{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "ffb0f1e5-557b-4b3a-892b-c6f0942e647b",
   "metadata": {},
   "source": [
    "This notebook shows the use of Macaulay2 with a SageMath kernel."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "84aceffa-7a07-4d04-8a48-1a8ad2e0845a",
   "metadata": {},
   "source": [
    "# 1. Resultants"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "66419cab-4d13-4f04-8ae7-fd2b1dd07b1c",
   "metadata": {},
   "source": [
    "We work with polynomials with integer coefficients in `x` and `y`.  So, our polynomial ring `R` is `ZZ[x,y]`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "90bb1048-58be-44a7-93ff-c0794657b2e9",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "R\n",
      "\n",
      "PolynomialRing\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "R = ZZ[x,y]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "45cdccf9-a38e-4d22-85b9-b12f9716701d",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "p = 4*y^2 - 4 + 4*x*y + x^2;"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d17af91e-aa39-4926-b304-7fc9c6d5d220",
   "metadata": {},
   "source": [
    "Using the semincolon at the end suppresses the output."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "e46eff6f-cb33-44b6-9fe5-f8c5b661fd00",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  2           2\n",
      "4x  - 4x*y + y  - 4\n",
      "\n",
      "R\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "q = 4*x^2 - 4*x*y + y^2 - 4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "defdc0d6-ccb2-4d44-b7cb-3843c6f0e76d",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "    4        2\n",
      "625y  - 1000y  + 144\n",
      "\n",
      "R\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "rx = resultant(p,q,x)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3e8e0554-50d0-4928-924f-6d64afac8798",
   "metadata": {},
   "source": [
    "The resultant is the determinant of the *Sylvester matrix*."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "913316cf-6927-4562-a20d-c30badd6948a",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{-3} | 1 4y  4y2-4 0     |\n",
      "{-2} | 0 1   4y    4y2-4 |\n",
      "{-3} | 4 -4y y2-4  0     |\n",
      "{-2} | 0 4   -4y   y2-4  |\n",
      "\n",
      "        4       4\n",
      "Matrix R  <--- R\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "M = sylvesterMatrix(p,q,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "db962950-f80d-4068-829e-205d90ba13d6",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "    4        2\n",
      "625y  - 1000y  + 144\n",
      "\n",
      "R\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "determinant(M)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "93627d26-6ac5-4e4e-8c76-83593fc91fa8",
   "metadata": {},
   "source": [
    "# 2. Solving Polynomial Systems Numerically"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8e61bd8f-d24f-4c04-9ac6-fd4e0a996cbd",
   "metadata": {},
   "source": [
    "For the commands in this section to work, the executable `phc` must be available in the path.  This executable can be found from the assets of the tags of the github release of PHCpack."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "8a322b6d-86e3-4ef3-89b4-de326d26454f",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "PHCpack\n",
      "\n",
      "Package\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "loadPackage(\"PHCpack\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "9c5d93a5-d725-418a-87af-b7b3040dcf29",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "R\n",
      "\n",
      "PolynomialRing\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2 \n",
    "R = CC[x,y]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "7124971d-bd61-4dce-9f3e-a0b736b1cd0a",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  2            2        2           2\n",
      "{x  + 4x*y + 4y  - 4, 4x  - 4x*y + y  - 4}\n",
      "\n",
      "List\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "F = {4*y^2 - 4 + 4*x*y + x^2,\n",
    "     4*x^2 - 4*x*y + y^2 - 4}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "86b9b151-3145-42df-ab01-5e4cc925c0cf",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{{.4, -1.2}, {-1.2, -.4}, {1.2, .4}, {-.4, 1.2}}\n",
      "\n",
      "List\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "s = solveSystem(F)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f1077ce0-28d0-4141-aeb7-b2fce927b693",
   "metadata": {},
   "source": [
    "# 3. The Ideal of a Monomial Curve"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2565edcb-a7e1-4be8-ba71-231f99b92c05",
   "metadata": {},
   "source": [
    "Following one of the tutorials of Macaulay2\n",
    "written by David Eisenbud and Michael Stillman.\n",
