From b61b3c6264988227fd20aa05456aff700f492534 Mon Sep 17 00:00:00 2001
From: Jan Snellman <jan.snellman@liu.se>
Date: Mon, 29 Apr 2024 11:12:20 +0200
Subject: [PATCH] =?UTF-8?q?uppgifter=20kedjebr=C3=A5k=20PPT?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit
---
.ipynb_checkpoints/Untitled1-checkpoint.ipynb | 6 +
.ipynb_checkpoints/t2-checkpoint.ipynb | 463 ++++++++++++++++++
Untitled1.ipynb | 463 ++++++++++++++++++
homepage/index.org | 2 +
homepage/sitemap.org | 2 +-
public/index.html | 33 +-
public/sitemap.html | 6 +-
t2.ipynb | 463 ++++++++++++++++++
8 files changed, 1422 insertions(+), 16 deletions(-)
create mode 100644 .ipynb_checkpoints/Untitled1-checkpoint.ipynb
create mode 100644 .ipynb_checkpoints/t2-checkpoint.ipynb
create mode 100644 Untitled1.ipynb
create mode 100644 t2.ipynb
diff --git a/.ipynb_checkpoints/Untitled1-checkpoint.ipynb b/.ipynb_checkpoints/Untitled1-checkpoint.ipynb
new file mode 100644
index 0000000..363fcab
--- /dev/null
+++ b/.ipynb_checkpoints/Untitled1-checkpoint.ipynb
@@ -0,0 +1,6 @@
+{
+ "cells": [],
+ "metadata": {},
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/.ipynb_checkpoints/t2-checkpoint.ipynb b/.ipynb_checkpoints/t2-checkpoint.ipynb
new file mode 100644
index 0000000..03d8318
--- /dev/null
+++ b/.ipynb_checkpoints/t2-checkpoint.ipynb
@@ -0,0 +1,463 @@
+{
+ "cells": [
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "id": "9ecab153",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "(x, y)"
+ ]
+ },
+ "execution_count": 1,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "var('x,y')"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "id": "ce0ab8a1",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "g(x,y) = x^2 + y^2 - 9"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "id": "d63bdcf7",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "f(x,y)= x^2 +x*y +y^2 -4*y"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "id": "220e5c14",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "(2*x, 2*y)"
+ ]
+ },
+ "execution_count": 5,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "gradg = g(x,y).gradient()\n",
+ "gradg"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 7,
+ "id": "2fb288a7",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "(2*x + y, x + 2*y - 4)"
+ ]
+ },
+ "execution_count": 7,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "gradf = f(x,y).gradient()\n",
+ "gradf"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 14,
+ "id": "54d7ffa7",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Multivariate Polynomial Ring in la, x, y over Algebraic Field"
+ ]
+ },
+ "execution_count": 14,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "R.<la,x,y> = PolynomialRing(QQbar,order='lex')\n",
+ "R"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 15,
+ "id": "fbe53eff",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "e1 = 2*x+y - la*2*x\n",
+ "e2 = x+2*y -4 - la*2*y\n",
+ "e3 = x^2 + y^2-9"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 16,
+ "id": "5635dddc",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "I = ideal(e1,e2,e3)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 17,
+ "id": "f79e370f",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "[la + 1/18*y^3 + (-1/36)*y - 1, x + 1/2*y^2 - 9/4, y^4 + (-5)*y^2 - 63/4]"
+ ]
+ },
+ "execution_count": 17,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "I.groebner_basis()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 19,
+ "id": "d3392e18",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "[{y: -2.681495060562937?, x: -1.345207879911715?, la: 1.996684267393276?},\n",
+ " {y: 2.681495060562937?, x: -1.345207879911715?, la: 0.003315732606724674?},\n",
+ " {y: -1.480005324255096?*I, x: 3.345207879911715?, la: 1 - 0.2212127582776941?*I},\n",
+ " {y: 1.480005324255096?*I, x: 3.345207879911715?, la: 1 + 0.2212127582776941?*I}]"
+ ]
+ },
+ "execution_count": 19,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "I.variety()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 24,
+ "id": "a4935561",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "x^2 - 2*x - 9/2"
+ ]
+ },
+ "execution_count": 24,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "x.subs(I.variety()[0]).minpoly()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "15a7d053",
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "5dcf93b0",
