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| { | |
| "cells": [ | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## Exercice 10.7" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 1, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "[h^2 + d^2 - b^2, h*a^2 - h*b^2 + 4*d*A - 2*c*A, h*c - 2*A, h*A - 1/4*d*a^2 + 1/4*d*b^2 - 1/8*a^2*c - 3/8*b^2*c + 1/8*c^3, d*c + 1/2*a^2 - 1/2*b^2 - 1/2*c^2, a^4 - 2*a^2*b^2 - 2*a^2*c^2 + b^4 - 2*b^2*c^2 + c^4 + 16*A^2]" | |
| ] | |
| }, | |
| "execution_count": 1, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "R.<h,d,a,b,c,A> = PolynomialRing(QQ, order='lex')\n", | |
| "I = R.ideal(c*h - 2*A,\n", | |
| " (c-d)^2 + h^2 - a^2,\n", | |
| " d^2 + h^2 - b^2)\n", | |
| "G = I.groebner_basis()\n", | |
| "G" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 2, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "a^4 - 2*a^2*b^2 - 2*a^2*c^2 + b^4 - 2*b^2*c^2 + c^4 + 16*A^2" | |
| ] | |
| }, | |
| "execution_count": 2, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "G[-1]" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "On conclut que la surface d'un triangle de côtés $a,b,c$ est égale à\n", | |
| "\n", | |
| "$$\\frac{\\sqrt{(-a + b - c) (-a + b + c) (a + b - c) (a + b + c)}}{4}$$" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 3, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "(-a + b - c) * (-a + b + c) * (a + b - c) * (a + b + c)" | |
| ] | |
| }, | |
| "execution_count": 3, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "(G[-1] - 16*A^2).factor()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## Exercice 11.4" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 4, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "[cosu - 1/4*x^2 - 1/4*y^2 - 1/4*z^2 + 5/4, sinu - z, cost^2 + sint^2 - 1, cost*x + sint*y - 1/4*x^2 - 1/4*y^2 - 1/4*z^2 - 3/4, cost*y - sint*x, cost*z^2 + 3*cost + 1/4*x^3 + 1/4*x*y^2 + 1/4*x*z^2 - 13/4*x, sint*x^2 + sint*y^2 - 1/4*x^2*y - 1/4*y^3 - 1/4*y*z^2 - 3/4*y, sint*z^2 + 3*sint + 1/4*x^2*y + 1/4*y^3 + 1/4*y*z^2 - 13/4*y, x^4 + 2*x^2*y^2 + 2*x^2*z^2 - 10*x^2 + y^4 + 2*y^2*z^2 - 10*y^2 + z^4 + 6*z^2 + 9]" | |
| ] | |
| }, | |
| "execution_count": 4, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "A.<cosu,sinu,cost,sint,x,y,z> = PolynomialRing(QQ, order='lex')\n", | |
| "I = A.ideal(x - (2 + cosu)*cost,\n", | |
| " y - (2 + cosu) * sint,\n", | |
| " z - sinu,\n", | |
| " cosu^2 + sinu^2 - 1,\n", | |
| " cost^2 + sint^2 - 1)\n", | |
| "G = I.groebner_basis()\n", | |
| "G" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 5, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "x^4 + 2*x^2*y^2 + 2*x^2*z^2 - 10*x^2 + y^4 + 2*y^2*z^2 - 10*y^2 + z^4 + 6*z^2 + 9" | |
| ] | |
| }, | |
| "execution_count": 5, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "p = G[-1]\n", | |
| "p" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Pour obtenir les relations algèbriques entre $\\cos(2s), \\sin(2s), \\cos(3s), \\sin(3s)$, on utilise le moteur symbolique de Sage:" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 7, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "2*cos(s)^2 - 1" | |
| ] | |
| }, | |
| "execution_count": 7, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "s = SR.var('s')\n", | |
| "cos(2*s).simplify_trig()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 8, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "sin(2*s) = 2*cos(s)*sin(s)\n", | |
| "sin(3*s) = (4*cos(s)^2 - 1)*sin(s)\n", | |
| "cos(2*s) = 2*cos(s)^2 - 1\n", | |
| "cos(3*s) = 4*cos(s)^3 - 3*cos(s)\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "for f in [sin, cos]:\n", | |
| " for n in [2,3]:\n", | |
| " print f(n*s), '=', f(n*s).simplify_trig()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 10, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "Ideal (4*Y^4 + 64*Y^2*Z^4 - 76*Y^2*Z^2 - 4*Y^2 + 16*Z^10 + 72*Z^8 + 9*Z^6 - 162*Z^4 + 81*Z^2, 16*X*Z^7 - 96*X*Z^5 + 67*X*Z^3 - 3*X*Z + 8*Y^3*Z^2 - 2*Y^3 + 64*Y*Z^6 - 88*Y*Z^4 - 6*Y*Z^2 + 2*Y, 8*X*Y*Z^4 - 42*X*Y*Z^2 + 2*X*Y - 32*Y^2*Z^3 + 24*Y^2*Z - 16*Z^9 - 80*Z^7 - 51*Z^5 + 126*Z^3 - 27*Z, 2*X*Y^2 - 8*X*Z^6 + 62*X*Z^4 - 42*X*Z^2 - 4*Y^3*Z - 36*Y*Z^5 + 59*Y*Z^3 + Y*Z, 25*X^2 + 72*X*Y*Z^3 - 374*X*Y*Z - 288*Y^2*Z^2 + 225*Y^2 - 144*Z^8 - 728*Z^6 - 505*Z^4 + 1074*Z^2 - 225) of Multivariate Polynomial Ring in coss, sins, cos2s, sin2s, cos3s, sin3s, X, Y, Z over Rational Field" | |
| ] | |
| }, | |
| "execution_count": 10, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "A.<coss,sins,cos2s,sin2s,cos3s,sin3s,X,Y,Z> = PolynomialRing(QQ, order='lex')\n", | |
| "I = A.ideal(X - (2 + cos2s) * cos3s,\n", | |
