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defeo/ace-2.ipynb

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MA2-ace TD 2
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ace-2.ipynb
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# TD 2\n",
"\n",
"## Exo 1.3"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"A.<x> = ZZ[]"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"x^7 + x^5 + 2*x^3 + 2*x^2 + 3*x + 2"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"P = x^7 + x^5 + 2*x^3 + 2*x^2 + 3*x + 2\n",
"P"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"x * (x^3 + x^2 + 1)^2"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"P.change_ring(GF(2)).factor()"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(x + 4) * (x^6 + 3*x^5 + 3*x^4 + 2*x^3 + x^2 + 5*x + 4)"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"P.change_ring(GF(7)).factor()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"On voit que soit $P$ est irreductible, soit il a une racine congrue à $-4 \\bmod 14$. On vérifie aisément que $P(x) > 0$ si $x>0$. On vérifie aussi que $P(-4)$ n'est pas racine:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-17514"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"P(-4)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Reste à vérifier que $P(-4 - c·14) ≠ 0$ pour tout $c>1$. Exercice."
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"P.is_irreducible()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Exo 1.4"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"x * (x + 1)\n",
"x^2 + x + 1\n"
]
}
],
"source": [
"p = 2\n",
"k = GF(p)\n",
"A.<X> = k[]\n",
"for a in k:\n",
" print((x^p - x + a).factor())"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"x * (x + 1) * (x + 2)\n",
"x^3 + 2*x + 1\n",
"x^3 + 2*x + 2\n"
]
}
],
"source": [
"p = 3\n",
"k = GF(p)\n",
"A.<X> = k[]\n",
"for a in k:\n",
" print((x^p - x + a).factor())"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"x * (x + 1) * (x + 2) * (x + 3) * (x + 4)\n",
"x^5 + 4*x + 1\n",
"x^5 + 4*x + 2\n",
"x^5 + 4*x + 3\n",
"x^5 + 4*x + 4\n"
]
}
],
"source": [
"p = 5\n",
"k = GF(p)\n",
"A.<X> = k[]\n",
"for a in k:\n",
" print((x^p - x + a).factor())"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"x * (x + 1) * (x + 2) * (x + 3) * (x + 4) * (x + 5) * (x + 6)\n",
"x^7 + 6*x + 1\n",
"x^7 + 6*x + 2\n",
"x^7 + 6*x + 3\n",
"x^7 + 6*x + 4\n",
"x^7 + 6*x + 5\n",
"x^7 + 6*x + 6\n"
]
}
],
"source": [
"p = 7\n",
"k = GF(p)\n",
"A.<X> = k[]\n",
"for a in k:\n",
" print((x^p - x + a).factor())"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"On conjecture que $x^p - x + c$ est scindé pour $c=0$ (facile) et irréductible pour $c≠0$ (plus dur)."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Exo 2.1"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [],
"source": [
"A.<x,y> = QQ[]"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3*x^3*y^3 + x*y^5 + 6*x^3*y + 4*x^2*y^2 + 2*y^4 + 5*x*y^2 + 7*y"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"P = x*y^5 + 2*y^4 + 3*y^3*x^3 + 4*x^2*y^2 + 5*x*y^2 + 6*y*x^3 + 7*y\n",
"P"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"6"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"P.degree()"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(3*y^3 + 6*y)*x^3 + 4*y^2*x^2 + (y^5 + 5*y^2)*x + 2*y^4 + 7*y"
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Q = P.polynomial(x)\n",
"Q"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Univariate Polynomial Ring in x over Univariate Polynomial Ring in y over Rational Field"
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Q.parent()"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Univariate Polynomial Ring in x over Univariate Polynomial Ring in y over Rational Field"
]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"B = QQ['y']['x']\n",
"B"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(3*y^3 + 6*y)*x^3 + 4*y^2*x^2 + (y^5 + 5*y^2)*x + 2*y^4 + 7*y"
]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"B(P)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Attention, le nom des variables doit être exactement le même pour que Sage accepte de faire la conversion."