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 26,
+ "id": "9081c07e",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "R.<x,y,z> = PolynomialRing(QQbar,order='lex')"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 27,
+ "id": "f33912f1",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "A = matrix(R,[[y*z,1,2*x],[x*z,1,2*y],[x*y,1,2*z]])"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 29,
+ "id": "4523617a",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "(-2)*x^2*y + 2*x^2*z + 2*x*y^2 + (-2)*x*z^2 + (-2)*y^2*z + 2*y*z^2"
+ ]
+ },
+ "execution_count": 29,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "f1=A.det()\n",
+ "f1"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 30,
+ "id": "fc38b38c",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "x + y + z - 1"
+ ]
+ },
+ "execution_count": 30,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "f2 = x+y+z-1\n",
+ "f2"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 31,
+ "id": "a2d7db27",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "x^2 + y^2 + z^2 - 1"
+ ]
+ },
+ "execution_count": 31,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "f3 = x^2 + y^2 + z^2-1\n",
+ "f3"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 32,
+ "id": "ffb34a3f",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Ideal ((-2)*x^2*y + 2*x^2*z + 2*x*y^2 + (-2)*x*z^2 + (-2)*y^2*z + 2*y*z^2, x + y + z - 1, x^2 + y^2 + z^2 - 1) of Multivariate Polynomial Ring in x, y, z over Algebraic Field"
+ ]
+ },
+ "execution_count": 32,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "J = ideal(f1,f2,f3)\n",
+ "J"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 33,
+ "id": "991507b1",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "[{z: 0, y: 0, x: 1},\n",
+ " {z: 0, y: 1, x: 0},\n",
+ " {z: 1, y: 0, x: 0},\n",
+ " {z: 0.6666666666666667?, y: 0.6666666666666667?, x: -0.3333333333333334?},\n",
+ " {z: 0.6666666666666667?, y: -0.3333333333333334?, x: 0.6666666666666667?},\n",
+ " {z: -0.3333333333333334?, y: 0.6666666666666667?, x: 0.6666666666666667?}]"
+ ]
+ },
+ "execution_count": 33,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "J.variety()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 35,
+ "id": "d950f08a",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "[x + y + z - 1, y^2 + y*z - y + z^2 - z, y*z^2 + (-2/3)*y*z + 1/2*z^3 + (-5/6)*z^2 + 1/3*z, z^4 + (-4/3)*z^3 + 1/9*z^2 + 2/9*z]"
+ ]
+ },
+ "execution_count": 35,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "jgb=J.groebner_basis()\n",
+ "jgb"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 37,
+ "id": "f71c0645",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "z^4 + (-4/3)*z^3 + 1/9*z^2 + 2/9*z"
+ ]
+ },
+ "execution_count": 37,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "jgb[3]"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 39,
+ "id": "4b26bc25",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "z * (z - 1) * (z - 2/3) * (z + 1/3)"
+ ]
+ },
+ "execution_count": 39,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "jgb[3].factor()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 40,
+ "id": "7945c62d",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Ideal ((-2)*x^2*y + 2*x^2*z + 2*x*y^2 + (-2)*x*z^2 + (-2)*y^2*z + 2*y*z^2, x + y + z - 1, x^2 + y^2 + z^2 - 1) of Multivariate Polynomial Ring in x, y, z over Algebraic Field"
+ ]
+ },
+ "execution_count": 40,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "J"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "5edad37f",
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "SageMath 10.1",
+ "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": "3.11.6"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/Untitled1.ipynb b/Untitled1.ipynb
new file mode 100644
index 0000000..19d9d5b
--- /dev/null
+++ b/Untitled1.ipynb
@@ -0,0 +1,463 @@
+{
+ "cells": [
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "id": "d07ea913",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "(x, y)"
+ ]
+ },
+ "execution_count": 1,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "var('x,y')"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "id": "30b7e0f3",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "g(x,y) = x^2 + y^2 - 9"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "id": "5f695aaa",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "f(x,y)= x^2 +x*y +y^2 -4*y"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "id": "701aed0e",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "(2*x, 2*y)"