| " Y - (2 + cos2s) * sin3s,\n", | |
| " Z - sin2s,\n", | |
| " sin2s - 2*coss*sins,\n", | |
| " sin3s - (4*coss^2 - 1)*sins,\n", | |
| " cos2s - (2*coss^2 - 1),\n", | |
| " cos3s - (4*coss^3 - 3*coss),\n", | |
| " sins^2 + coss^2 - 1)\n", | |
| "J = I.elimination_ideal([coss,sins,cos2s,sin2s,cos3s,sin3s])\n", | |
| "J" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "On vérifie que la courbe est bien contenue dans la surface, i.e. que l'ideal de la surface (engendré par `p`, calculé plus haut) est contenu dans l'idéal de la courbe.\n", | |
| "\n", | |
| "On convertit `p` en un polynôme du nouvel anneau en l'évaluant en `X,Y,Z`." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 11, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "True" | |
| ] | |
| }, | |
| "execution_count": 11, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "p(x=X, y=Y, z=Z) in J" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "La surface paramétrèe est un tore. Le code ci-dessous permet de dessinner la surface et la courbe en 3 dimensions. À l'heure actuelle (versions de Sage jusqu'à 8.0), l'interface Jupyter de Sage ne supporte pas le dessin 3d. Cette instruction peut néanmoins être utilisée en ligne de commande." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": null, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "s, u, t = SR.var('s, u, t')\n", | |
| "(\n", | |
| " parametric_plot3d([(2 + cos(u))*cos(t), (2 + cos(u))*sin(t), sin(u)],\n", | |
| " (u,-pi,pi), (t,-pi,pi))\n", | |
| " + parametric_plot3d([(2 + cos(2*s))*cos(3*s), (2 + cos(2*s))*sin(3*s), sin(2*s)],\n", | |
| " (s,0,6*pi), color='red')\n", | |
| ")" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## Exercice 12.5" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 12, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "Ideal (4*x^3*y^2 + 2*x*y^4 - 2*x*y^2, 2*x^4*y + 4*x^2*y^3 - 2*x^2*y) of Multivariate Polynomial Ring in x, y, t over Rational Field" | |
| ] | |
| }, | |
| "execution_count": 12, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "A.<x,y,t> = PolynomialRing(QQ, order='lex')\n", | |
| "f = x^4*y^2 + x^2*y^4 - x^2*y^2\n", | |
| "I = A.ideal(f.derivative(x), f.derivative(y))\n", | |
| "I" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "On voit clairement que l'idéal a dimension 1, et que donc le lieu critique est infini." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 13, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "[x^4*y + 2*x^2*y^3 - x^2*y, x^3*y^2 + 1/2*x*y^4 - 1/2*x*y^2, x^2*y^4 - 1/3*x^2*y^2, x*y^6 - 4/3*x*y^4 + 1/3*x*y^2]" | |
| ] | |
| }, | |
| "execution_count": 13, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "I.groebner_basis()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 14, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "Ideal (x^4*y^2 + x^2*y^4 - x^2*y^2 - t, 4*x^3*y^2 + 2*x*y^4 - 2*x*y^2, 2*x^4*y + 4*x^2*y^3 - 2*x^2*y) of Multivariate Polynomial Ring in x, y, t over Rational Field" | |
| ] | |
| }, | |
| "execution_count": 14, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "I = A.ideal(f - t, f.derivative(x), f.derivative(y))\n", | |
| "I" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "On voit qu'il y a uniquement deux valeurs critques: $0$ et $-1/27$. L'idéal est par contre toujours de dimension 1, car il y a une infinité de points associés à ces valeurs critiques." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 15, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "[x^4*y - x^2*y - 6*y*t, x^2*y^2 + 3*t, x^2*t - 1/3*t, x*y^4 - x*y^2 - 6*x*t, y^2*t - 1/3*t, t^2 + 1/27*t]" | |
| ] | |
| }, | |
| "execution_count": 15, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "I.groebner_basis()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 16, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "1" | |
| ] | |
| }, | |
| "execution_count": 16, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "I.dimension()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "On peut raffiner l'analyse en séparant les deux valeurs critiques. Pour la valeur critique $t=0$, il suffit de considérer l'idéal $〈f,f_x,f_y〉$. On voit qu'il s'agit d'un idéal de dimension 1, dont la variété correspond aux droites $x=0$ et $y=0$." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 17, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "[x^4*y - x^2*y, x^2*y^2, x*y^4 - x*y^2]" | |
| ] | |
| }, | |
| "execution_count": 17, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "I = A.ideal(f, f.derivative(x), f.derivative(y))\n", | |