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Univariate Polynomial Ring in X over Univariate Polynomial Ring in Y over Rational Field"
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"C = QQ['Y']['X']\n",
"C"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {},
"outputs": [
{
"ename": "TypeError",
"evalue": "not a constant polynomial",
"output_type": "error",
"traceback": [
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[0;31mTypeError\u001b[0m Traceback (most recent call last)",
"\u001b[0;32m<ipython-input-25-d7215608f13a>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mC\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mP\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
"\u001b[0;32m/opt/conda/lib/python3.6/site-packages/sage/structure/parent.pyx\u001b[0m in \u001b[0;36msage.structure.parent.Parent.__call__ (build/cythonized/sage/structure/parent.c:9727)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 918\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mmor\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 919\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mno_extra_args\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 920\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mmor\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_call_\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 921\u001b[0m \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 922\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mmor\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_call_with_args\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mkwds\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m/opt/conda/lib/python3.6/site-packages/sage/structure/coerce_maps.pyx\u001b[0m in \u001b[0;36msage.structure.coerce_maps.NamedConvertMap._call_ (build/cythonized/sage/structure/coerce_maps.c:6016)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 269\u001b[0m \u001b[0;32mraise\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"Cannot coerce {} to {}\"\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mformat\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 270\u001b[0m \u001b[0mcdef\u001b[0m \u001b[0mMap\u001b[0m \u001b[0mm\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 271\u001b[0;31m \u001b[0mcdef\u001b[0m \u001b[0mElement\u001b[0m \u001b[0me\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mmethod\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 272\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0me\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 273\u001b[0m \u001b[0;32mraise\u001b[0m \u001b[0mRuntimeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"BUG in coercion model: {} method of {} returned None\"\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mformat\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmethod_name\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mtype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m/opt/conda/lib/python3.6/site-packages/sage/rings/polynomial/multi_polynomial.pyx\u001b[0m in \u001b[0;36msage.rings.polynomial.multi_polynomial.MPolynomial._polynomial_ (build/cythonized/sage/rings/polynomial/multi_polynomial.c:5669)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 195\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mR\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpolynomial\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_parent\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvar\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 196\u001b[0m \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 197\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mR\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 198\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 199\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mcoefficients\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m/opt/conda/lib/python3.6/site-packages/sage/structure/parent.pyx\u001b[0m in \u001b[0;36msage.structure.parent.Parent.__call__ (build/cythonized/sage/structure/parent.c:9727)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 918\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mmor\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 919\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mno_extra_args\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 920\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mmor\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_call_\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 921\u001b[0m \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 922\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mmor\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_call_with_args\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mkwds\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m/opt/conda/lib/python3.6/site-packages/sage/structure/coerce_maps.pyx\u001b[0m in \u001b[0;36msage.structure.coerce_maps.DefaultConvertMap_unique._call_ (build/cythonized/sage/structure/coerce_maps.c:4554)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 143\u001b[0m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mtype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 144\u001b[0m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mtype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_element_constructor\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_element_constructor\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 145\u001b[0;31m \u001b[0;32mraise\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 146\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 147\u001b[0m \u001b[0mcpdef\u001b[0m \u001b[0mElement\u001b[0m \u001b[0m_call_with_args\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mkwds\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m{\u001b[0m\u001b[0;34m}\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m/opt/conda/lib/python3.6/site-packages/sage/structure/coerce_maps.pyx\u001b[0m in \u001b[0;36msage.structure.coerce_maps.DefaultConvertMap_unique._call_ (build/cythonized/sage/structure/coerce_maps.c:4422)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 138\u001b[0m \u001b[0mcdef\u001b[0m \u001b[0mParent\u001b[0m \u001b[0mC\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_codomain\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 139\u001b[0m \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 140\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_element_constructor\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 141\u001b[0m \u001b[0;32mexcept\u001b[0m \u001b[0mException\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 142\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mprint_warnings\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m/opt/conda/lib/python3.6/site-packages/sage/rings/polynomial/polynomial_ring.py\u001b[0m