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port initial notes on commutative algebra (ideals, modules, Noetherian rings) (#1)

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* port initial notes on commutative algebra: ideals & modules, Nakayama's lemma, etc

* port notes on Noetherian rings&modules

* add ideals related definitions

* improve Cayley-Hamilton proof (specially determinant trick explanation)

* polishing

* add typos detection

* add some exercises, and proof of Z and K[X} being PID
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arnaucube
2025-12-25 13:02:49 +01:00
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notes_fri_stir.tex
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@@ -76,7 +76,7 @@ Consider the following protocol:
\end{enumerate}
%/// TODO tabulate this next lines
With high probablility, $\alpha$ will not cancel the coeffs with $deg \geq d+1$. % TODO check which is the name of this theorem or why this is true
With high probability, $\alpha$ will not cancel the coeffs with $deg \geq d+1$. % TODO check which is the name of this theorem or why this is true
Let $g(x)=a \cdot x^{d+1}, ~~ h(x)=b \cdot x^{d+1}$, and set $f(x) = g(x) + \alpha h(x)$.
Imagine that P can chose $\alpha$ such that $a x^{d+1} + \alpha \cdot b x^{d+1} = 0$, then, in $f(x)$ the coefficients of degree $d+1$ would cancel.
@@ -314,14 +314,15 @@ $$g(z) = (f(z)-f(r))\cdot (z-r)^{-1}$$
\section{STIR (main idea)}
\emph{Update from 2024-03-22, notes from Héctor Masip Ardevol (\href{https://hecmas.github.io/}{https://hecmas.github.io}) explanations.}
\emph{Update from 2024-03-22, notes from Héctor Masip Ardevol\\
(\href{https://hecmas.github.io/}{https://hecmas.github.io}) explanations.}
\vspace{0.3cm}
Let $p \in \mathbb{F}[x]^{<n}$.
In FRI we decompose $p(x)$ as
$$p(x) = p_e(x^2) + x \cdot p_o(x^2)$$
with $p_e, p_o \in \mahtbb{F}[x]^{<n}$ containing the even and odd powers respectively.
with $p_e, p_o \in \mathbb{F}[x]^{<n}$ containing the even and odd powers respectively.
The next FRI polynomial is
$$p_1(x) = p_e(x) + \alpha p_o(x)$$
@@ -329,7 +330,7 @@ for $\alpha \in^R \mathbb{F}$.
In STIR, this would be $q(x)=x^2$,
$$Q(z,y) = p_e(y) + z \cdot p_o(y)$$
and then, $p(x) = Q(x, q(x))$. And $Q$ fullfills the degree from Fact 4.6 from the STIR paper.
and then, $p(x) = Q(x, q(x))$. And $Q$ fulfills the degree from Fact 4.6 from the STIR paper.
We can generalize to a $q$ with bigger degree, or with another shape, and adapting $Q$ on the choice of $q$.