| "I.groebner_basis()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Pour la valeur critique $-1/27$, nous utilisons ajoutons le polynôme $ft-1$, qui force $f$ et $t$ à être différents de $0$. Avec cette formulation, $t$ sera alors l'inverse de l'unique valeur critique non-nulle." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 18, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "[x^2 - 1/3, y^2 - 1/3, t + 27]" | |
| ] | |
| }, | |
| "execution_count": 18, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "I = A.ideal(f*t - 1, f.derivative(x), f.derivative(y))\n", | |
| "I.groebner_basis()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Cette fois-ci, nous trouvons un idéal de dimension $0$, qui correspond à un nombre fini de points critiques. Ces points sont définis dans $ℚ[\\sqrt{1/3}]$, ce que nous vérifions (dans la suite $a=\\sqrt{1/3}$)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 19, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "[]" | |
| ] | |
| }, | |
| "execution_count": 19, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "I.variety()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 20, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "Number Field in a with defining polynomial x^2 - 1/3" | |
| ] | |
| }, | |
| "execution_count": 20, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "k.<a> = QuadraticField(1/3)\n", | |
| "k" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 21, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "[{y: -a, t: -27, x: -a},\n", | |
| " {y: -a, t: -27, x: a},\n", | |
| " {y: a, t: -27, x: -a},\n", | |
| " {y: a, t: -27, x: a}]" | |
| ] | |
| }, | |
| "execution_count": 21, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "I.variety(k)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 22, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "(-1/27, 0, 0)" | |
| ] | |
| }, | |
| "execution_count": 22, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "f(x=a, y=a), f.derivative(x)(x=a, y=a), f.derivative(y)(x=a, y=a)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## Exercice 14.3" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 23, | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "A.<x,y> = QQ[]\n", | |
| "I = A.ideal(x^3*y, x*y^2)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 24, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "[x^3*y, x*y^2]" | |
| ] | |
| }, | |
| "execution_count": 24, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "I.groebner_basis()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "On affiche tous les monomes, de degrés jusqu'à 5 en $x$ ou $y$, qui **n'appartiennent pas** à $I$.\n", | |
| "\n", | |
| "Pour aider la lecture, on ordonne la liste (l'ordre par défaut est *degrevlex*, ce qui nous arrange bien)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 25, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "[1, y, x, y^2, x*y, x^2, y^3, x^2*y, x^3, y^4, x^4, y^5, x^5]" | |
| ] | |
| }, | |
| "execution_count": 25, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "sorted([m for m in monomials([x,y], [6,6]) if m not in I ])" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "On compte les monômes par **degré total**, et on lit bien les premières valeures de la fonction de Hilbert:\n", | |
| "```\n", | |
| "H(1) = 1,\n", | |
| "H(2) = 3,\n", | |
| "H(3) = 6,\n", | |
| "H(4) = 9,\n", | |
| "H(5) = 11\n", | |
| "H(6) = 13\n", | |
| "```\n", | |
| "on voit bien que $H(i) = 2 + H(i-1)$ pour tout $i>4$. On en déduit que $H(s) = 2s+1$, et que la dimension de la variété est 1." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 26, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "1" | |
| ] | |
| }, | |
| "execution_count": 26, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "I.dimension()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Sage possède une méthode `hilbert_polynomial()`, mais attention: cette méthode calcule le polynôme d'Hilbert gradué d'un idéal homogène, ce qui ne correspond pas à notre définition:" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 27, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "2" | |
| ] | |
| }, | |
| "execution_count": 27, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "I.hilbert_polynomial()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## Exercice 14.4" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 28, | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "A.<x,y,z> = QQ[]\n", | |
| "I = A.ideal(x^3*y*z^5, x*y^3*z^2)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 29, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "[1,\n", | |
| " z,\n", | |
| " y,\n", | |
| " x,\n", | |
| " z^2,\n", | |
| " y*z,\n", | |
