diff --git a/notebooks/todo.ipynb b/notebooks/todo.ipynb
index 70ed6be..f3c0727 100644
--- a/notebooks/todo.ipynb
+++ b/notebooks/todo.ipynb
@@ -177,7 +177,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 6,
+   "execution_count": 2,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -327,7 +327,42 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 7,
+   "execution_count": 3,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Funciones añadidas"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "    def get_multidegree(self, p):\n",
+    "        a3 = p.parent()\n",
+    "        n = len(a3.gens())/2\n",
+    "        multidegrees = {}\n",
+    "        for k in p.dict().keys():\n",
+    "            multideg = vector(QQ,n*[0])\n",
+    "            for i in range(n):\n",
+    "                multideg[i] = k[i]+k[i+n]\n",
+    "            multidegrees[k] = multideg\n",
+    "        return multidegrees\n",
+    "        "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
    "metadata": {},
    "outputs": [
     {
@@ -343,7 +378,7 @@
        "0"
       ]
      },
-     "execution_count": 7,
+     "execution_count": 5,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -366,7 +401,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 50,
+   "execution_count": 6,
    "metadata": {},
    "outputs": [
     {
@@ -377,7 +412,7 @@
        " (1, 0, 0, 0, 0, 1, 1, 0): 1}"
       ]
      },
-     "execution_count": 50,
+     "execution_count": 6,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -389,54 +424,14 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 51,
+   "execution_count": null,
    "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/html": [
-       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrr}\n",
-       "1 & 0 & 0 & -2 & -\\frac{1}{2} \\\\\n",
-       "0 & 1 & 0 & -2 & -\\frac{1}{2} \\\\\n",
-       "0 & 0 & 1 & -1 & 0 \\\\\n",
-       "0 & 0 & 0 & 0 & 0 \\\\\n",
-       "0 & 0 & 0 & 0 & 0 \\\\\n",
-       "0 & 0 & 0 & 0 & 0\n",
-       "\\end{array}\\right)</script></html>"
-      ],
-      "text/latex": [
-       "\\begin{math}\n",
-       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrr}\n",
-       "1 & 0 & 0 & -2 & -\\frac{1}{2} \\\\\n",
-       "0 & 1 & 0 & -2 & -\\frac{1}{2} \\\\\n",
-       "0 & 0 & 1 & -1 & 0 \\\\\n",
-       "0 & 0 & 0 & 0 & 0 \\\\\n",
-       "0 & 0 & 0 & 0 & 0 \\\\\n",
-       "0 & 0 & 0 & 0 & 0\n",
-       "\\end{array}\\right)\n",
-       "\\end{math}"
-      ],
-      "text/plain": [
-       "[   1    0    0   -2 -1/2]\n",
-       "[   0    1    0   -2 -1/2]\n",
-       "[   0    0    1   -1    0]\n",
-       "[   0    0    0    0    0]\n",
-       "[   0    0    0    0    0]\n",
-       "[   0    0    0    0    0]"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "M = Matrix(QQ,6,5,[[-2,2,0,0,0],[-1,0,2,0,1/2],[-1,0,0,2,1/2],[0,-1,2,0,1/2],[0,-1,0,2,1/2],[0,0,-1,1,0]])\n",
-    "show(M.echelon_form())"
-   ]
+   "outputs": [],
+   "source": []
   },
   {
    "cell_type": "code",
-   "execution_count": 17,
+   "execution_count": 7,
    "metadata": {},
    "outputs": [
     {
@@ -452,7 +447,7 @@
        "True"
       ]
      },
-     "execution_count": 17,
+     "execution_count": 7,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -475,7 +470,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 57,
+   "execution_count": 8,
    "metadata": {},
    "outputs": [
     {
@@ -484,7 +479,7 @@
        "[t2^2*y0*y1, t1*y0*y2, t0*y1*y2]"
       ]
      },
-     "execution_count": 57,
+     "execution_count": 8,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -496,7 +491,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 67,
+   "execution_count": 9,
    "metadata": {},
    "outputs": [
     {
@@ -505,7 +500,7 @@
        "5*t2^2*y0*y1 + 5*y0*y1"
