{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "%display latex"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "<h1> <center> Interactive tutorials, <br> an example on symmetric functions </center> </h1>\n",
    "\n",
    "<p> <center> Pauline Hubert <br>\n",
    "Joint work with Mélodie Lapointe </center>  </p>\n",
    "<p> <center> ACA July 2019 - Montreal </center>  </p>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "<h2>1. Sage</h2>\n",
    "<center><img src=\"logo.png\" alt=\"Sage logo\" style=\"width:216px;height:60px;\"></center>\n",
    "\n",
    "<ul>\n",
    "  <li>Free, open source, mathematical software</li>\n",
    "  <li>Started by William Stein in 2005</li>\n",
    "  <li>Hundreds of worldwide contributors</li>\n",
    "  <li>Based on Python and build on top of many open source softwares (GAP, Maxima, R, ...)</li>\n",
    "  <li>Can be installed on Linux, Mac or Windows or used online (CoCalc.com)</li>\n",
    "    <li>Jupyter notebook</li>\n",
    "</ul> \n",
    "\n",
    "<center><img src=\"jupyter_logo.png\" alt=\"Jupyter logo\" style=\"width:100px;height:112px;\"></center>\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "<h2> 2. Symmetric Functions Tutorial </h2>\n",
    "\n",
    "<p>Two reference pages for symmetric functions in Sage.</p>\n",
    "\n",
    "<ul>\n",
    "    <li> Documentation <br>\n",
    "        <a href=\"http://doc.sagemath.org/html/en/reference/combinat/sage/combinat/sf/sfa.html\">http://doc.sagemath.org/html/en/reference/combinat/sage/combinat/sf/sfa.html</a> <br> <br> </li>\n",
    "    <li> Previous tutorial <br>\n",
    "        <a href=\"http://doc.sagemath.org/html/en/reference/combinat/sage/combinat/sf/sf.html#sage.combinat.sf.sf.SymmetricFunctions\">http://doc.sagemath.org/html/en/reference/combinat/sage/combinat/sf/sf.html#sage.combinat.sf.sf.SymmetricFunctions</a></li>\n",
    "</ul>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "<h3> Why a new tutorial on symmetric functions? </h3>\n",
    "\n",
    "<ul>\n",
    "    <li>More maths and more definitions : accessible with few knowledge on symmetric functions</li>\n",
    "    <li>Add exercises in addition to examples</li>\n",
    "    <li>Add interactivity</li>\n",
    "</ul>\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "<center> <font size=\"10\"><a href=\"https://more-sagemath-tutorials.readthedocs.io/en/latest/tutorial-symmetric-functions.html\">The new tutorial ! </a> </font> </center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "<h2> 3. Find the discriminant of a polynomial using symmetric functions </h2>\n",
    "\n",
    "<br>\n",
    "\n",
    "Reference about [discriminant](https://fr.wikipedia.org/wiki/Discriminant)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "<h3> Exercise </h3>\n",
    "\n",
    "Let $f$ by a monic polynomial on $z$ and $x_0, x_1, \\dots, x_{n-1}$ its roots. Then we have $$f(z) = (z-x_0)(z-x_1)\\dots (z-x_{n-1}).$$\n",
    "\n",
    "Our goal is to find the discriminant of $f$ for $n=2$ and $n=3$ using symmetric functions. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "<h3> Part 1 :  $n=2$ </h3>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "n = 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Defining z\n",
      "Defining x0, x1\n"
     ]
    }
   ],
   "source": [
    "F = QQ['z']\n",
    "F.inject_variables()\n",
    "R = PolynomialRing(F,'x',n)\n",
    "R.inject_variables()\n",
    "x = R.gens()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "*Question* : Define the polynomial $f$ with the indeterminates $z, x_0$ and $x_1$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "scrolled": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x_{0} x_{1} + \\left(-z\\right) x_{0} + \\left(-z\\right) x_{1} + z^{2}</script></html>"
      ],
      "text/plain": [
       "x0*x1 + (-z)*x0 + (-z)*x1 + z^2"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = prod((z-x[i]) for i in range(0,n))\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "if n==2 :\n",