| " x*z,\n", | |
| " y^2,\n", | |
| " x*y,\n", | |
| " x^2,\n", | |
| " y*z^2,\n", | |
| " x*z^2,\n", | |
| " y^2*z,\n", | |
| " x*y*z,\n", | |
| " x^2*z,\n", | |
| " x*y^2,\n", | |
| " x^2*y,\n", | |
| " y^2*z^2,\n", | |
| " x*y*z^2,\n", | |
| " x^2*z^2,\n", | |
| " x*y^2*z,\n", | |
| " x^2*y*z,\n", | |
| " x^2*y^2,\n", | |
| " x*y^2*z^2,\n", | |
| " x^2*y*z^2,\n", | |
| " x^2*y^2*z,\n", | |
| " x^2*y^2*z^2]" | |
| ] | |
| }, | |
| "execution_count": 29, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "sorted([m for m in monomials([x,y,z], [3,3,3]) if m not in I])" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Il y a ici trop de monomes pour compter à la main, il va falloir automatiser." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 30, | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "HF = lambda s: len([m for m in monomials([x,y,z], [s,s,s]) if m.degree() < s and m not in I])" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 31, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "1 --> 1\n", | |
| "2 --> 4\n", | |
| "3 --> 10\n", | |
| "4 --> 20\n", | |
| "5 --> 35\n", | |
| "6 --> 56\n", | |
| "7 --> 83\n", | |
| "8 --> 116\n", | |
| "9 --> 155\n", | |
| "10 --> 199\n", | |
| "11 --> 247\n", | |
| "12 --> 299\n", | |
| "13 --> 355\n", | |
| "14 --> 415\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "for i in range(1,15):\n", | |
| " print(\"%d --> %d\" % (i, HF(i)))" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "On sait que la dimension de l'idéal ne peut pas être plus que 3, en supposant qu'on a calculé asséz de valeurs de la fonction d'Hilbert (et c'est bien le cas : il suffit de regarder les générateurs de $I$), on peut obtenir le polynôme d'Hilbert en interpolant sur les quatre dernières valeurs" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 32, | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "B.<s> = QQ[]" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 33, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "2*s^2 + 6*s - 61" | |
| ] | |
| }, | |
| "execution_count": 33, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "B.lagrange_polynomial([(11, 247), (12, 299), (13, 355), (14, 415)])" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "L'idéal est de dimension 2, comme attendu" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 34, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "2" | |
| ] | |
| }, | |
| "execution_count": 34, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "I.dimension()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## Exercice 14.7\n", | |
| "\n", | |
| "Il est important ici d'utiliser un ordre gradué (par ex., degrevlex) pour calculer la base de Gröbner" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 35, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "A.<x,y,z> = QQ[]\n", | |
| "I = A.ideal([x*z, x*y - 1])" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 36, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "[x*y - 1, z]" | |
| ] | |
| }, | |
| "execution_count": 36, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "G = I.groebner_basis()\n", | |
| "G" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "On construit l'idéal monomial de $I$" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 37, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "Ideal (x*y, z) of Multivariate Polynomial Ring in x, y, z over Rational Field" | |
| ] | |
| }, | |
| "execution_count": 37, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "M = A.ideal([p.lm() for p in G])\n", | |
| "M" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Et on évalue la fonction d'Hilbert" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 38, | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "HF = lambda s: len([m for m in monomials([x,y,z], [s,s,s]) if m.degree() < s and m not in M])" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 39, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| "1 --> 1\n", | |
| "2 --> 3\n", | |
| "3 --> 5\n", | |
| "4 --> 7\n", | |
| "5 --> 9\n", | |
| "6 --> 11\n", | |
| "7 --> 13\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "for i in range(1,8):\n", | |
| " print(\"%d --> %d\" % (i, HF(i)))" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "source": [ | |
| "On voit immédiatement que le polynôme d'Hilbert vaut $2s-1$, ce qui nous confime que $I$ et $M$ sont de dimension 1.\n", | |
| "\n", | |
| "Encore une fois, la méthode `hilbert_polynomial()` de Sage ne calcule pas ce que l'on souhaite. En effet, elle n'est même pas conçue pour être utilisée avec des idéaux non-homogènes." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 40, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "ename": "TypeError", | |