in \u001b[0;36m_element_constructor_\u001b[0;34m(self, x, check, is_gen, construct, **kwds)\u001b[0m\n\u001b[1;32m 405\u001b[0m \u001b[0mC\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0melement_class\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 406\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mlist\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mtuple\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 407\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcheck\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mcheck\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mis_gen\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mFalse\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mconstruct\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mconstruct\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 408\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 409\u001b[0m return C(self, list(x), check=check, is_gen=False,\n",
"\u001b[0;32m/opt/conda/lib/python3.6/site-packages/sage/rings/polynomial/polynomial_element.pyx\u001b[0m in \u001b[0;36msage.rings.polynomial.polynomial_element.Polynomial_generic_dense.__init__ (build/cythonized/sage/rings/polynomial/polynomial_element.c:96766)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 10489\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlist\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 10490\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mcheck\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m> 10491\u001b[0;31m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__coeffs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mR\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mt\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mt\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 10492\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__normalize\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 10493\u001b[0m \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m/opt/conda/lib/python3.6/site-packages/sage/structure/parent.pyx\u001b[0m in \u001b[0;36msage.structure.parent.Parent.__call__ (build/cythonized/sage/structure/parent.c:9727)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 918\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mmor\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 919\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mno_extra_args\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 920\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mmor\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_call_\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 921\u001b[0m \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 922\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mmor\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_call_with_args\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mkwds\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m/opt/conda/lib/python3.6/site-packages/sage/structure/coerce_maps.pyx\u001b[0m in \u001b[0;36msage.structure.coerce_maps.NamedConvertMap._call_ (build/cythonized/sage/structure/coerce_maps.c:6016)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 269\u001b[0m \u001b[0;32mraise\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"Cannot coerce {} to {}\"\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mformat\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 270\u001b[0m \u001b[0mcdef\u001b[0m \u001b[0mMap\u001b[0m \u001b[0mm\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 271\u001b[0;31m \u001b[0mcdef\u001b[0m \u001b[0mElement\u001b[0m \u001b[0me\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mmethod\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 272\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0me\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 273\u001b[0m \u001b[0;32mraise\u001b[0m \u001b[0mRuntimeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"BUG in coercion model: {} method of {} returned None\"\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mformat\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmethod_name\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mtype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m/opt/conda/lib/python3.6/site-packages/sage/rings/polynomial/multi_polynomial.pyx\u001b[0m in \u001b[0;36msage.rings.polynomial.multi_polynomial.MPolynomial._polynomial_ (build/cythonized/sage/rings/polynomial/multi_polynomial.c:5669)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 195\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mR\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpolynomial\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_parent\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvar\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 196\u001b[0m \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 197\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mR\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 198\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 199\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mcoefficients\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m/opt/conda/lib/python3.6/site-packages/sage/structure/parent.pyx\u001b[0m in \u001b[0;36msage.structure.parent.Parent.__call__ (build/cythonized/sage/structure/parent.c:9727)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 918\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mmor\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 919\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mno_extra_args\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 920\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mmor\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_call_\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 921\u001b[0m \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 922\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mmor\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_call_with_args\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mkwds\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m/opt/conda/lib/python3.6/site-packages/sage/structure/coerce_maps.pyx\u001b[0m in \u001b[0;36msage.structure.coerce_maps.DefaultConvertMap_unique._call_ (build/cythonized/sage/structure/coerce_maps.c:4554)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 143\u001b[0m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mtype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 144\u001b[0m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mtype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_element_constructor\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_element_constructor\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 