ec: simple intersection method

script/research/ec/valuation.ipynb

@@ -689,6 +689,51 @@
     "d = [i for i in range(1, 10) if f in M^i]\n",
     "max(d)"
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3b1b13e9",
+   "metadata": {},
+   "source": [
+    "# Simple Method by Parameterizing\n",
+    "\n",
+    "We parameterize our line $f$ so that $(x(0), y(0)) = P$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "02ae75dc",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "10*t^3 + 9*t^2"
+      ]
+     },
+     "execution_count": 1,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "K.<t> = GF(11)[]\n",
+    "x = t + 2\n",
+    "# y - 2x = 0\n",
+    "# => y = 2x\n",
+    "y = 2*x\n",
+    "# Now just write equation in terms of parameterized x(t) and y(t)\n",
+    "y^2 - x^3 - 4*x"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7381d05b",
+   "metadata": {},
+   "source": [
+    "As observed we have a uniformizer $t$ with $k = 2$."
+   ]
   }
  ],
  "metadata": {