    "\n",
    "Our first example considers the rational quartic curve\n",
    "defined parametrically in affine coordinates as\n",
    "\n",
    "$$\n",
    "   t \\mapsto (t, t^3, t^4).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a2740a90-bcbd-492f-be1d-30bbb628b705",
   "metadata": {},
   "source": [
    "Let us first compute its ideal as follows:\n",
    "\n",
    "1. We work modulo 31991, in ${\\mathbb K} = {\\mathbb Z}_{31991}$.\n",
    "\n",
    "2. In projective 3-space, we make a polynomial ring in\n",
    "   four variables ${\\mathbb R} = {\\mathbb K}[a,b,c,d]$.\n",
    "\n",
    "3. With the degrees 1, 3, and 4 of the parametric representation\n",
    "   of the quartic, we can ask Macaulay2 for the ideal of the quartic:\n",
    "\n",
    "   $$\n",
    "      I = \\langle b c - a d, c^3  - b d^2, a c^2  - b^2 d, b^3  - a^2 c \\rangle. \n",
    "   $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "b4951dd2-bab0-48c0-a52a-d72d13bb42c2",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "KK\n",
      "\n",
      "QuotientRing\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "KK = ZZ/31991"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "a9b4a055-c832-4490-88d8-06fa2f1ca824",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "R\n",
      "\n",
      "PolynomialRing\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "R = KK[a..d]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "b6104cc4-601f-4373-87dc-75e989d0a690",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                   3      2     2    2    3    2\n",
      "ideal (b*c - a*d, c  - b*d , a*c  - b d, b  - a c)\n",
      "\n",
      "Ideal of R\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "I = monomialCurveIdeal(R,{1,3,4})"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a36cf8fc-683f-4e46-973b-7e9d268ab12d",
   "metadata": {},
   "source": [
    "In ${\\mathbb K}[a,b,c,d]$, the ideal\n",
    "\n",
    "$$\n",
    "    I = \\langle b c - a d, c^3  - b d^2, a c^2  - b^2 d, b^3  - a^2 c \\rangle\n",
    "$$\n",
    "\n",
    "contains all polynomials that vanish on the curve $(t, t^3, t^4)$,\n",
    "\n",
    "or $[1:t:t^3:t^4] \\in {\\mathbb P}^3$, set $a=1$, $b=t$, $c=t^3$, $d=t^4$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "8c601ad6-c443-4ee7-82b3-23355676c75d",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "| bc-ad c3-bd2 ac2-b2d b3-a2c |\n",
      "\n",
      "        1       4\n",
      "Matrix R  <--- R\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "J = groebnerBasis I"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "ae9f9538-2d08-4b64-94e8-a7a5720e9f15",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "dim I"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "e68ec1f1-1a39-45b9-8f08-94ba7e20cfac",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "degree I"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3069ee1a-7209-42bc-ab6c-f7528e8553dd",
   "metadata": {},
   "source": [
    "Let us compute the kernel and co-kernel of the generators of this ideal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "98702749-5f90-4001-bc0e-7575740621a2",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "| bc-ad c3-bd2 ac2-b2d b3-a2c |\n",
      "\n",
      "        1       4\n",
      "Matrix R  <--- R\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "gens I"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "c98ccef4-3ea5-445f-b8cd-26efeb7c7032",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "cokernel | bc-ad c3-bd2 ac2-b2d b3-a2c |\n",
      "\n",
      "                       1\n",
      "R-module, quotient of R\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "M = coker gens I"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "ca49abf2-2787-4b7e-8789-082fcaf33f18",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "image {2} | c2 bd ac b2 |\n",
      "      {3} | -b -a 0  0  |\n",
      "      {3} | d  c  -b -a |\n",
      "      {3} | 0  0  -d -c |\n",
      "\n",
      "                        4\n",
      "R-module, submodule of R\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "ker gens I"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "adbbfbe9-f43a-4dab-8bdb-23fa8e66a939",
   "metadata": {},
   "source": [
    "The degree of the Hilbert polynomial gives the dimension."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "b0b0a01b-2c0a-4bb2-bf79-7a3209ed2ee0",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "- 3*P  + 4*P\n",
      "     0      1\n",
      "\n",
      "ProjectiveHilbertPolynomial\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "hilbertPolynomial M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "a48f97ec-1ddc-4613-8a1f-f131387534fd",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "         2     3     4    5\n",