145\u001b[0;31m \u001b[0;32mraise\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 146\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 147\u001b[0m \u001b[0mcpdef\u001b[0m \u001b[0mElement\u001b[0m \u001b[0m_call_with_args\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mkwds\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m{\u001b[0m\u001b[0;34m}\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m/opt/conda/lib/python3.6/site-packages/sage/structure/coerce_maps.pyx\u001b[0m in \u001b[0;36msage.structure.coerce_maps.DefaultConvertMap_unique._call_ (build/cythonized/sage/structure/coerce_maps.c:4422)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 138\u001b[0m \u001b[0mcdef\u001b[0m \u001b[0mParent\u001b[0m \u001b[0mC\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_codomain\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 139\u001b[0m \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 140\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_element_constructor\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 141\u001b[0m \u001b[0;32mexcept\u001b[0m \u001b[0mException\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 142\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mprint_warnings\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m/opt/conda/lib/python3.6/site-packages/sage/rings/polynomial/polynomial_ring.py\u001b[0m in \u001b[0;36m_element_constructor_\u001b[0;34m(self, x, check, is_gen, construct, **kwds)\u001b[0m\n\u001b[1;32m 405\u001b[0m \u001b[0mC\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0melement_class\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 406\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mlist\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mtuple\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 407\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcheck\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mcheck\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mis_gen\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mFalse\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mconstruct\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mconstruct\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 408\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 409\u001b[0m return C(self, list(x), check=check, is_gen=False,\n",
"\u001b[0;32m/opt/conda/lib/python3.6/site-packages/sage/rings/polynomial/polynomial_rational_flint.pyx\u001b[0m in \u001b[0;36msage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint.__init__ (build/cythonized/sage/rings/polynomial/polynomial_rational_flint.cpp:5083)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 236\u001b[0m \u001b[0;32mreturn\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 237\u001b[0m \u001b[0;32melif\u001b[0m \u001b[0mlen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 238\u001b[0;31m Polynomial_rational_flint.__init__(self, parent, x[0],\n\u001b[0m\u001b[1;32m 239\u001b[0m check=check, is_gen=False, construct=construct)\n\u001b[1;32m 240\u001b[0m \u001b[0;32mreturn\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m/opt/conda/lib/python3.6/site-packages/sage/rings/polynomial/polynomial_rational_flint.pyx\u001b[0m in \u001b[0;36msage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint.__init__ (build/cythonized/sage/rings/polynomial/polynomial_rational_flint.cpp:6066)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 285\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 286\u001b[0m \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 287\u001b[0;31m \u001b[0mx\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mparent\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mbase_ring\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 288\u001b[0m Polynomial_rational_flint.__init__(self, parent, x, check=check,\n\u001b[1;32m 289\u001b[0m is_gen=is_gen, construct=construct)\n",
"\u001b[0;32m/opt/conda/lib/python3.6/site-packages/sage/structure/parent.pyx\u001b[0m in \u001b[0;36msage.structure.parent.Parent.__call__ (build/cythonized/sage/structure/parent.c:9727)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 918\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mmor\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 919\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mno_extra_args\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 920\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mmor\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_call_\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 921\u001b[0m \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 922\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mmor\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_call_with_args\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mkwds\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m/opt/conda/lib/python3.6/site-packages/sage/rings/polynomial/polynomial_element.pyx\u001b[0m in \u001b[0;36msage.rings.polynomial.polynomial_element.ConstantPolynomialSection._call_ (build/cythonized/sage/rings/polynomial/polynomial_element.c:105936)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 11377\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0;34m<\u001b[0m\u001b[0mElement\u001b[0m\u001b[0;34m>\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m<\u001b[0m\u001b[0mPolynomial\u001b[0m\u001b[0;34m>\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mconstant_coefficient\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 11378\u001b[0m \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m> 11379\u001b[0;31m \u001b[0;32mraise\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"not a constant polynomial\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 11380\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 11381\u001b[0m \u001b[0mcdef\u001b[0m \u001b[0;32mclass\u001b[0m \u001b[0mPolynomialBaseringInjection\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mMorphism\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;31mTypeError\u001b[0m: not a constant polynomial"
]
}
],
"source": [
"C(P)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Exo 2.2"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"x^3 + 2*x^2*y + x*y^2 + x^2 + 2*x*y + y^2"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A.<x,y> = QQ[]\n",
"P = (x^2 + x*y + x + y) * (x + y)\n",
"P"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(x + 1) * (x + y)^2"
]
},
"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"fac = P.factor()\n",
"fac"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Attention: en Sage, une \"factorisation\" est un objet différent d'un polynôme."
