Thesis_Appendix_B.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "bbc051d4",
   "metadata": {},
   "source": [
    "## Appendix B: Matroid realization\n",
    "\n",
    "To demonstrate some of OSCAR's capabilities for studying matroid realization, we preproduce the content of Appendix B."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "dc5760ff",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  ___   ____   ____    _    ____\n",
      " / _ \\ / ___| / ___|  / \\  |  _ \\   |  Combining ANTIC, GAP, Polymake, Singular\n",
      "| | | |\\___ \\| |     / _ \\ | |_) |  |  Type \"?Oscar\" for more information\n",
      "| |_| | ___) | |___ / ___ \\|  _ <   |  Manual: https://docs.oscar-system.org\n",
      " \\___/ |____/ \\____/_/   \\_\\_| \\_\\  |  1.2.0-DEV #polytrope 3e9a246 2024-08-03\n"
     ]
    }
   ],
   "source": [
    "using Oscar"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8d53a519",
   "metadata": {},
   "source": [
    "**1.** Define a $(3,12)$-matroid $Q$ via nonbases."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "3a91d0f2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matroid of rank 3 on 12 elements"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "NBs = [[1, 2, 3], [1, 4, 5], [1, 6, 10], [2, 4, 6],[2, 5, 8],\n",
    "[2, 9, 10], [3, 4, 7], [3, 5, 9],[3, 6, 11],[4, 10, 12],\n",
    "[5, 6, 7], [5, 11, 12],[7, 8, 11],[8, 9, 12]]; #define nonbases\n",
    "\n",
    "Q = matroid_from_nonbases(NBs,12)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0d4f2f80",
   "metadata": {},
   "source": [
    "**2.** Compute realization space of $Q$ over characteristic 0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "f33ee590",
   "metadata": {},
   "outputs": [],
   "source": [
    "R = realization_space(Q,B = [2,5,6],char= 0);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "80b0c273",
   "metadata": {},
   "source": [
    "**3.** `realization_space()` computes the data to produce the coordinate ring of $\\mathcal{R}(Q;\\mathbb{Q})$. That is, ambient ring $B_{Q}$, the ideal $I_{Q}$, and the semigroup $S_{Q}$. We see that $I_{Q}$ is a principal ideal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "f09b2c39",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Multivariate polynomial ring in 3 variables x1, x2, x3\n",
       "  over rational field"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R.ambient_ring"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "ac50f912",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Ideal generated by\n",
       "  x1^2*x2 + x1^2*x3 - x1^2 - x1*x2^2 - 2*x1*x2*x3 - 2*x1*x3^2 + x1*x3 + x1 + x2^2 + x2*x3^2 + x2*x3 - x2 + x3^3 - x3"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R.defining_ideal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "91f56b43",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "35-element Vector{RingElem}:\n",
       " x3 - 1\n",
       " x1 - x2\n",
       " x3\n",
       " x1 - x3\n",
       " x1 - 1\n",
       " x2 - 1\n",
       " x1*x2 - x1*x3 + x1 - x2\n",
       " x1^2*x3 - x1^2 + x1*x2 - x1*x3^2 + x1 - x2\n",
       " x1*x2 - x2 - x3 + 1\n",
       " x2\n",
       " x1*x2^2 - x1*x2 - x1*x3 + x1 - x2^2 + x2 + x3^2 - x3\n",
       " x1\n",
       " x1 - x2 - x3 + 1\n",
       " ⋮\n",
       " x1^2*x2 + x1^2*x3 - x1^2 - x1*x2*x3 - x1*x2 - 2*x1*x3 + 2*x1 + x2*x3 + x3^2 - x3\n",
       " x1*x2 + x1*x3 - x1 - x2*x3 - x3 + 1\n",
       " x1*x2 + 2*x1*x3 - 2*x1 - x2*x3 - x3^2 + 1\n",
       " x1^2 - x1*x3 - x1 + 1\n",
       " x1*x3 - x1 + x2\n",
       " x1^2 - x1*x3 - 2*x1 + 1\n",
       " x2 + x3 - 1\n",
       " x1^2*x2 - x1*x2*x3 - 2*x1*x2 + x2*x3 + x2 + x3^2 - x3\n",
       " x1*x2 + x1*x3 - x1 - x2*x3 - x2 - x3^2 - x3 + 2\n",
       " x1*x2 + x1*x3 - x1 - x2*x3 - x3^2 + 1\n",
       " x1*x3 - x1 - x2*x3 - x3^2 + 1\n",
       " x1*x3 - x1 + x2 + x3 - 1"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R.inequations"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ca1fc96a",
   "metadata": {},
   "source": [
