Skip to content

Commit ccbee27

Browse files
Gordie NovakGordie Novak
Gordie Novak
authored and
Gordie Novak
committed
adding notes for june 16th
1 parent 164a8f2 commit ccbee27

6 files changed

Lines changed: 209 additions & 1 deletion

File tree

‎.gitignore‎

Lines changed: 1 addition & 1 deletion
Original file line numberDiff line numberDiff line change
@@ -112,4 +112,4 @@ codechat_config.yaml
112112
*.fls
113113
*.out
114114
*.synctex.gz
115-
LateX-notes/*.log
115+
Notes/LateX-notes/*.log

‎Notes/June 16th.md‎

Lines changed: 7 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -0,0 +1,7 @@
1+
## June 16th Goals
2+
* Start writing the Gröbner basis section in the textbook
3+
* Everyone should have at least one commit to the chatper.
4+
* We can utilized Strumfeld's slides and notes.
5+
* Additionally, we can pull from Strumfeld's exercises.
6+
* We need to export to BibTeX and get and example in LaTeX and PreTeXt.
7+

‎Notes/LaTeX-notes/June 16th.pdf‎

95.7 KB
Binary file not shown.

‎Notes/LaTeX-notes/June 16th.tex‎

Lines changed: 100 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -0,0 +1,100 @@
1+
\documentclass[10pt,oneside]{article}
2+
%%%% Page Info + Commands %%%%%{
3+
4+
%packages
5+
\usepackage{geometry}
6+
\usepackage{latexsym}
7+
\usepackage{amssymb}
8+
\usepackage{amsfonts}
9+
\usepackage{amstext}
10+
\usepackage{amsmath}
11+
\usepackage{amsthm}
12+
\usepackage{multicol}
13+
\usepackage{hyperref}
14+
\usepackage{enumerate}
15+
\usepackage{tikz}
16+
\usepackage{enumitem}
17+
\usepackage{xcolor}
18+
19+
20+
\setlength{\footskip}{-5mm}
21+
22+
% a good babble textwidth is 5.75in
23+
\newcommand{\babblewidth}{\setlength\textwidth{5.75in}}
24+
25+
26+
% This will stretch out the page
27+
\newcommand{\bigpage}{ \setlength \oddsidemargin{-.25in}
28+
\setlength \textwidth{6.75in}
29+
\setlength \topmargin{-1in}
30+
\setlength \textheight{9.75in}}
31+
32+
33+
%This will shrink the page
34+
\newcommand{\smallpage}{ \setlength \oddsidemargin{.5in}
35+
\setlength \textwidth{5in}
36+
\setlength \topmargin{0in}
37+
\setlength \textheight{9in}}
38+
39+
\newcommand{\separator}{\vglue .1in\hrule\vglue .1in}
40+
41+
\newcommand{\pause}{\vglue .1in\hrulefill {\tiny Pause here}\hrulefill \vglue .1in}
42+
43+
%%general stuff
44+
\newcommand{\caret}{\textasciicircum}
45+
46+
%This will put a circle around something.
47+
\newcommand*\circled[1]{\tikz[baseline=(char.base)]{
48+
\node[shape=circle,draw,inner sep=2pt] (char) {#1};}}
49+
50+
51+
% Commands for abstract
52+
\newcommand{\Z}{\mathbb{Z}}
53+
\newcommand{\R}{\mathbb{R}}
54+
\newcommand{\C}{\mathbb{C}}
55+
\newcommand{\normal}{\triangleleft}
56+
\newcommand{\Q}{\mathbb{Q}}
57+
\newcommand{\F}{\mathbb{F}}
58+
\newcommand{\N}{\mathbb{N}}
59+
\newcommand{\K}{\mathbb{K}}
60+
\newcommand{\aut}[1]{{\rm Aut}(#1)}
61+
\newcommand{\Ker}{{\rm Ker}\,}
62+
\newcommand{\im}{{\rm Im}\,}
63+
\newcommand{\cyclic}[1]{\langle #1 \rangle}
64+
\newcommand{\isom}{\cong}
65+
\newcommand{\autc}[1]{{\rm Aut_c}(#1)}
66+
\newcommand{\autsub}[2]{{\rm Aut}_{#1}(#2)}
67+
68+
\newcommand{\vp}{\vspace{0.15cm}\\}
69+
\newcommand{\vpp}{\vspace{0.25cm}\\}
70+
\newcommand{\vpn}{\vspace{0.05cm}\\}
71+
\newcommand{\rmv}[1]{\,\backslash\{#1\}}
72+
\newcommand{\rmvs}[1]{\,\backslash{#1}}
73+
\newcommand{\md}[1]{\,\text{mod } #1}
74+
75+
%%%%%%%% command for graphics %%%%%%%%%%%%%
76+
%}
77+
\definecolor{darkgreen}{rgb}{0.0, 0.5, 0.0}
78+
\definecolor{sasha}{rgb}{0.0, 0.5, 0.5}
79+
\definecolor{marcus}{rgb}{0.7, 0.3, 0.3}
80+
\definecolor{sam}{rgb}{0.2, 0.2, 0.8}
81+
82+
83+
\begin{document}
84+
\subsection*{Gandini Notes on Invariants}
85+
$$S = \K[x]^G$$
86+
$J_G = $IDEAL generated by all POSITIVE degree invariants.
87+
\begin{align*}
88+
x\mapsto -x, \quad f = x^2\\
89+
\K[x^2] \text{ vs } (x^2)
90+
\end{align*}
91+
To compute $R^G$, first compute $J_G = (f_1,\cdots, f_2)$ then $R^G=\L[R_G(f_1),\cdots, R_G(f_2)]$ where $R_G$ is the map $f\mapsto \frac{1}{|G|}\sum g\cdot f$.\vpp
92+
\textit{Note: $J_G$ is the hilbert ideal.}\vpp
93+
$J_G=(\mathbb I (A_G) + (y_1,\cdots, y_n)\cap \K[x_1,\cdots, x_n])$. \vpp
94+
Now consider $\underline x, \underline y$.\\
95+
$G$ acts on $\underline x$.
96+
$$\{(\underline x, g\cdots \underline x)\}$$
97+
on all $(\underline x, \underline y)$ such that $\underline y = g \cdots \underline x$.
98+
$$A_G = \bigcup_{g\in G}\{(\underline x, g\cdot \underline x)\}$$
99+
100+
\end{document}

