,0,170]',\n",
       "  'select atomno = 6',\n",
       "  'color atom  [0,0,0]',\n",
       "  'label &quot;\\sigma_0&quot;',\n",
       "  'select atomno = 7',\n",
       "  'color atom  [0,0,0]',\n",
       "  'label &quot;\\sigma_1&quot;',\n",
       "  'select atomno = 8',\n",
       "  'color atom  [0,0,0]',\n",
       "  'label &quot;\\sigma_2&quot;',\n",
       "  'select atomno = 9',\n",
       "  'color atom  [0,0,0]',\n",
       "  'label &quot;\\sigma_3&quot;',\n",
       "  'select atomno = 10',\n",
       "  'color atom  [0,0,0]',\n",
       "  'label &quot;\\sigma_4&quot;',\n",
       "  'select atomno = 11',\n",
       "  'color atom  [0,0,0]',\n",
       "  'label &quot;\\sigma_5&quot;',\n",
       "  'select atomno = 12',\n",
       "  'color atom  [0,0,0]',\n",
       "  'label &quot;\\sigma_6&quot;',\n",
       "  'select atomno = 13',\n",
       "  'color atom  [0,0,0]',\n",
       "  'label &quot;\\sigma_7&quot;',\n",
       "  'select atomno = 14',\n",
       "  'color atom  [0,0,0]',\n",
       "  'label &quot;\\sigma_8&quot;',\n",
       "  'isosurface fullylit; pmesh o* fullylit; set antialiasdisplay on;',\n",
       "].join('\\n');;\n",
       "    var Info = {\n",
       "      width: '100%',\n",
       "      height: '500',\n",
       "      debug: false,\n",
       "      disableInitialConsole: true,   // very slow when used with inline mesh\n",
       "      color: '#3131ff',\n",
       "      addSelectionOptions: false,\n",
       "      use: 'HTML5',\n",
       "      j2sPath: '/nbextensions/jsmol/j2s',\n",
       "      script: script,\n",
       "    };\n",
       "    var jmolApplet0 = Jmol.getApplet('jmolApplet0', Info);\n",
       "  </script>\n",
       "</body>\n",
       "</html>\n",
       "\" \n",
       "        width=\"100%\"\n",
       "        height=\"500\"\n",
       "        style=\"border: 0;\">\n",
       "</iframe>\n"
      ],
      "text/plain": [
       "Graphics3d Object"
      ]
     },
     "execution_count": 135,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P.plot()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Can we convert a M2 HashTable to a Sage object?\n",
    "\n",
    "Mostly, but there are some caveats.\n",
    "\n",
    "The keys of the first HastTable below (`ht1`) are strings, and this works well.\n",
    "\n",
    "The keys of the second HastTable below (`ht2`) are M2 \"symbols\", and this does not work so well because Sage does not know how to deal with these symbols (they should just be turned into strings; this should be a quick fix)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 159,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "HashTable{key1 => 12}\n",
       "          key2 => 1"
      ]
     },
     "execution_count": 159,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ht1 = m2('new HashTable from {\"key1\"=>12, \"key2\"=>1}')\n",
    "ht1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 160,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'key1': 12, 'key2': 1}"
      ]
     },
     "execution_count": 160,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ht1.sage()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 161,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "HashTable{key1 => 12}\n",
       "          key2 => 1"
      ]
     },
     "execution_count": 161,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ht2 = m2('new HashTable from {key1=>12, key2=>1}')\n",
    "ht2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 162,
   "metadata": {},
   "outputs": [
    {
     "ename": "NotImplementedError",
     "evalue": "cannot convert key2 to a Sage object",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNotImplementedError\u001b[0m                       Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-162-14087f630959>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mht2\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msage\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m/home/saliola/Applications/sage-devel/local/lib/python2.7/site-packages/sage/interfaces/interface.pyc\u001b[0m in \u001b[0;36msage\u001b[0;34m(self, *args, **kwds)\u001b[0m\n\u001b[1;32m   1079\u001b[0m             \u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m \u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1080\u001b[0m         \"\"\"\n\u001b[0;32m-> 1081\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_sage_\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwds\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1082\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1083\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0m__repr__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/saliola/Applications/sage-devel/local/lib/python2.7/site-packages/sage/interfaces/macaulay2.pyc\u001b[0m in \u001b[0;36m_sage_\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m   1217\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0mmatrix\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mbase_ring\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mentries\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1218\u001b[0m             \u001b[0;32melif\u001b[0m \u001b[0mcls_str\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0;34m'HashTable'\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1219\u001b[0;31m                 \u001b[0;32mreturn\u001b[0m \u001b[0;34m{\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_sage_\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m \u001b[0my\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_sage_\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0my\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpairs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m}\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1220\u001b[0m             \u001b[0;32melif\u001b[0m \u001b[0mcls_str\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0;34m\"Ideal\"\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1221\u001b[0m                 \u001b[0mparent\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mring\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_sage_\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/saliola/Applications/sage-devel/local/lib/python2.7/site-packages/sage/interfaces/macaulay2.pyc\u001b[0m in \u001b[0;36m<dictcomp>\u001b[0;34m((x, y))\u001b[0m\n\u001b[1;32m   1217\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0mmatrix\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mbase_ring\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mentries\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1218\u001b[0m             \u001b[0;32melif\u001b[0m \u001b[0mcls_str\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0;34m'HashTable'\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1219\u001b[0;31m                 \u001b[0;32mreturn\u001b[0m \u001b[0;34m{\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_sage_\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m \u001b[0my\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_sage_\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0my\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpairs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m}\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1220\u001b[0m             \u001b[0;32melif\u001b[0m \u001b[0mcls_str\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0;34m\"Ideal\"\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1221\u001b[0m                 \u001b[0mparent\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mring\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_sage_\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/saliola/Applications/sage-devel/local/lib/python2.7/site-packages/sage/interfaces/macaulay2.pyc\u001b[0m in \u001b[0;36m_sage_\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m   1308\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0msage_eval\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mrepr_str\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1309\u001b[0m         \u001b[0;32mexcept\u001b[0m \u001b[0mException\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1310\u001b[0;31m             \u001b[0;32mraise\u001b[0m \u001b[0mNotImplementedError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"cannot convert %s to a Sage object\"\u001b[0m\u001b[0;34m%\u001b[0m\u001b[0mrepr_str\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1311\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1312\u001b[0m     \u001b[0;32mfrom\u001b[0m \u001b[0msage\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmisc\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msuperseded\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mdeprecated_function_alias\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mNotImplementedError\u001b[0m: cannot convert key2 to a Sage object"
     ]
    }
   ],
   "source": [
    "ht2.sage()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Some miscellaneous stuff"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 163,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "pk = X.peek()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 164,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "NormalToricVariety{cache => CacheTable{...3...}                                             }\n",
       "                   max => {{0, 1, 3}, {0, 1, 4}, {0, 2, 3}, {0, 2, 4}, {1, 2, 3}, {1, 2, 4}}\n",
       "                   rays => {{1, 0, 0}, {0, 1, 0}, {-1, -1, 0}, {0, 0, 1}, {0, 1, -1}}"
      ]
     },
     "execution_count": 164,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pk"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 165,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Net"
      ]
     },
     "execution_count": 165,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m2(\"class %s\" % pk._name)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 166,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Thing : HashTable : MutableHashTable : Variety : NormalToricVariety"
      ]
     },
     "execution_count": 166,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m2(\"showStructure class %s\" % X._name)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### M2 can be used for some computations in Sage already (provided M2 is installed)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 198,
   "metadata": {},
   "outputs": [],
   "source": [
    "R = PolynomialRing(QQ, 3, \"a,b,c\", order='lex')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 199,
   "metadata": {},
   "outputs": [],
   "source": [
    "I = sage.rings.ideal.Katsura(R, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 200,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Ideal (a + 2*b + 2*c - 1, a^2 - a + 2*b^2 + 2*c^2, 2*a*b + 2*b*c - b) of Multivariate Polynomial Ring in a, b, c over Rational Field"
      ]
     },
     "execution_count": 200,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 204,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]"
      ]
     },
     "execution_count": 204,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I.groebner_basis(algorithm='macaulay2:gb')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The above dispatchs the computation to the method `_groebner_basis_macaulay2`, and then does some post-processing."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 205,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[84*c^4 - 40*c^3 + c^2 + c, 7*b + 210*c^3 - 79*c^2 + 3*c, 7*a - 420*c^3 + 158*c^2 + 8*c - 7]"
      ]
     },
     "execution_count": 205,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I._groebner_basis_macaulay2()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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   "codemirror_mode": {
    "name": "ipython",
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   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
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