      "    1 - T  - 3T  + 4T  - T\n",
      "i11:-----------------------\n",
      "                   4\n",
      "            (1 - T)\n",
      "\n",
      "Expression of class BinaryOperation\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "i11 : hilbertSeries M"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "faf7d6a1-20f8-4e20-95a1-fda69f60c84f",
   "metadata": {},
   "source": [
    "To conclude this tutorial section, we compute free resolutions and Betti tables."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "137883cd-4d7c-4754-b49d-8fbc0d9ad789",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " 1      4      4      1\n",
      "R  <-- R  <-- R  <-- R  <-- 0\n",
      "                             \n",
      "0      1      2      3      4\n",
      "\n",
      "ChainComplex\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "mres = res M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "949f3dbf-c4a6-4692-bef6-bafdf0ea9403",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "       0 1 2 3\n",
      "total: 1 4 4 1\n",
      "    0: 1 . . .\n",
      "    1: . 1 . .\n",
      "    2: . 3 4 1\n",
      "\n",
      "BettiTally\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "betti mres"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "100c9e59-8a90-4c41-beb5-142b1fb8bbd5",
   "metadata": {},
   "source": [
    "# 4. Elimination: Projecting the Twisted Cubic"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d7b3daa2-accb-4d0f-8010-bc72c19e6d7f",
   "metadata": {},
   "source": [
    "The twisted cubic is a space curve defined parametrically as\n",
    "\n",
    "$$\n",
    "   t \\mapsto (t, t^2, t^3)\n",
    "$$\n",
    "or by the equations $y = x^2$ and $z = x^3$, in affine coordinates.\n",
    "\n",
    "Consider the three 2-by-2 minors of the matrix\n",
    "\n",
    "$$\n",
    "   \\left[\n",
    "      \\begin{array}{ccc}\n",
    "         x_0 & x_1 & x_2 \\\\\n",
    "         x_1 & x_2 & x_3\n",
    "      \\end{array}\n",
    "   \\right]\n",
    "$$\n",
    "\n",
    "in the ring ${\\mathbb R} = {\\mathbb K}[x_0,x_1,x_2,x_3]$ and define\n",
    "the ideal\n",
    "\n",
    "$$\n",
    "   I = \\langle\n",
    "   \\left|\n",
    "      \\begin{array}{cc}\n",
    "         x_0 & x_1 \\\\\n",
    "         x_1 & x_2\n",
    "      \\end{array}\n",
    "   \\right|,\n",
    "   \\left|\n",
    "      \\begin{array}{cc}\n",
    "         x_0 & x_2 \\\\\n",
    "         x_1 & x_3\n",
    "      \\end{array}\n",
    "   \\right|,\n",
    "   \\left|\n",
    "      \\begin{array}{cc}\n",
    "         x_1 & x_2 \\\\\n",
    "         x_2 & x_3\n",
    "      \\end{array}\n",
    "   \\right| \\rangle.\n",
    "$$\n",
    "\n",
    "Then we work in projective coordinates $[x_0:x_1:x_2:x_3] \\in {\\mathbb P}^3$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "15320443-b85d-430a-9c3d-f0321e78a734",
   "metadata": {},
   "source": [
    "Let us make first the equations of the twisted cubic."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "70227951-702b-4e55-95f8-2816f316dbb7",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "R\n",
      "\n",
      "PolynomialRing\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "R = KK[x0, x1, x2, x3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "e790f818-f3a0-4838-a811-5ae00ec61b91",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "| x0 x1 x2 x3 |\n",
      "\n",
      "        1       4\n",
      "Matrix R  <--- R\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "v = vars(R)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "c16563cf-3aa8-4ce6-a7b5-8894f350bd43",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "| x0 x1 x2 |\n",
      "| x1 x2 x3 |\n",
      "\n",
      "        2       3\n",
      "Matrix R  <--- R\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "M = map(R^2, 3, (i,j)->v_(0,i+j))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "bd13cafe-2ec9-4a99-ade4-c188889df36e",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "| -x1^2+x0x2 -x1x2+x0x3 -x2^2+x1x3 |\n",
      "\n",
      "        1       3\n",
      "Matrix R  <--- R\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "I = gens minors(2, M)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2a3d99b3-4b7a-40d1-9875-d79b12acd0ca",
   "metadata": {},
   "source": [
    "To project the twisted cubic, we do the following:\n",
    "\n",
    "1. We define the center of the projection, as the point defined by\n",
    "\n",
    "   $$\n",
    "         \\langle x_0 + x_3, x_1, x_2 \\rangle.\n",
    "   $$\n",
    "\n",
    "2. We set up a ring map\n",
    "\n",
    "   * from the polynomial ring in the three variables, representing the plane,\n",
    "\n",
    "   * to the quotient ring,\n",