+ ]
+ },
+ "execution_count": 5,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "gradg = g(x,y).gradient()\n",
+ "gradg"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 7,
+ "id": "ebb00f48",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "(2*x + y, x + 2*y - 4)"
+ ]
+ },
+ "execution_count": 7,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "gradf = f(x,y).gradient()\n",
+ "gradf"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 14,
+ "id": "dbc0ed3b",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Multivariate Polynomial Ring in la, x, y over Algebraic Field"
+ ]
+ },
+ "execution_count": 14,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "R.<la,x,y> = PolynomialRing(QQbar,order='lex')\n",
+ "R"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 15,
+ "id": "e82d9224",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "e1 = 2*x+y - la*2*x\n",
+ "e2 = x+2*y -4 - la*2*y\n",
+ "e3 = x^2 + y^2-9"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 16,
+ "id": "c51a6b16",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "I = ideal(e1,e2,e3)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 17,
+ "id": "5a9e061c",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "[la + 1/18*y^3 + (-1/36)*y - 1, x + 1/2*y^2 - 9/4, y^4 + (-5)*y^2 - 63/4]"
+ ]
+ },
+ "execution_count": 17,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "I.groebner_basis()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 19,
+ "id": "a51b0491",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "[{y: -2.681495060562937?, x: -1.345207879911715?, la: 1.996684267393276?},\n",
+ " {y: 2.681495060562937?, x: -1.345207879911715?, la: 0.003315732606724674?},\n",
+ " {y: -1.480005324255096?*I, x: 3.345207879911715?, la: 1 - 0.2212127582776941?*I},\n",
+ " {y: 1.480005324255096?*I, x: 3.345207879911715?, la: 1 + 0.2212127582776941?*I}]"
+ ]
+ },
+ "execution_count": 19,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "I.variety()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 24,
+ "id": "ff87ebbf",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "x^2 - 2*x - 9/2"
+ ]
+ },
+ "execution_count": 24,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "x.subs(I.variety()[0]).minpoly()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "45778ed0",
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "a7f12dbb",
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 26,
+ "id": "aa794406",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "R.<x,y,z> = PolynomialRing(QQbar,order='lex')"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 27,
+ "id": "f941187e",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "A = matrix(R,[[y*z,1,2*x],[x*z,1,2*y],[x*y,1,2*z]])"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 29,
+ "id": "3ed1478f",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "(-2)*x^2*y + 2*x^2*z + 2*x*y^2 + (-2)*x*z^2 + (-2)*y^2*z + 2*y*z^2"
+ ]
+ },
+ "execution_count": 29,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "f1=A.det()\n",
+ "f1"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 30,
+ "id": "60c173ea",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "x + y + z - 1"
+ ]
+ },
+ "execution_count": 30,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "f2 = x+y+z-1\n",
+ "f2"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 31,
+ "id": "9d9d2f92",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "x^2 + y^2 + z^2 - 1"
+ ]
+ },
+ "execution_count": 31,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "f3 = x^2 + y^2 + z^2-1\n",
+ "f3"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 32,
+ "id": "c5388074",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Ideal ((-2)*x^2*y + 2*x^2*z + 2*x*y^2 + (-2)*x*z^2 + (-2)*y^2*z + 2*y*z^2, x + y + z - 1, x^2 + y^2 + z^2 - 1) of Multivariate Polynomial Ring in x, y, z over Algebraic Field"
+ ]
+ },
+ "execution_count": 32,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "J = ideal(f1,f2,f3)\n",
+ "J"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 33,
+ "id": "4c83a33e",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "[{z: 0, y: 0, x: 1},\n",
+ " {z: 0, y: 1, x: 0},\n",
+ " {z: 1, y: 0, x: 0},\n",