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"<class 'sage.structure.factorization.Factorization'>"
]
},
"execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"parent(fac)"
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"<class 'sage.structure.factorization.Factorization'>"
]
},
"execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"type(fac)"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"False"
]
},
"execution_count": 30,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"fac in A"
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"x^3 + 2*x^2*y + x*y^2 + x^2 + 2*x*y + y^2"
]
},
"execution_count": 31,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"e = fac.expand()\n",
"e"
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 32,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"e in A"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"x^3 + (2*y + 1)*x^2 + (y^2 + 2*y)*x + y^2"
]
},
"execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"P.polynomial(x)"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(x + 1)*y^2 + (2*x^2 + 2*x)*y + x^3 + x^2"
]
},
"execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"P.polynomial(y)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Exo 3.1"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3*x^3*y^3 + x*y^5 + 6*x^3*y + 4*x^2*y^2 + 2*y^4 + 5*x*y^2 + 7*y"
]
},
"execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A.<x,y> = QQ[]\n",
"P = x*y^5 + 2*y^4 + 3*y^3*x^3 + 4*x^2*y^2 + 5*x*y^2 + 6*y*x^3 + 7*y\n",
"P"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Multivariate Polynomial Ring in x, y over Rational Field"
]
},
"execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"B.<x,y> = PolynomialRing(QQ, order='lex')\n",
"B"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Lexicographic term order"
]
},
"execution_count": 37,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"B.term_order()"
]
},
{
"cell_type": "code",
"execution_count": 38,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Degree reverse lexicographic term order"
]
},
"execution_count": 38,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A.term_order()"
]
},
{
"cell_type": "code",
"execution_count": 39,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3*x^3*y^3 + 6*x^3*y + 4*x^2*y^2 + x*y^5 + 5*x*y^2 + 2*y^4 + 7*y"
]
},
"execution_count": 39,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"B(P)"
]
},
{
"cell_type": "code",
"execution_count": 40,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Multivariate Polynomial Ring in x, y over Rational Field"
]
},
"execution_count": 40,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"C = A.change_ring(order='deglex')\n",
"C"
]
},
{
"cell_type": "code",
"execution_count": 41,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Degree lexicographic term order"
]
},
"execution_count": 41,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"C.term_order()"
]
},
{
"cell_type": "code",
"execution_count": 42,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3*x^3*y^3 + x*y^5 + 6*x^3*y + 4*x^2*y^2 + 2*y^4 + 5*x*y^2 + 7*y"
]
},
"execution_count": 42,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"C(P)"
]
},
{
"cell_type": "code",
"execution_count": 43,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3*x^3*y^3 + x*y^5 + 6*x^3*y + 4*x^2*y^2 + 2*y^4 + 5*x*y^2 + 7*y"