‎Notes/LaTeX-notes/grobner.pdf‎

62.6 KB
Binary file not shown.

‎Notes/LaTeX-notes/grobner.tex‎

Lines changed: 101 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -0,0 +1,101 @@
1+
\documentclass[10pt,oneside]{article}
2+
%%%% Page Info + Commands %%%%%{
3+
4+
%packages
5+
\usepackage{geometry}
6+
\usepackage{latexsym}
7+
\usepackage{amssymb}
8+
\usepackage{amsfonts}
9+
\usepackage{amstext}
10+
\usepackage{amsmath}
11+
\usepackage{amsthm}
12+
\usepackage{multicol}
13+
\usepackage{hyperref}
14+
\usepackage{enumerate}
15+
\usepackage{tikz}
16+
\usepackage{enumitem}
17+
\usepackage{xcolor}
18+
19+
20+
\setlength{\footskip}{-5mm}
21+
22+
% a good babble textwidth is 5.75in
23+
\newcommand{\babblewidth}{\setlength\textwidth{5.75in}}
24+
25+
26+
% This will stretch out the page
27+
\newcommand{\bigpage}{ \setlength \oddsidemargin{-.25in}
28+
\setlength \textwidth{6.75in}
29+
\setlength \topmargin{-1in}
30+
\setlength \textheight{9.75in}}
31+
32+
33+
%This will shrink the page
34+
\newcommand{\smallpage}{ \setlength \oddsidemargin{.5in}
35+
\setlength \textwidth{5in}
36+
\setlength \topmargin{0in}
37+
\setlength \textheight{9in}}
38+
39+
\newcommand{\separator}{\vglue .1in\hrule\vglue .1in}
40+
41+
\newcommand{\pause}{\vglue .1in\hrulefill {\tiny Pause here}\hrulefill \vglue .1in}
42+
43+
%%general stuff
44+
\newcommand{\caret}{\textasciicircum}
45+
46+
%This will put a circle around something.
47+
\newcommand*\circled[1]{\tikz[baseline=(char.base)]{
48+
\node[shape=circle,draw,inner sep=2pt] (char) {#1};}}
49+
50+
51+
% Commands for abstract
52+
\newcommand{\Z}{\mathbb{Z}}
53+
\newcommand{\R}{\mathbb{R}}
54+
\newcommand{\C}{\mathbb{C}}
55+
\newcommand{\normal}{\triangleleft}
56+
\newcommand{\Q}{\mathbb{Q}}
57+
\newcommand{\F}{\mathbb{F}}
58+
\newcommand{\N}{\mathbb{N}}
59+
\newcommand{\K}{\mathbb{K}}
60+
\newcommand{\aut}[1]{{\rm Aut}(#1)}
61+
\newcommand{\Ker}{{\rm Ker}\,}
62+
\newcommand{\im}{{\rm Im}\,}
63+
\newcommand{\cyclic}[1]{\langle #1 \rangle}
64+
\newcommand{\isom}{\cong}
65+
\newcommand{\autc}[1]{{\rm Aut_c}(#1)}
66+
\newcommand{\autsub}[2]{{\rm Aut}_{#1}(#2)}
67+
68+
\newcommand{\vp}{\vspace{0.15cm}\\}
69+
\newcommand{\vpp}{\vspace{0.25cm}\\}
70+
\newcommand{\vpn}{\vspace{0.05cm}\\}
71+
\newcommand{\rmv}[1]{\,\backslash\{#1\}}
72+
\newcommand{\rmvs}[1]{\,\backslash{#1}}
73+
\newcommand{\md}[1]{\,\text{mod } #1}
74+
75+
%%%%%%%% command for graphics %%%%%%%%%%%%%
76+
%}
77+
\definecolor{darkgreen}{rgb}{0.0, 0.5, 0.0}
78+
\definecolor{sasha}{rgb}{0.0, 0.5, 0.5}
79+
\definecolor{marcus}{rgb}{0.7, 0.3, 0.3}
80+
\definecolor{sam}{rgb}{0.2, 0.2, 0.8}
81+
82+
83+
\begin{document}
84+
\subsection*{Gandini Notes on Gröbner Bases}
85+
\begin{align*}
86+
\begin{bmatrix}2&3&4&\mid &5 \\ 3&4&5&\mid &2\end{bmatrix}
87+
\approx
88+
\begin{bmatrix}1&0&-1&\mid &-14 \\ 0&1&2&\mid &11\end{bmatrix}
89+
\end{align*}
90+
$F[x]$ is a PID: principal ideal domain. So we have that:
91+
\begin{align*}
92+
(f_1,f_2)=(\gcd(f_1,f_2))
93+
\end{align*}
94+
In $F[x]$ the smallest monic polynomial in $I$ generates $I$.\\
95+
$\implies F[x]$ is a PID.\vpp
96+
Remember Bezout's:\\
97+
$\implies \gcd(f_1,f_2)=af_1+bf_2$\\
98+
$\implies \gcd \in I = (f_1,f_2)$
99+
100+
101+
\end{document}

0 commit comments

Comments
 (0)