+ " {z: 0.6666666666666667?, y: 0.6666666666666667?, x: -0.3333333333333334?},\n",
+ " {z: 0.6666666666666667?, y: -0.3333333333333334?, x: 0.6666666666666667?},\n",
+ " {z: -0.3333333333333334?, y: 0.6666666666666667?, x: 0.6666666666666667?}]"
+ ]
+ },
+ "execution_count": 33,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "J.variety()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 35,
+ "id": "db8474fb",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "[x + y + z - 1, y^2 + y*z - y + z^2 - z, y*z^2 + (-2/3)*y*z + 1/2*z^3 + (-5/6)*z^2 + 1/3*z, z^4 + (-4/3)*z^3 + 1/9*z^2 + 2/9*z]"
+ ]
+ },
+ "execution_count": 35,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "jgb=J.groebner_basis()\n",
+ "jgb"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 37,
+ "id": "dacdffed",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "z^4 + (-4/3)*z^3 + 1/9*z^2 + 2/9*z"
+ ]
+ },
+ "execution_count": 37,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "jgb[3]"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 39,
+ "id": "ac01d945",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "z * (z - 1) * (z - 2/3) * (z + 1/3)"
+ ]
+ },
+ "execution_count": 39,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "jgb[3].factor()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 40,
+ "id": "586ad09b",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Ideal ((-2)*x^2*y + 2*x^2*z + 2*x*y^2 + (-2)*x*z^2 + (-2)*y^2*z + 2*y*z^2, x + y + z - 1, x^2 + y^2 + z^2 - 1) of Multivariate Polynomial Ring in x, y, z over Algebraic Field"
+ ]
+ },
+ "execution_count": 40,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "J"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "d35170a9",
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "SageMath 10.1",
+ "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": "3.11.6"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/homepage/index.org b/homepage/index.org
index d6f96cb..0038e55 100644
--- a/homepage/index.org
+++ b/homepage/index.org
@@ -23,6 +23,8 @@ Skriftlig tentamen.
| TATA54/TEN1 | Talteori | 2024-08-22 | 14-18 | Linköping | 2024-07-23 - 2024-08-12 |
* Allra senaste nytt VT2024 <<allra-senaste-nytt>>
+** 2024-04-29
+Till nästa gång så tittar vi på 12.4.3b, 12.4.8, 13.1.2, 13.1.3.
** 2024-04-23
Jag pratade om "Eulers regel" som uttrycker kedjebråkskonvergenterna
q_n/p_n till [x_0;x_1,x_2,\dots] som rationella funktioner i
diff --git a/homepage/sitemap.org b/homepage/sitemap.org
index c1e5cb8..aa4e7f7 100644
--- a/homepage/sitemap.org
+++ b/homepage/sitemap.org
@@ -2,7 +2,6 @@
- [[file:newlectures/newlecture.org][Föreläsningar i Talteori]]
- [[file:newlectures/new-lect-0.org][Talteori översiktsföreläsning]]
-- [[file:index.org][TATA54 Talteori]]
- [[file:lectures/Henselfaktorisering.org][Henselfaktorisering]]
- [[file:senaste-nytt.org][senaste-nytt]]
- [[file:labs/lecture5.org][lecture5]]
@@ -11,5 +10,6 @@
- [[file:labs/lecture2.org][lecture2]]
- [[file:labs/lecture1.org][lecture1]]
- [[file:labs/lecture0.org][lecture0]]
+- [[file:index.org][TATA54 Talteori]]
- [[file:labs/kedjebraklab.org][Sagemath-övningar på kedjebråk]]
- [[file:labs/HenselLyftLabHT2023.org][SageMath-övningar på Hensellyft, primitiva rötter, och KRS]]
\ No newline at end of file
diff --git a/public/index.html b/public/index.html
index 960fa0f..e267400 100644
--- a/public/index.html
+++ b/public/index.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="sv">
<head>
-<!-- 2024-04-23 -->
+<!-- 2024-04-29 -->
<meta charset="utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>TATA54 Talteori</title>
@@ -74,17 +74,18 @@
</li>
<li><a href="#orgda7b5cb">Allra senaste nytt VT2024 </a>
<ul>
+<li><a href="#org6916c72">2024-04-29</a></li>
<li><a href="#org2fc933f">2024-04-23</a></li>
<li><a href="#org0e96bfa">2024-04-22</a></li>
<li><a href="#org796ce06">2024-04-14</a>
<ul>
-<li><a href="#orge6da327">Uppgifter att räkna till nästa gång</a></li>
+<li><a href="#org7d0c4fd">Uppgifter att räkna till nästa gång</a></li>
<li><a href="#org87e8642">Datorlaboration</a></li>
</ul>
</li>
<li><a href="#orgff00c35">2024-04-07</a>
<ul>
-<li><a href="#org7d0c4fd">Uppgifter att räkna till nästa gång</a></li>
+<li><a href="#org1367de2">Uppgifter att räkna till nästa gång</a></li>
<li><a href="#orgcb6d72a">Beviset för kvadratisk reciprocitet</a></li>
</ul>
</li>
@@ -257,6 +258,14 @@ Skriftlig tentamen.