    "\n",
    "   taking the variables to the three linear forms\n",
    "   that define the projection center.\n",
    "\n",
    "3. We compute the kernel of this map."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "10c43c85-3b20-4adb-86f4-db102dcc497d",
   "metadata": {},
   "source": [
    "Defining the center, variables, quotient ring is done below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "ce2f0a46-a234-40b8-8a32-954cea024513",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ideal (x0 + x3, x1, x2)\n",
      "\n",
      "Ideal of R\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "pideal = ideal(x0+x3, x1, x2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "1ce8d570-e1ec-4d77-8c96-10c2271e2516",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "S1\n",
      "\n",
      "PolynomialRing\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "S1 = KK[y1, y2, y3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "0ea9b3cc-1372-4690-aba4-b881af83ec94",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Rbar\n",
      "\n",
      "QuotientRing\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "Rbar = R/(ideal I)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0192f2cf-3e0f-4e75-a53e-3e72bca75bf6",
   "metadata": {},
   "source": [
    "Then, we let Macaulay2 define the map and compute its kernel."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "697d2e4c-5cc1-493a-a2a9-ecc4d9537db4",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "map (Rbar, S1, {x0 + x3, x1, x2})\n",
      "\n",
      "RingMap Rbar <--- S1\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "f = map(Rbar, S1, matrix(Rbar,{{x0+x3, x1,x2}}))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "1032992a-c281-4596-a158-3fdb09f3bc3f",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "        3                3\n",
      "ideal(y2  - y1*y2*y3 + y3 )\n",
      "\n",
      "Ideal of S1\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "J1 = ker f"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1ab34119-1323-4751-bab5-1dc8ce528dbb",
   "metadata": {},
   "source": [
    "# 5. quotients and saturation"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "beb33320-a701-437d-8c47-103742faf4e7",
   "metadata": {},
   "source": [
    "For $I$ and $J$ ideals in some ring $R$, the ideal quotient is\n",
    "\n",
    "$$\n",
    "  (I:J) = \\{ \\ f \\in R \\ | f J \\subset I \\ \\}.\n",
    "$$\n",
    "\n",
    "We consider the case where $J$ is generated by one polynomial $g$.\n",
    "\n",
    "This case arises in the problem of homogenizing an ideal.\n",
    "\n",
    "Consider again the twisted cubic $t \\mapsto (t, t^2, t^3)$.\n",
    "\n",
    "Using $(a,b,c)$ as affine coordinates, the homogenization of $b = a^2$\n",
    "and $c = a^3$ happens with the addition of a fourth variable $d$ and\n",
    "leads to\n",
    "\n",
    "$$\n",
    "   b d - a = 0, \\quad c d^2 - a^3 = 0.\n",
    "$$\n",
    "    \n",
    "Problem: these two forms do not define the projective closure\n",
    "of the twisted cubic, because it contains $d = 0$ as well.\n",
    "\n",
    "To get the ideal of the closure we must remove unwanted components."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "d08dd184-8723-4d24-bc2d-52e6b954af51",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "R\n",
      "\n",
      "PolynomialRing\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "R = KK[a,b,c,d]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "60ecdc59-0709-4f3a-a694-69f336a5bd97",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "          2           3      2\n",
      "ideal (- a  + b*d, - a  + c*d )\n",
      "\n",
      "Ideal of R\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "I1 = ideal(d*b-a^2, d^2*c-a^3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "fb38c533-9eef-4ea6-9f62-3011e551d36f",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "| a2-bd abd-cd2 b2d2-acd2 |\n",
      "\n",
      "        1       3\n",
      "Matrix R  <--- R\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "gens gb I1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "48644872-38a1-403b-9d0c-8752533faa14",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(| a2-bd ab-cd b2-ac |, 2)\n",
      "\n",
      "Sequence\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "I2 = divideByVariable(gens gb I1,d)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "0fb9bff7-3ec5-45d5-a039-83787ef7177e",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "        2                    2\n",
      "ideal (b  - a*c, a*b - c*d, a  - b*d)\n",
      "\n",
      "Ideal of R\n"
     ]
    }
   ],
   "source": [
    "%%macaulay2\n",
    "saturate(I1, d)"
   ]
  }
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