]
},
"execution_count": 43,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"P"
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Degree reverse lexicographic term order"
]
},
"execution_count": 44,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"P.parent().term_order()"
]
},
{
"cell_type": "code",
"execution_count": 45,
"metadata": {},
"outputs": [],
"source": [
"%display latex"
]
},
{
"cell_type": "code",
"execution_count": 46,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x^{2} y z^{3} t + x^{3} z^{2} t^{2} + x y^{3} z t + x^{2} y z^{2} t</script></html>"
],
"text/plain": [
"x^2*y*z^3*t + x^3*z^2*t^2 + x*y^3*z*t + x^2*y*z^2*t"
]
},
"execution_count": 46,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A.<x,y,z,t> = QQ[]\n",
"Q = x*y^3*z*t + x^2*y*z^3*t + x^2*y*z^2*t + x^3*z^2*t^2\n",
"Q"
]
},
{
"cell_type": "code",
"execution_count": 47,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x^{3} z^{2} t^{2} + x^{2} y z^{3} t + x^{2} y z^{2} t + x y^{3} z t</script></html>"
],
"text/plain": [
"x^3*z^2*t^2 + x^2*y*z^3*t + x^2*y*z^2*t + x*y^3*z*t"
]
},
"execution_count": 47,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A.change_ring(order='lex')(Q)"
]
},
{
"cell_type": "code",
"execution_count": 48,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x^{3} z^{2} t^{2} + x^{2} y z^{3} t + x^{2} y z^{2} t + x y^{3} z t</script></html>"
],
"text/plain": [
"x^3*z^2*t^2 + x^2*y*z^3*t + x^2*y*z^2*t + x*y^3*z*t"
]
},
"execution_count": 48,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A.change_ring(order='deglex')(Q)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Exo 3.4"
]
},
{
"cell_type": "code",
"execution_count": 49,
"metadata": {},
"outputs": [],
"source": [
"A.<x,y> = PolynomialRing(QQ, order='lex')\n",
"g = x - y\n",
"h = x - y^2\n",
"p = x*y - x"
]
},
{
"cell_type": "code",
"execution_count": 50,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(x - y, x - y^{2}, x y - x\\right)</script></html>"
],
"text/plain": [
"(x - y, x - y^2, x*y - x)"
]
},
"execution_count": 50,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"g, h, p"
]
},
{
"cell_type": "code",
"execution_count": 51,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}y^{3} - y^{2}</script></html>"
],
"text/plain": [
"y^3 - y^2"
]
},
"execution_count": 51,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"p.reduce([g, h])"
]
},
{
"cell_type": "code",
"execution_count": 52,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}y^{2} - y</script></html>"
],
"text/plain": [
"y^2 - y"
]
},
"execution_count": 52,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"p.reduce([h, g])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Le résultat de `reduce` n'est pas uniquement défini si la liste des polynômes passés ne forme pas une base de Gröbner. Ceci est précisé dans la doc"
]
},
{
"cell_type": "code",
"execution_count": 53,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Help on built-in function reduce:\n",
"\n",
"reduce(...) method of sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular instance\n",
" MPolynomial_libsingular.reduce(self, I)\n",
" File: sage/rings/polynomial/multi_polynomial_libsingular.pyx (starting at line 4482)\n",
" \n",
" Return a remainder of this polynomial modulo the \n",
" polynomials in ``I``.\n",
" \n",
" INPUT:\n",
" \n",