| "evalue": "Ideal must be homogeneous.", | |
| "output_type": "error", | |
| "traceback": [ | |
| "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", | |
| "\u001b[0;31mTypeError\u001b[0m Traceback (most recent call last)", | |
| "\u001b[0;32m<ipython-input-40-51a93ff31666>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mI\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mhilbert_polynomial\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m", | |
| "\u001b[0;32m/local/SageMath-8.0/local/lib/python2.7/site-packages/sage/rings/polynomial/multi_polynomial_ideal.pyc\u001b[0m in \u001b[0;36m__call__\u001b[0;34m(self, *args, **kwds)\u001b[0m\n\u001b[1;32m 294\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mR\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[0mis_field\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 295\u001b[0m \u001b[0;32mraise\u001b[0m \u001b[0mValueError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"Coefficient ring must be a field for function '%s'.\"\u001b[0m\u001b[0;34m%\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__name__\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 296\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_instance\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0margs\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[0m\n\u001b[0m\u001b[1;32m 297\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 298\u001b[0m \u001b[0mrequire_field\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mRequireField\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", | |
| "\u001b[0;32m/local/SageMath-8.0/local/lib/python2.7/site-packages/sage/rings/polynomial/multi_polynomial_ideal.pyc\u001b[0m in \u001b[0;36mhilbert_polynomial\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m 2401\u001b[0m \"\"\"\n\u001b[1;32m 2402\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_homogeneous\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-> 2403\u001b[0;31m \u001b[0;32mraise\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"Ideal must be homogeneous.\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 2404\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 2405\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0msage\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlibs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msingular\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfunction_factory\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", | |
| "\u001b[0;31mTypeError\u001b[0m: Ideal must be homogeneous." | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "I.hilbert_polynomial()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 41, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "True" | |
| ] | |
| }, | |
| "execution_count": 41, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "I.dimension() == M.dimension() == 1" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "### 2)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 42, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "A.<x,y,z,w> = QQ[]\n", | |
| "I = A.ideal(z*w-y^2, x*y-z^3)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 43, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "Ideal (z^3, y^2) of Multivariate Polynomial Ring in x, y, z, w over Rational Field" | |
| ] | |
| }, | |
| "execution_count": 43, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "M = A.ideal([p.lm() for p in I.groebner_basis()])\n", | |
| "M" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Ici, même pas la peine d'évaluer la fonction d'Hilbert : on voit immédiatement que la variété de $M$ est l'hyperplan $z=0,y=0$, la dimension de $I$ et de $M$ est donc 2." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 44, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "True" | |
| ] | |
| }, | |
| "execution_count": 44, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "I.dimension() == M.dimension() == 2" | |
| ] | |
| } | |
| ], | |
| "metadata": { | |
| "kernelspec": { | |
| "display_name": "SageMath 8.0", | |
| "language": "", | |
| "name": "sagemath" | |
| }, | |
| "language_info": { | |
| "codemirror_mode": { | |
| "name": "ipython", | |
| "version": 2 | |
| }, | |
| "file_extension": ".py", | |
| "mimetype": "text/x-python", | |
| "name": "python", | |
| "nbconvert_exporter": "python", | |
| "pygments_lexer": "ipython2", | |
| "version": "2.7.13" | |
| } | |
| }, | |
| "nbformat": 4, | |
| "nbformat_minor": 1 | |
| } |
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