    "    A = QQ['z,x0,x1']\n",
    "    assert A(f)(z=x[0]) == 0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Remark that $f$ is symmetric w.r.t the variables $x_0$ and $x_1$. Then [the fundamental theorem of symmetric polynomials](https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial#Fundamental_theorem_of_symmetric_polynomials) tells us that we can express $f$ in terms of elementary symmetric polynomials. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "**Definition:** The elementary symmetric polynomial $e_k$ is defined to be the sum of all monomials of degree $k$ with no squares. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}1</script></html>"
      ],
      "text/plain": [
       "1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x_{0} + x_{1} + x_{2} + x_{3}</script></html>"
      ],
      "text/plain": [
       "x0 + x1 + x2 + x3"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x_{0} x_{1} + x_{0} x_{2} + x_{1} x_{2} + x_{0} x_{3} + x_{1} x_{3} + x_{2} x_{3}</script></html>"
      ],
      "text/plain": [
       "x0*x1 + x0*x2 + x1*x2 + x0*x3 + x1*x3 + x2*x3"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x_{0} x_{1} x_{2} + x_{0} x_{1} x_{3} + x_{0} x_{2} x_{3} + x_{1} x_{2} x_{3}</script></html>"
      ],
      "text/plain": [
       "x0*x1*x2 + x0*x1*x3 + x0*x2*x3 + x1*x2*x3"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x_{0} x_{1} x_{2} x_{3}</script></html>"
      ],
      "text/plain": [
       "x0*x1*x2*x3"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "e = SymmetricFunctions(QQ[z]).e()\n",
    "\n",
    "for k in range(5):\n",
    "    show(e[k].expand(4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "*Question* : Show that the coefficients of $f = z^2+bz+c$ in terms of elementary symmetric functions are $b=-e_1$ and $c=e_2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}z^{2}e_{} - ze_{1} + e_{2}</script></html>"
      ],
      "text/plain": [
       "z^2*e[] - z*e[1] + e[2]"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g = e.from_polynomial(f)\n",
    "g"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "scrolled": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x_{0} x_{1} + \\left(-z\\right) x_{0} + \\left(-z\\right) x_{1} + z^{2}</script></html>"
      ],
      "text/plain": [
       "x0*x1 + (-z)*x0 + (-z)*x1 + z^2"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.expand(n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "**Definition :** The discriminant of $f$ is given by\n",
    "$$\\Delta(f) = \\prod_{i>j} (x_i-x_j)^2$$ \n",
    "where $x_i$ are the roots of $f$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Recall that the Vandermonde matrix \n",
    "$$ M = \\begin{bmatrix} \n",
    "1 & 1 & \\dots & 1 \\\\\n",
    "x_0 & x_1 & \\dots & x_{n-1} \\\\ \n",
    "x_0^2 & x_1^2 & \\dots & x_{n-1}^2 \\\\\n",
    "\\vdots & \\vdots & \\ddots & \\vdots \\\\\n",
    "x_0^{n-1} & x_1^{n-1} & \\dots & x_{n-1}^{n-1}  \n",
    "\\end{bmatrix} $$\n",
    "\n",
    "has determinant $\\det M = \\prod_{i>j} (x_i-x_j)$. So $\\Delta(f) = (\\det M)^2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "*Question* : Check that $\\Delta(f) = \\det(M^2)$ for $n=2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n",
       "1 & x_{0} \\\\\n",
       "1 & x_{1}\n",
       "\\end{array}\\right)</script></html>"
      ],
      "text/plain": [
       "[ 1 x0]\n",
       "[ 1 x1]"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = matrix.vandermonde(x)\n",
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x_{0}^{2} + \\left(-2\\right) x_{0} x_{1} + x_{1}^{2}</script></html>"
      ],
      "text/plain": [
       "x0^2 + (-2)*x0*x1 + x1^2"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(M.determinant())^2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(M.determinant())^2 == prod((x[i]-x[j])^2 for i in range(n) for j in range(n) if i>j)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "**Proposition:** Let $A$ be a $n \\times n $ matrix, then we have $\\det(A)^2 = \\det(A^\\bot A)$. \n",
