{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "### Example 5.3 (Smooth Critical Points and the Amoeba Contour)\n", "Compute and plot points on the contour of an amoeba. \n", "*Requirements: None*" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[-x^2*y^2 - 6*x*y - x - y + 1, (2*x^2*y + 6*x + 1)*t*y - (2*x*y^2 + 6*y + 1)*x]" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Define the polynomial under consideration and the system defining contour points\n", "var('x,y,t')\n", "H = 1-x-y-6*x*y-x^2*y^2\n", "sys = [H,diff(H,x)*x-t*diff(H,y)*y]\n", "sys" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "At a point of the contour y and t are roots of\n" ] }, { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}y^{4} t^{2} + \\left(-4\\right) y^{4} t + 4 y^{4} + \\left(-4\\right) y^{3} t^{2} + 48 y^{3} t + \\left(-44\\right) y^{3} + 46 y^{2} t^{2} + \\left(-68\\right) y^{2} t + 28 y^{2} + 14 y t^{2} + \\left(-23\\right) y t + 11 y + \\left(-2\\right) t + 1\n", "\\end{math}" ], "text/plain": [ "y^4*t^2 + (-4)*y^4*t + 4*y^4 + (-4)*y^3*t^2 + 48*y^3*t + (-44)*y^3 + 46*y^2*t^2 + (-68)*y^2*t + 28*y^2 + 14*y*t^2 + (-23)*y*t + 11*y + (-2)*t + 1" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "and x = (56*t^4 - 36*(t^4 - 6*t^3 + 13*t^2 - 12*t + 4)*y^3 + 16*t^3 + 6*(26*t^4 - 347*t^3 + 885*t^2 - 832*t + 268)*y^2 - 390*t^2 - (1708*t^4 - 6428*t^3 + 9009*t^2 - 5564*t + 1276)*y + 506*t - 188)/(56*t^4 - 172*t^3 + 186*t^2 - 85*t + 14)\n" ] } ], "source": [ "# Parametrize the contour by Pyt(y,t) = 0 and x = R(y,t), \n", "# where Pyt is a polynomial and R is a rational function\n", "R = PolynomialRing(QQbar,3,'x,y,t',order='lex')\n", "GB = R.ideal([R(k) for k in sys]).groebner_basis()\n", "Pyt = GB[-1]\n", "X = SR(GB[-2]).solve(x)[0].rhs()\n", "\n", "print(\"At a point of the contour y and t are roots of\")\n", "show(Pyt)\n", "print(\"and x = {}\".format(X))" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "# Determine the \"bad\" values of t, which cause the denominator of x to be zero\n", "bad = [k.rhs() for k in X.denominator().solve(t,multiplicities=false)]" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "# Find points on the contour when t lies between -5 and 5\n", "pt = plot([])\n", "for tt in range(-500,500):\n", " T = tt/100\n", " if T not in bad:\n", " ys = CC['y'](Pyt.subs(t=T)).roots(multiplicities=false)\n", " for k in ys:\n", " xs = X.subs(y=k,t=T)\n", " pt += point([log(abs(xs)),log(abs(k))])" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 3995 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Plot the computed points on the contour\n", "show(pt, xmin=-4, xmax=4, ymin=-4, ymax=4)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "SageMath 9.2", "language": "sage", "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": "3.8.5" } }, "nbformat": 4, "nbformat_minor": 2 }