       ]
      },
-     "execution_count": 67,
+     "execution_count": 9,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -515,6 +510,29 @@
     "p"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "{(0, 0, 2, 0, 1, 1, 0, 0): 5,\n",
+       " (0, 1, 0, 0, 1, 0, 1, 0): -1,\n",
+       " (1, 0, 0, 0, 0, 1, 1, 0): 1}"
+      ]
+     },
+     "execution_count": 10,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "p = 5*t2^2*y0*y1-t1*y0*y2+t0*y1*y2\n",
+    "p.dict()"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": null,
@@ -522,6 +540,322 @@
    "outputs": [],
    "source": []
   },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def get_term(coef, exp, gens):\n",
+    "    return coef*prod(list(map(lambda exp, gen: gen^exp, exp, gens)))\n",
+    "\n",
+    "def get_terms(p):\n",
+    "    a = p.parent()\n",
+    "    n = len(a.gens())/2\n",
+    "    ts_only_gens = a.gens()[:n]+tuple(n*[1])\n",
+    "    ys_only_gens = tuple(n*[1])+a.gens()[n:]\n",
+    "    terms = {}\n",
+    "    pol = p.dict()\n",
+    "    for exp in pol.keys():\n",
+    "        key =  get_term(1, exp,ys_only_gens)\n",
+    "        if key not in terms.keys():\n",
+    "            coef = get_term(pol[exp],exp,ts_only_gens)\n",
+    "            terms[key] = coef\n",
+    "        else:\n",
+    "            coef = get_term(pol[exp],exp,ts_only_gens)\n",
+    "            terms[key] += coef    \n",
+    "    return terms\n",
+    "    "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 12,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "5*t2^2*y0*y1 - t1*y0*y2 + t0*y1*y2 + 3*y0*y1\n"
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "{y0*y1: 5*t2^2 + 3, y0*y2: -t1, y1*y2: t0}"
+      ]
+     },
+     "execution_count": 12,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "p = 5*t2^2*y0*y1-t1*y0*y2+t0*y1*y2+3*y0*y1\n",
+    "print(p)\n",
+    "terms = get_terms(p)\n",
+    "terms"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "True"
+      ]
+     },
+     "execution_count": 13,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "# comprobamos que la suma de los términos nos da el polinomio original\n",
+    "p2 = sum([terms[key]*key for key in terms.keys()])\n",
+    "p2==p"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 81,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "3"
+      ]
+     },
+     "execution_count": 81,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "\n",
+    "\n",
+    "        \n",
+    "    "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 80,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "{y1: 5*t0*t2^2 + 3*t0}"
+      ]
+     },
+     "execution_count": 80,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "lead_term = get_leading_term(p)\n",
+    "symbolic_partial_integral(lead_term, 0)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 114,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def get_leading_pol(pol):\n",
+    "    terms = get_terms(pol)\n",
+    "    keys = list(terms.keys())\n",
+    "    keys.sort(reverse=True)\n",
+    "    return terms[keys[0]]*keys[0]\n",
+    "\n",
+    "def get_integration_idx(exp):\n",
+    "    # gets the idx 'i' of the first generator 'y_i'  that appears in exp\n",
+    "    n = len(exp)/2\n",
+    "    filter_ts = n*[0]+n*[1]\n",
+    "    diff_forms = list(map(lambda v, f: v*f, exp, filter_ts))\n",
+    "    return next((i for i, x in enumerate(list(diff_forms)) if x), None)-n\n",
+    "\n",
+    "def symbolic_partial_monomial_integral(monomial, idx):\n",
+    "    exp = list(monomial.dict().keys())[0]\n",
+    "    coef = monomial.dict()[exp]\n",
+    "    n = len(exp)/2\n",
+    "    if (exp[n+idx]==0):\n",
+    "        raise IndexError(\"El término no se puede integrar respecto a esa variable\")\n",
