ComputerAlgebra/28_apr.md

Monomial ideals

we always will try to reduce a polynomial ideal to a monomial ideal.

3.1. Definition

A set Г together with an operation * Г * Г -> Г is called commutative monoid if the following conditions are satisfied:

  1. The operation * is associative: $(a * b) * c = a * (b * c)$ for all a, b, c in Г.
  2. The operation * is commutative: $a * b = b * a$ for all a, b in Г.
  3. There exists an element e in Г such that for all a in Г, $a * e = a$

3.3 Propposition

the map $log: T^n \mapsto \mathbb{N}^n$, meaning $ x_1^{a_1} x_2^{a_2} ... x_n^{a_n} \mapsto (a_1, a_2, ..., a_n)$ is an isomorphism of commutative monoids.

Remark: in the book we look at the $P^r = {f_1, ..., f_r}\ |\ f_i \in P$ and submodules of $U <= P^r$, i.e.

3.4 Definition

  1. A subset \triangle <= T^n is called a monoideal if t - \triangle <= \triangle for all t in T^n.

  2. Let $\delta$ be a monoideal. A set of terms is called a ystem of generators of \delta if $\delta = T^n * n_1 u_1 + ... + n_r u_r$ for all $u_i \in \delta$ and $n_i \in \mathbb{N}$.

$\delta = _{T^n}$ - t-s are generated by $T^n$.

3.6. Proposition __important__

For n >= 1, the monoid $N^n$ is is noetherian (?), i.e. every monoideal is finitely generated.

proove it later.

  1. Every monoideal in $N^n$ is finitely generated.
  2. Let $p_1, ..$ be a sequence of terms. then there exists a number $k >= 1 that every term in the sequence is a multiple of one of the first k terms.

3.8 Dicksons Lemma

Let $P = K[x]$ be a polynomial ring and $I <= P$ be an ideal. Then there exists a finite set of monomials $m_1, ..., m_r$ such that $I = _P$.

Dicksons Lemma on wiki

3.10. Proposition

Let $t_1... t_k \in T^n$ and $I = < t_1, ..., t_k >_p$.

  1. Then there exists minimal monomial system of generators of I, say $t'_1, ..., t'_r$ <= $T^n$ ie a system of generator of I such that any system of generators satisfies the propperty that each $t'_i$ is a multiple of one of the $t_i$ and no $t_i$ can be dropped.
  2. given any system of generators (s.o.g.) $t_1, ..., t_k$ of I, the minimal monomial system of generators can be calculated as follows:
    1. for j in [1, K] check wether $t_j$ is a multiple of one of the $t_i$ for i != j. If it is, drop $t_j$
    2. repeat step 1 until no more terms can be dropped.

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