<h2 id="orgda7b5cb">Allra senaste nytt VT2024 <a id="org53d52c1"></a></h2>
<div class="outline-text-2" id="text-orgda7b5cb">
</div>
+<div id="outline-container-org6916c72" class="outline-3">
+<h3 id="org6916c72">2024-04-29</h3>
+<div class="outline-text-3" id="text-org6916c72">
+<p>
+Till nästa gång så tittar vi på 12.4.3b, 12.4.8, 13.1.2, 13.1.3.
+</p>
+</div>
+</div>
<div id="outline-container-org2fc933f" class="outline-3">
<h3 id="org2fc933f">2024-04-23</h3>
<div class="outline-text-3" id="text-org2fc933f">
@@ -286,9 +295,9 @@ Till på fredag 12.4.5b, 12.4.6b
<h3 id="org796ce06">2024-04-14</h3>
<div class="outline-text-3" id="text-org796ce06">
</div>
-<div id="outline-container-orge6da327" class="outline-4">
-<h4 id="orge6da327">Uppgifter att räkna till nästa gång</h4>
-<div class="outline-text-4" id="text-orge6da327">
+<div id="outline-container-org7d0c4fd" class="outline-4">
+<h4 id="org7d0c4fd">Uppgifter att räkna till nästa gång</h4>
+<div class="outline-text-4" id="text-org7d0c4fd">
<p>
12.2.1bf, 12.2.2cd, 12.2.3c, 12.2.11
</p>
@@ -309,9 +318,9 @@ gränsvärdet av a(k,m) då m går mot oändligheten?
<h3 id="orgff00c35">2024-04-07</h3>
<div class="outline-text-3" id="text-orgff00c35">
</div>
-<div id="outline-container-org7d0c4fd" class="outline-4">
-<h4 id="org7d0c4fd">Uppgifter att räkna till nästa gång</h4>
-<div class="outline-text-4" id="text-org7d0c4fd">
+<div id="outline-container-org1367de2" class="outline-4">
+<h4 id="org1367de2">Uppgifter att räkna till nästa gång</h4>
+<div class="outline-text-4" id="text-org1367de2">
<p>
11.1.1d, 11.1.4, 11.1.25ab, 11.2.1cd, 11.2.2, 11.2.3,11.2.4
</p>
@@ -439,7 +448,7 @@ H_L_tree(f,p,r).plot(layout='tree')
</div>
-<figure id="orgb640d28">
+<figure id="orga443756">
<img src="img/Henseluppgift.png" alt="Henseluppgift.png">
</figure>
@@ -1665,7 +1674,7 @@ Another text is <a href="https://wstein.org/ent/">Elementary Number Theory</a> a
</ul>
-<figure id="org296c115">
+<figure id="orgd70112c">
<img src="img/sieveE.jpg" alt="sieveE.jpg">
</figure>
@@ -1686,7 +1695,7 @@ Another text is <a href="https://wstein.org/ent/">Elementary Number Theory</a> a
</main>
<footer id="postamble" class="status">
<p class="author">Författare: Jan Snellman Jan Snellman</p>
-<p class="date">Created: 2024-04-23</p>
+<p class="date">Created: 2024-04-29</p>
<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
</footer>
</body>
diff --git a/public/sitemap.html b/public/sitemap.html
index c466128..7372682 100644
--- a/public/sitemap.html
+++ b/public/sitemap.html
@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
-<!-- 2024-04-23 -->
+<!-- 2024-04-29 -->
<meta charset="utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Kurshemsida</title>
@@ -208,7 +208,6 @@
</header><ul class="org-ul">
<li><a href="newlectures/newlecture.html">Föreläsningar i Talteori</a></li>
<li><a href="newlectures/new-lect-0.html">Talteori översiktsföreläsning</a></li>
-<li><a href="index.html">TATA54 Talteori</a></li>
<li><a href="lectures/Henselfaktorisering.html">Henselfaktorisering</a></li>
<li><a href="senaste-nytt.html">senaste-nytt</a></li>
<li><a href="labs/lecture5.html">lecture5</a></li>
@@ -217,13 +216,14 @@
<li><a href="labs/lecture2.html">lecture2</a></li>
<li><a href="labs/lecture1.html">lecture1</a></li>
<li><a href="labs/lecture0.html">lecture0</a></li>
+<li><a href="index.html">TATA54 Talteori</a></li>
<li><a href="labs/kedjebraklab.html">Sagemath-övningar på kedjebråk</a></li>
<li><a href="labs/HenselLyftLabHT2023.html">SageMath-övningar på Hensellyft, primitiva rötter, och KRS</a></li>
</ul>
</main>
<footer id="postamble" class="status">
