ec: add example of using sage valuation ring

script/research/ec/valuation.ipynb

@@ -734,11 +734,107 @@
    "source": [
     "As observed we have a uniformizer $t$ with $k = 2$."
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a05bc126",
+   "metadata": {},
+   "source": [
+    "# Valuation with Places\n",
+    "\n",
+    "Using sage valuation rings since local ring is a DVR"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "3da404fc",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[Place (x, y),\n",
+       " Place (x + 3, y + 4),\n",
+       " Place (x + 3, y + 7),\n",
+       " Place (x + 5, y + 3),\n",
+       " Place (x + 5, y + 8),\n",
+       " Place (x + 7, y + 5),\n",
+       " Place (x + 7, y + 6),\n",
+       " Place (x + 9, y + 4),\n",
+       " Place (x + 9, y + 7),\n",
+       " Place (x + 10, y + 4),\n",
+       " Place (x + 10, y + 7)]"
+      ]
+     },
+     "execution_count": 2,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "K.<x> = FunctionField(GF(11))\n",
+    "_.<Y> = K[]\n",
+    "L.<y> = K.extension(Y^2 - x^3 - 4*x)\n",
+    "L.places_finite()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e4719879",
+   "metadata": {},
+   "source": [
+    "We are interested in $P = (2, 4)$ so we want the place $(x - 2, x - 4) = (x + 9, y + 7)$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "f87c8b23",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "Place (x + 9, y + 7)"
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "# I don't know how to actually construct this yet\n",
+    "p = L.places_finite()[-3]\n",
+    "p"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "3d007b0b",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "2"
+      ]
+     },
+     "execution_count": 4,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(y - 2*x).valuation(p)"
+   ]
   }
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "SageMath 9.6",
+   "display_name": "SageMath 9.5",
    "language": "sage",
    "name": "sagemath"
   },
@@ -752,7 +848,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.10.5"
+   "version": "3.10.7"
   }
  },
  "nbformat": 4,