add BKK checker to a public repo

BKKchecker.ipynb

+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "pressing-loading",
+   "metadata": {},
+   "source": [
+    "Starting with a set $ A \\subseteq \\mathbb{Z}^d, $ we figure out a set of equations defining $ X^A, $ and then compute the degree of $ X^A. $"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 155,
+   "id": "disabled-forth",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "5"
+      ]
+     },
+     "execution_count": 155,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "A = [ [1,0], [0,1], [2,0], [1,1], [1,2] ]\n",
+    "# Warning: Do not input [0,...,0] as a vector in A;\n",
+    "# it is automatically added to homogenize throughout\n",
+    "# the code execution.\n",
+    "\n",
+    "parameters = [ \"t\" + str(i+1) for i in range(len(A[0])) ]\n",
+    "variables = [ \"x\" + str(i+1) for i in range(len(A)) ]\n",
+    "\n",
+    "R = PolynomialRing(CC, parameters + variables)\n",
+    "\n",
+    "def make_term(i, exponent):\n",
+    "    term = \"x\" + str(i+1) + \" - \"\n",
+    "    second = \"1\"\n",
+    "    for j in range(len(exponent)):\n",
+    "        e = exponent[j]\n",
+    "        if e > 0:\n",
+    "            second = second + \" * \" + \"t\" + str(j+1) + \"^(\" + str(e) + \")\"\n",
+    "    return (term + second)\n",
+    "\n",
+    "# this ideal is the affine variety; now we homogenize\n",
+    "I = R.ideal([make_term(i, A[i]) for i in range(len(variables))])\n",
+    "\n",
+    "S = PolynomialRing(CC, variables)\n",
+    "\n",
+    "# By elimination theory, if we take a Groebner basis of I, \n",
+    "# and then remove all the terms containing a 'ti' parameter,\n",
+    "# we get an equation for our affine variety.\n",
+    "def filter_basis(basis):\n",
+    "    filtered_basis = []\n",
+    "    for poly in basis:\n",
+    "        poly_str = str(poly)\n",
+    "        if not ('t' in poly_str):\n",
+    "            filtered_basis.append(poly_str)\n",
+    "    return filtered_basis\n",
+    "\n",
+    "# So, take the ideal of our affine variety,\n",
+    "# and then homogenize with a new variable x0.\n",
+    "J = S.ideal(filter_basis(I.groebner_basis()))\n",
+    "\n",
+    "K = J.homogenize(\"x0\")\n",
+    "\n",
+    "# now, we can compute deg(X^A), since\n",
+    "# the Hilbert polynomial of X^A is\n",
+    "# deg(X^A) * t^d/d! + O(t^{d-1})\n",
+    "hilbert_poly = K.hilbert_polynomial()\n",
+    "\n",
+    "# output degree!\n",
+    "\n",
+    "dimension = hilbert_poly.degree()\n",
+    "factorial(dimension) * hilbert_poly.coefficients()[-1]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "disabled-uncle",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
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+   "name": "sagemath"
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+    "version": 3
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+   "file_extension": ".py",
+   "mimetype": "text/x-python",
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+ "nbformat_minor": 5
+}