" - ``I`` - an ideal or a list/set/iterable of polynomials. \n",
" \n",
" OUTPUT:\n",
" \n",
" A polynomial ``r`` such that:\n",
" \n",
" - ``self`` - ``r`` is in the ideal generated by ``I``.\n",
" \n",
" - No term in ``r`` is divisible by any of the leading monomials\n",
" of ``I``.\n",
" \n",
" The result ``r`` is canonical if:\n",
" \n",
" - ``I`` is an ideal, and Sage can compute a Groebner basis of it.\n",
" \n",
" - ``I`` is a list/set/iterable that is a (strong) Groebner basis\n",
" for the term order of ``self``. (A strong Groebner basis is \n",
" such that for every leading term ``t`` of the ideal generated\n",
" by ``I``, there exists an element ``g`` of ``I`` such that the\n",
" leading term of ``g`` divides ``t``.)\n",
" \n",
" The result ``r`` is implementation-dependent (and possibly \n",
" order-dependent) otherwise. If ``I`` is an ideal and no Groebner\n",
" basis can be computed, its list of generators ``I.gens()`` is\n",
" used for the reduction.\n",
" \n",
" EXAMPLES::\n",
" \n",
" sage: P.<x,y,z> = QQ[]\n",
" sage: f1 = -2 * x^2 + x^3\n",
" sage: f2 = -2 * y + x* y\n",
" sage: f3 = -x^2 + y^2\n",
" sage: F = Ideal([f1,f2,f3])\n",
" sage: g = x*y - 3*x*y^2\n",
" sage: g.reduce(F)\n",
" -6*y^2 + 2*y\n",
" sage: g.reduce(F.gens())\n",
" -6*y^2 + 2*y\n",
" \n",
" `\\ZZ` is also supported. ::\n",
" \n",
" sage: P.<x,y,z> = ZZ[]\n",
" sage: f1 = -2 * x^2 + x^3\n",
" sage: f2 = -2 * y + x* y\n",
" sage: f3 = -x^2 + y^2\n",
" sage: F = Ideal([f1,f2,f3])\n",
" sage: g = x*y - 3*x*y^2\n",
" sage: g.reduce(F)\n",
" -6*y^2 + 2*y\n",
" sage: g.reduce(F.gens())\n",
" -6*y^2 + 2*y\n",
" \n",
" sage: f = 3*x\n",
" sage: f.reduce([2*x,y])\n",
" 3*x\n",
" \n",
" The reduction is not canonical when ``I`` is not a Groebner \n",
" basis::\n",
" \n",
" sage: A.<x,y> = QQ[]\n",
" sage: (x+y).reduce([x+y, x-y])\n",
" 2*y\n",
" sage: (x+y).reduce([x-y, x+y])\n",
" 0\n",
"\n"
]
}
],
"source": [
"help(p.reduce)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"On retrouve facilement quel est le calcul qui a été fait par `reduce`"
]
},
{
"cell_type": "code",
"execution_count": 54,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}y^{3} - y^{2}</script></html>"
],
"text/plain": [
"y^3 - y^2"
]
},
"execution_count": 54,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"p.mod(h)"
]
},
{
"cell_type": "code",
"execution_count": 55,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}y^{2} - y</script></html>"
],
"text/plain": [
"y^2 - y"
]
},
"execution_count": 55,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"p.mod(g)"
]
},
{
"cell_type": "code",
"execution_count": 56,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}y^{3} - y^{2}</script></html>"
],
"text/plain": [
"y^3 - y^2"
]
},
"execution_count": 56,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"p.mod(h).mod(g)"
]
},
{
"cell_type": "code",
"execution_count": 57,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}y^{2} - y</script></html>"
],
"text/plain": [
"y^2 - y"
]
},
"execution_count": 57,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"p.mod(g).mod(h)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"On pourrait croire que `reduce` reduit modulo la liste de polynômes dans l'ordre dans lequel ils sont donnés, c'est plus compliqué que cela, cependant. Dans cet exemple, on voit que le résultat n'est pas canonique, mais ne depend pas de l'ordre."