<p class="author">Author: Jan Snellman</p>
-<p class="date">Created: 2024-04-23</p>
+<p class="date">Created: 2024-04-29</p>
<p class="validation"><a href="https://validator.w3.org/check?uri=referer">Validate</a></p>
</footer>
</body>
diff --git a/t2.ipynb b/t2.ipynb
new file mode 100644
index 0000000..03d8318
--- /dev/null
+++ b/t2.ipynb
@@ -0,0 +1,463 @@
+{
+ "cells": [
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "id": "9ecab153",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "(x, y)"
+ ]
+ },
+ "execution_count": 1,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "var('x,y')"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "id": "ce0ab8a1",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "g(x,y) = x^2 + y^2 - 9"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "id": "d63bdcf7",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "f(x,y)= x^2 +x*y +y^2 -4*y"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "id": "220e5c14",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "(2*x, 2*y)"
+ ]
+ },
+ "execution_count": 5,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "gradg = g(x,y).gradient()\n",
+ "gradg"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 7,
+ "id": "2fb288a7",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "(2*x + y, x + 2*y - 4)"
+ ]
+ },
+ "execution_count": 7,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "gradf = f(x,y).gradient()\n",
+ "gradf"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 14,
+ "id": "54d7ffa7",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Multivariate Polynomial Ring in la, x, y over Algebraic Field"
+ ]
+ },
+ "execution_count": 14,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "R.<la,x,y> = PolynomialRing(QQbar,order='lex')\n",
+ "R"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 15,
+ "id": "fbe53eff",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "e1 = 2*x+y - la*2*x\n",
+ "e2 = x+2*y -4 - la*2*y\n",
+ "e3 = x^2 + y^2-9"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 16,
+ "id": "5635dddc",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "I = ideal(e1,e2,e3)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 17,
+ "id": "f79e370f",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "[la + 1/18*y^3 + (-1/36)*y - 1, x + 1/2*y^2 - 9/4, y^4 + (-5)*y^2 - 63/4]"
+ ]
+ },
+ "execution_count": 17,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "I.groebner_basis()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 19,
+ "id": "d3392e18",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "[{y: -2.681495060562937?, x: -1.345207879911715?, la: 1.996684267393276?},\n",
+ " {y: 2.681495060562937?, x: -1.345207879911715?, la: 0.003315732606724674?},\n",
+ " {y: -1.480005324255096?*I, x: 3.345207879911715?, la: 1 - 0.2212127582776941?*I},\n",
+ " {y: 1.480005324255096?*I, x: 3.345207879911715?, la: 1 + 0.2212127582776941?*I}]"
+ ]
+ },
+ "execution_count": 19,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "I.variety()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 24,
+ "id": "a4935561",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "x^2 - 2*x - 9/2"
+ ]
+ },
+ "execution_count": 24,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "x.subs(I.variety()[0]).minpoly()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "15a7d053",
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "5dcf93b0",
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 26,
+ "id": "9081c07e",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "R.<x,y,z> = PolynomialRing(QQbar,order='lex')"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 27,