    " By default, this data is presented after applying the elimination homomorphism. We could have computed the presentation in terms of determinants using `simplify = false`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "60f81060",
   "metadata": {},
   "outputs": [],
   "source": [
    "R2 = realization_space(Q,simplify = false,B = [2,5,6],char = 0);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "02d03cdf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Ideal generated by\n",
       "  x11 - x13\n",
       "  x8 - 1\n",
       "  x7 - x10\n",
       "  x3 - 1\n",
       "  x2 - x7\n",
       "  x1 - x7\n",
       "  x9*x12 - x12 - x13 + 1\n",
       "  x6*x12 + x6*x13 - x6 - x10*x12\n",
       "  -x5*x13 + x5 + x6*x13 + x10^2 - x10*x12 - x10*x13 - x10\n",
       "  x5*x10 - x5*x13 + x6*x13 - x10*x12\n",
       "  x4*x10 - x10 + 1\n",
       "  x6*x9 - x10\n",
       "  x4*x5 - x4*x10 + x13\n",
       "  x5*x12*x13 - x5*x12 + x5*x13^2 - 2*x5*x13 + x5 - x10*x12*x13 - x10*x13^2 + x10*x13 + x12^2 + x12*x13^2 + x12*x13 - x12 + x13^3 - x13\n",
       "  x5*x9*x13 - x5*x9 - x9*x10*x13 + x9*x13^2 + x9*x13 + x12 - 1\n",
       "  x4*x6*x13 + x5*x13 - x5 - x6*x13 - x10*x13 + x12 + x13^2 + x13\n",
       "  x5^2*x13 - x5^2 - x5*x6*x13 + x5*x12 + x5*x13 + x6*x10*x13 - x6*x13 - x10*x12*x13\n",
       "  x4*x12^2 + x4*x12*x13^2 + x4*x12*x13 - x4*x12 + x4*x13^3 - x4*x13 - x10*x12 - x10*x13 + x10 - x12*x13^2 + x12*x13 + x12 - x13^3 + 2*x13^2 - 1\n",
       "  x4*x9*x13^2 + x4*x9*x13 + x4*x12 - x4 - x9*x10 - x9*x13^2 + x9*x13 + x9"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R2.defining_ideal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "f400803f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "149-element Vector{RingElem}:\n",
       " x10*x13 - x10 - x11*x12 + x11 + x12 - x13\n",
       " x10*x13 - x11*x12\n",
       " x1*x11 - x1*x13 - x2*x10 + x2*x12 + x10*x13 - x11*x12\n",
       " x3*x10 - x3*x12 - x10*x13 + x11*x12\n",
       " x10 - x12\n",
       " x4*x10 - x4*x12 - x11 + x13\n",
       " x5*x11 - x5*x13 + x10*x13 - x11*x12\n",
       " x6*x11 - x6*x13 - x7*x10 + x7*x12 + x10*x13 - x11*x12\n",
       " x8*x11 - x8*x13 - x9*x10 + x9*x12 + x10*x13 - x11*x12\n",
       " x8*x13 - x8 - x9*x12 + x9 + x12 - x13\n",
       " x8*x13 - x9*x12\n",
       " x1*x9 - x1*x13 - x2*x8 + x2*x12 + x8*x13 - x9*x12\n",
       " x9 - x13\n",
       " ⋮\n",
       " x3*x5 - x3 - x5\n",
       " x1*x3 + x2*x5 - x3*x5\n",
       " x1 + x2*x5 - x2 - x5\n",
       " x3 + x4 - 1\n",
       " x1*x4 - x2 - x4 + 1\n",
       " x1*x4 - x2\n",
       " x4 - 1\n",
       " x1\n",
       " x1 - 1\n",
       " x2 - x3\n",
       " x2 - 1\n",
       " x1*x3 - x1 + x2 - x3"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R2.inequations"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ae20b13b",
   "metadata": {},
   "source": [
    "We return to our study for the simplified case.\n",
    "\n",
    "**4.**  We see $\\mathcal{R}(Q;\\mathbb{Q})$ has two connected components by computing the minimal primes of $I_{Q}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "b1c62683",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Vector{MPolyIdeal{QQMPolyRingElem}}:\n",
       " Ideal (x1*x2 + x1*x3 - x1 - x2 - x3^2 + 1)\n",
       " Ideal (x1 - x2 - x3)"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ms = minimal_primes(R.defining_ideal)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "afa14fe9",
   "metadata": {},
   "source": [
    "**5.** We veritfy the dimension below by directly computing the affine scheme."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "e503f5db",
   "metadata": {},
   "outputs": [],
   "source": [
    "S = reduce(product,powers_of_element.(R.inequations));#compute multiplicative semigroup"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "6630fe6f",
   "metadata": {},
   "outputs": [],
   "source": [
    "Rs = affine_scheme(R.ambient_ring,R.defining_ideal,S);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "cc4b842f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dim(Rs)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "30de4485",
   "metadata": {},