]
},
{
"cell_type": "code",
"execution_count": 58,
"metadata": {},
"outputs": [],
"source": [
"g = x^2*y^2 - x\n",
"h = x*y^2 + y"
]
},
{
"cell_type": "code",
"execution_count": 59,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}- x y - x</script></html>"
],
"text/plain": [
"-x*y - x"
]
},
"execution_count": 59,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"g.reduce([g,h])"
]
},
{
"cell_type": "code",
"execution_count": 60,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}0</script></html>"
],
"text/plain": [
"0"
]
},
"execution_count": 60,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"h.reduce([g,h])"
]
},
{
"cell_type": "code",
"execution_count": 61,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}- x y - x</script></html>"
],
"text/plain": [
"-x*y - x"
]
},
"execution_count": 61,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"g.reduce([h,g])"
]
},
{
"cell_type": "code",
"execution_count": 62,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}0</script></html>"
],
"text/plain": [
"0"
]
},
"execution_count": 62,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A.ideal([g,h]).reduce(g)"
]
},
{
"cell_type": "code",
"execution_count": 63,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}0</script></html>"
],
"text/plain": [
"0"
]
},
"execution_count": 63,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"g.mod([g,h])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Exo 3.6"
]
},
{
"cell_type": "code",
"execution_count": 64,
"metadata": {},
"outputs": [],
"source": [
"%display plain"
]
},
{
"cell_type": "code",
"execution_count": 65,
"metadata": {},
"outputs": [],
"source": [
"A.<x,y> = PolynomialRing(QQ, order='lex')\n",
"f = x^7*y^2 + x^3*y^2 - y - 1"
]
},
{
"cell_type": "code",
"execution_count": 66,
"metadata": {},
"outputs": [],
"source": [
"f1 = x*y^2 - x\n",
"f2 = x - y^3"
]
},
{
"cell_type": "code",
"execution_count": 67,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Ideal (x*y^2 - x, x - y^3) of Multivariate Polynomial Ring in x, y over Rational Field"
]
},
"execution_count": 67,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"I = A.ideal([f1, f2])\n",
"I"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Cette fois-ci la réduction calcule bien une base de Gröbner"
]
},
{
"cell_type": "code",
"execution_count": 69,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2*y^3 - y - 1"
]
},
"execution_count": 69,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"I.reduce(f)"
]
},
{
"cell_type": "code",
"execution_count": 70,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2*y^3 - y - 1"
]
},
"execution_count": 70,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f.mod(I)"
]
},
{
"cell_type": "code",
"execution_count": 71,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2*y^3 - y - 1"
]
},
"execution_count": 71,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f.mod([f1, f2])"
]
},
{
"cell_type": "code",
"execution_count": 72,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Multivariate Polynomial Ring in x, y over Rational Field"
]
},
"execution_count": 72,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"B = A.change_ring(order='deglex')\n",
"B"
]
},
{
"cell_type": "code",
"execution_count": 73,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Ideal (x*y^2 - x, -y^3 + x) of Multivariate Polynomial Ring in x, y over Rational Field"
]
},
"execution_count": 73,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"J = I.change_ring(B)\n",
"J"
]
},
{
"cell_type": "code",
"execution_count": 74,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2*x - y - 1"
]
},
"execution_count": 74,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"g = B(f).mod(J)\n",
"g"
]
},
{
"cell_type": "code",
"execution_count": 75,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Degree lexicographic term order"
]
},
"execution_count": 75,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"g.parent().term_order()"
]
},
{
"cell_type": "code",
"execution_count": 76,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2*y^3 - y - 1"
]
},
"execution_count": 76,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f.mod([f2, f1])"
]
},
{
"cell_type": "code",
"execution_count": 77,
"metadata": {},
"outputs": [],
"source": [
"I2 = A.ideal([f2, f1])"
]
},
{
"cell_type": "code",
"execution_count": 78,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2*y^3 - y - 1"
]
},
"execution_count": 78,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f.mod(I2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Exo 3.7"
]
},
{
"cell_type": "code",
"execution_count": 79,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Ideal (-x^2 + y, -x^3 + z) of Multivariate Polynomial Ring in x, y, z over Rational Field"
]
},
"execution_count": 79,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A.<x,y,z> = QQ[]\n",
"p = z^2 - x^4*y\n",
"g1, g2 = y - x^2, z - x^3\n",
"I = A.ideal([g1, g2])\n",
"I"
]
},
{
"cell_type": "code",
"execution_count": 80,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 80,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"p in I"
]
},
{
"cell_type": "code",
"execution_count": 81,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0"
]
},
"execution_count": 81,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"p.mod(I)"
]
},
{
"cell_type": "code",
"execution_count": 82,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(-x*z, x*y + z)"
]
},
"execution_count": 82,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a, b = p.lift(I)\n",
"a, b"
]
},
{
"cell_type": "code",
"execution_count": 83,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 83,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a*g1 + b*g2 == p"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "SageMath 8.3",
"language": "",
"name": "sagemath"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.6.3"
}
},
"nbformat": 4,
"nbformat_minor": 2
}
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