+    "    integral = list(copy.deepcopy(exp))\n",
+    "    integral[n+idx] = 0\n",
+    "    integral[idx] += 1\n",
+    "    new_coef = coef/integral[idx]\n",
+    "    a = monomial.parent()\n",
+    "    return get_term(new_coef, integral, a.gens())\n",
+    "    \n",
+    "def symbolic_partial_integral(polynomial):\n",
+    "    # necessary condition: all monomials have apl.gens[n+integration_idx] factor\n",
+    "    exp = list(polynomial.dict().keys())[0]\n",
+    "    integration_idx = get_integration_idx(exp)\n",
+    "    aGens = polynomial.parent().gens()\n",
+    "    integral = 0\n",
+    "    for exp in polynomial.dict().keys():\n",
+    "        coef = polynomial.dict()[exp]\n",
+    "        monomial = get_term(coef, exp, aGens)\n",
+    "        integral += symbolic_partial_monomial_integral(monomial, integration_idx)\n",
+    "    terms = get_terms(integral)\n",
+    "    result = sum([terms[key]*key for key in terms.keys()])\n",
+    "    return result"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 115,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "5*t0*t2^2*y1 + 3*t0*y1"
+      ]
+     },
+     "execution_count": 115,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "symbolic_partial_integral(terms[y0*y1]*y0*y1)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 143,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def find_preimage_diff(pol):\n",
+    "    if pol.differential()!=0 :\n",
+    "        raise ValueError(\"This polynomial is not in the kernel of the differential\")\n",
+    "    lead_pol = get_leading_pol(pol)\n",
+    "    w = symbolic_partial_integral(lead_pol)\n",
+    "    theta = pol - w.differential()\n",
+    "    if (theta == 0):\n",
+    "        return w\n",
+    "    else:\n",
+    "        return w+find_preimage_diff(theta)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 144,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "t0*t1"
+      ]
+     },
+     "execution_count": 144,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "# Ejemplo 1\n",
+    "q = t0*y1+t1*y0\n",
+    "find_preimage_diff(q)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 149,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "True"
+      ]
+     },
+     "execution_count": 149,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "#Ejemplo 2\n",
+    "q1 = (-1/2)*(t0*t1^2*t2*t3*y0)\n",
+    "q2 =  (-1/2)*(t0^2*t1*t2*t3*y1)\n",
+    "q = q1+q2\n",
+    "dq = q.differential()\n",
+    "w = find_preimage_diff(dq)\n",
+    "w.differential() == dq"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 157,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|-1/2*t0*t1^2*t2*t3*y0|\\phantom{\\verb!x!}\\verb|-|\\phantom{\\verb!x!}\\verb|1/2*t0^2*t1*t2*t3*y1|</script></html>"
+      ],
+      "text/latex": [
+       "\\begin{math}\n",
+       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|-1/2*t0*t1^2*t2*t3*y0|\\phantom{\\verb!x!}\\verb|-|\\phantom{\\verb!x!}\\verb|1/2*t0^2*t1*t2*t3*y1|\n",
+       "\\end{math}"
+      ],
+      "text/plain": [
+       "-1/2*t0*t1^2*t2*t3*y0 - 1/2*t0^2*t1*t2*t3*y1"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/html": [
+       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|1/4*t0^2*t1^2*t3*y2|\\phantom{\\verb!x!}\\verb|+|\\phantom{\\verb!x!}\\verb|1/4*t0^2*t1^2*t2*y3|</script></html>"
+      ],
+      "text/latex": [
+       "\\begin{math}\n",
+       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|1/4*t0^2*t1^2*t3*y2|\\phantom{\\verb!x!}\\verb|+|\\phantom{\\verb!x!}\\verb|1/4*t0^2*t1^2*t2*y3|\n",
+       "\\end{math}"
+      ],
+      "text/plain": [
+       "1/4*t0^2*t1^2*t3*y2 + 1/4*t0^2*t1^2*t2*y3"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "show(q)\n",
+    "show(w)"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},