    "\n",
    "We can check that on a random $3 \\times 3$ matrix on $\\mathbb{Q}$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = matrix.random(QQ, 3, algorithm='diagonalizable')\n",
    "(A.determinant())^2 == (A.transpose()*A).determinant()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "In the case of the Vandermonde matrix, we get the following. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n",
       "2 & x_{0} + x_{1} \\\\\n",
       "x_{0} + x_{1} & x_{0}^{2} + x_{1}^{2}\n",
       "\\end{array}\\right)</script></html>"
      ],
      "text/plain": [
       "[          2     x0 + x1]\n",
       "[    x0 + x1 x0^2 + x1^2]"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T = M.transpose()*M\n",
    "T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "We recognize the power sum symmetric polynomials. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "**Definition:** The power sum symmetric polynomials $p_k$ are defined to be $\\sum_i x_i^k$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}1</script></html>"
      ],
      "text/plain": [
       "1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x_{0} + x_{1} + x_{2} + x_{3}</script></html>"
      ],
      "text/plain": [
       "x0 + x1 + x2 + x3"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x_{0}^{2} + x_{1}^{2} + x_{2}^{2} + x_{3}^{2}</script></html>"
      ],
      "text/plain": [
       "x0^2 + x1^2 + x2^2 + x3^2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x_{0}^{3} + x_{1}^{3} + x_{2}^{3} + x_{3}^{3}</script></html>"
      ],
      "text/plain": [
       "x0^3 + x1^3 + x2^3 + x3^3"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = SymmetricFunctions(QQ[z]).p()\n",
    "\n",
    "for k in range(4):\n",
    "    show(p[k].expand(4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "*Question* : Express the coefficients of $ T := M^\\bot M$ in terms of power sum symmetric polynomials. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n",
       "2p_{} & p_{1} \\\\\n",
       "p_{1} & p_{2}\n",
       "\\end{array}\\right)</script></html>"
      ],
      "text/plain": [
       "[2*p[]  p[1]]\n",
       "[ p[1]  p[2]]"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T_p = matrix([[p.from_polynomial(T[i,j]) for j in range(n)] for i in range(n)])\n",
    "T_p"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "*Question* : Compute the determinant of the matrix $T_p$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-p_{1,1} + 2p_{2}</script></html>"
      ],
      "text/plain": [
       "-p[1, 1] + 2*p[2]"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T_p.determinant()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "*Question* : Express the result in terms of elementary symmetric polynomials and find the discriminant of $f$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
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   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{1,1} - 4e_{2}</script></html>"
      ],
      "text/plain": [
       "e[1, 1] - 4*e[2]"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e(T_p.determinant())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Remember $a=1$, $b=-e_1$ and $c=e_2$. As expected we obtain that $\\Delta = b^2 - 4c$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "<h3> Part 2 : $n=3$ </h3>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "n = 3 \n",
    "F = QQ['z']\n",
    "F.inject_variables(verbose=False)\n",
    "R = PolynomialRing(F,'x',n)\n",
    "R.inject_variables(verbose=False)\n",
    "x = R.gens()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|f|\\phantom{\\verb!x!}\\verb|in|\\phantom{\\verb!x!}\\verb|terms|\\phantom{\\verb!x!}\\verb|of|\\phantom{\\verb!x!}\\verb|elementary|\\phantom{\\verb!x!}\\verb|symmetric|\\phantom{\\verb!x!}\\verb|functions|\\phantom{\\verb!x!}\\verb|:|\\phantom{\\verb!x!}\\verb|\t| z^{3}e_{} - z^{2}e_{1} + ze_{2} - e_{3}</script></html>"