+ "id": "f33912f1",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "A = matrix(R,[[y*z,1,2*x],[x*z,1,2*y],[x*y,1,2*z]])"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 29,
+ "id": "4523617a",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "(-2)*x^2*y + 2*x^2*z + 2*x*y^2 + (-2)*x*z^2 + (-2)*y^2*z + 2*y*z^2"
+ ]
+ },
+ "execution_count": 29,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "f1=A.det()\n",
+ "f1"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 30,
+ "id": "fc38b38c",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "x + y + z - 1"
+ ]
+ },
+ "execution_count": 30,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "f2 = x+y+z-1\n",
+ "f2"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 31,
+ "id": "a2d7db27",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "x^2 + y^2 + z^2 - 1"
+ ]
+ },
+ "execution_count": 31,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "f3 = x^2 + y^2 + z^2-1\n",
+ "f3"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 32,
+ "id": "ffb34a3f",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Ideal ((-2)*x^2*y + 2*x^2*z + 2*x*y^2 + (-2)*x*z^2 + (-2)*y^2*z + 2*y*z^2, x + y + z - 1, x^2 + y^2 + z^2 - 1) of Multivariate Polynomial Ring in x, y, z over Algebraic Field"
+ ]
+ },
+ "execution_count": 32,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "J = ideal(f1,f2,f3)\n",
+ "J"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 33,
+ "id": "991507b1",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "[{z: 0, y: 0, x: 1},\n",
+ " {z: 0, y: 1, x: 0},\n",
+ " {z: 1, y: 0, x: 0},\n",
+ " {z: 0.6666666666666667?, y: 0.6666666666666667?, x: -0.3333333333333334?},\n",
+ " {z: 0.6666666666666667?, y: -0.3333333333333334?, x: 0.6666666666666667?},\n",
+ " {z: -0.3333333333333334?, y: 0.6666666666666667?, x: 0.6666666666666667?}]"
+ ]
+ },
+ "execution_count": 33,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "J.variety()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 35,
+ "id": "d950f08a",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "[x + y + z - 1, y^2 + y*z - y + z^2 - z, y*z^2 + (-2/3)*y*z + 1/2*z^3 + (-5/6)*z^2 + 1/3*z, z^4 + (-4/3)*z^3 + 1/9*z^2 + 2/9*z]"
+ ]
+ },
+ "execution_count": 35,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "jgb=J.groebner_basis()\n",
+ "jgb"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 37,
+ "id": "f71c0645",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "z^4 + (-4/3)*z^3 + 1/9*z^2 + 2/9*z"
+ ]
+ },
+ "execution_count": 37,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "jgb[3]"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 39,
+ "id": "4b26bc25",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "z * (z - 1) * (z - 2/3) * (z + 1/3)"
+ ]
+ },
+ "execution_count": 39,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "jgb[3].factor()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 40,
+ "id": "7945c62d",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Ideal ((-2)*x^2*y + 2*x^2*z + 2*x*y^2 + (-2)*x*z^2 + (-2)*y^2*z + 2*y*z^2, x + y + z - 1, x^2 + y^2 + z^2 - 1) of Multivariate Polynomial Ring in x, y, z over Algebraic Field"
+ ]
+ },
+ "execution_count": 40,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "J"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "5edad37f",
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "SageMath 10.1",
+ "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": "3.11.6"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
--
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