   "source": [
    "**6.** Verify $\\mathcal{R}(Q,\\mathbb{Q})$ is singular."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "070de701",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "false"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "is_smooth(R)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a94c9dc3",
   "metadata": {},
   "source": [
    "6. Produce realization matrix. Evaluating at appropriate points yields a realization of $Q$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "2fa57000",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1   1    1   1   0   0        0              x1 - 1          x2 + x3 - 1            x2    1    1]\n",
       "[1   0   x1   0   1   0       x1   x1^2 - x1*x3 - x1                x1*x2            x2   x1   x2]\n",
       "[1   0   x1   1   0   1   x1 - 1                   0   x1*x2 + x1*x3 - x1   x2 + x3 - 1   x3   x3]"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = R.realization_matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b09adefa",
   "metadata": {},
   "source": [
    "**7**. We obtain a realization for $Q$ by evaluating $A$ at the point  $t=\\left(\\frac{5}{3},2,-\\frac{1}{3}\\right)$. We construct a homomorphism $\\rho:B_{Q}\\to \\mathbb{Q}$ such that $\\rho(f) = f(t)$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "d9914b70",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Ring homomorphism\n",
       "  from multivariate polynomial ring in 3 variables over QQ\n",
       "  to rational field\n",
       "defined by\n",
       "  x1 -> 5//3\n",
       "  x2 -> 2\n",
       "  x3 -> -1//3"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rho = hom(R.ambient_ring,QQ,[5//3,2,-1//3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "e0f6b3f5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1   1      1   1   0   0      0   2//3    2//3      2       1       1]\n",
       "[1   0   5//3   0   1   0   5//3   5//3   10//3      2    5//3       2]\n",
       "[1   0   5//3   1   0   1   2//3      0   10//9   2//3   -1//3   -1//3]"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Arho = matrix(QQ,[rho(a) for a in A])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5c7e0ac5",
   "metadata": {},
   "source": [
    "**8.** We obtain a matroid $Q_{A}$ from Arho and verify it is isomorphic to $Q$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "75b4eb63",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matroid of rank 3 on 12 elements"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "QA = matroid_from_matrix_columns(Arho)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "5c09d38a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "is_isomorphic(Q,QA)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fe4b2773",
   "metadata": {},
   "source": [
    "**9.** Finally, we verify `Arho` corresponds to a point in the singular locus of $\\mathcal{R}(Q;\\mathbb{Q})$. That is, a point which annihilates the generators of both minimal primes of $I_{Q}$, and nothing in the semigroup $S_{Q}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "c5213881",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x1*x2 + x1*x3 - x1 - x2 - x3^2 + 1"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f1 = gens(ms[1])[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "b537808e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x1 - x2 - x3"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f2 = gens(ms[2])[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "ab30e055",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rho(f1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "id": "15537795",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rho(f2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "id": "563769d7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "false"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "0 in [rho(s) for s in R.inequations]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3d1db748",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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