      ],
      "text/plain": [
       "'f in terms of elementary symmetric functions : \\t' z^3*e[] - z^2*e[1] + z*e[2] - e[3]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f = prod((z-x[i]) for i in range(0,3))\n",
    "show(\"f in terms of elementary symmetric functions : \\t\",e.from_polynomial(f))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "def discriminant_with_sym_func(n) :\n",
    "    M = matrix.vandermonde(x)\n",
    "    T = M.transpose()*M\n",
    "    M_p = matrix([[p.from_polynomial(T[i,j]) for j in range(n)] for i in range(n)])\n",
    "    return e(M_p.determinant())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|the|\\phantom{\\verb!x!}\\verb|discriminant|\\phantom{\\verb!x!}\\verb|of|\\phantom{\\verb!x!}\\verb|f|\\phantom{\\verb!x!}\\verb|:|\\phantom{\\verb!x!}\\verb|\t| e_{2,2,1,1} - 4e_{2,2,2} - 4e_{3,1,1,1} + 18e_{3,2,1} - 27e_{3,3} - 8e_{4,1,1} + 24e_{4,2}</script></html>"
      ],
      "text/plain": [
       "'the discriminant of f : \\t' e[2, 2, 1, 1] - 4*e[2, 2, 2] - 4*e[3, 1, 1, 1] + 18*e[3, 2, 1] - 27*e[3, 3] - 8*e[4, 1, 1] + 24*e[4, 2]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "show(\"the discriminant of f : \\t\",discriminant_with_sym_func(3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Remember that the discriminant of $z^3+bz^2+cz+d$ is $$b^2c^2+18bcd-4c^3-4b^3d-27d^2$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}b^{2} c^{2} - 4 \\, b^{3} d - 4 \\, c^{3} + 18 \\, b c d - 27 \\, d^{2}</script></html>"
      ],
      "text/plain": [
       "b^2*c^2 - 4*b^3*d - 4*c^3 + 18*b*c*d - 27*d^2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "var('b,c,d')\n",
    "D = b^2*c^2+18*b*c*d-4*c^3-4*b^3*d-27*d^2\n",
    "show(D)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{2,2,1,1} - 4e_{2,2,2} - 4e_{3,1,1,1} + 18e_{3,2,1} - 27e_{3,3} - 8e_{4,1,1} + 24e_{4,2}</script></html>"
      ],
      "text/plain": [
       "e[2, 2, 1, 1] - 4*e[2, 2, 2] - 4*e[3, 1, 1, 1] + 18*e[3, 2, 1] - 27*e[3, 3] - 8*e[4, 1, 1] + 24*e[4, 2]"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D = D.substitute({b:e[1].expand(n), c:e[2].expand(n), d:e[3].expand(n)})\n",
    "e.from_polynomial(R((D))) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e.from_polynomial(R((D))) == discriminant_with_sym_func(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "A = QQ['y']['x0','x1']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "B = QQ['y','x0','x1']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "A.random_element()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "f = _"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "B(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "B(f).subs(y=B.gens()[1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "A(B(f))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "<h2> Conclusion </h2>\n",
    "\n",
    "<br>\n",
    "\n",
    "<ul>\n",
    "    <li> The new tutorial just got a positive review on the Sage development server and will be integrated in the next version of Sage. </li>\n",
    "</ul>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "<ul>\n",
    "  <li> Some nice tools to learn/teach mathematics : Jupyter, Rise, Binder </li>\n",
    "  <ul>\n",
    "        <li> Jupyter : Learn, experiment, practice, take note, write maths, etc, in the same environment </li>\n",
    "        <li> Rise : Interactive slides in Jupyter from a worksheet </li>\n",
    "        <li> Binder : Access notebooks online </li>\n",
    "    </ul>\n",
    "</ul>\n",
    "\n",
    "For more on binder and rise : https://opendreamkit.org/2018/07/23/live-online-slides-with-sagemath-jupyter-rise-binder/"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "<center> <b> <font size=\"10\"> Thank you ! </font> </b> </center>"
   ]
  }
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