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性質和定理

Equivalent Bases

Q : 我們怎麼知道兩個 basis \mathbf{B}, \mathbf{B'} 是不是產生同樣的 lattice ?

A : 檢查 \mathbf{B'B^{-1}} 是不是 unimodular matrix

小定理

下面兩個敘述是等價的

  • L(\mathbf{B}) = L(\mathbf{B'})
  • \exists unimodular matrix \mathbf{U} such that \mathbf{B'} = \mathbf{BU}

某定理

Let \mathbf{b_1}, \cdots \mathbf{b_n} \in \mathbb{R}^n denote linear independent vectors in L ( full-rank n-dimensional lattice )

\mathbf{b_1}, \cdots \mathbf{b_n} form a basis of L \Leftrightarrow P(\mathbf{b_1}, \cdots, \mathbf{b_n}) \cap L = \{\mathbf{0}\}

Minkowski Theorem

Any convex, centrally symmetric body S

\text{vol}(S) > 2^n\text{det}(L)

convex set ( convex body )

A set S

滿足 x, y \in S \Rightarrow \alpha x + (1-\alpha)y \in S 就是 convex

滿足 x \in S \Leftrightarrow -x \in S 就是 centrally symmetric

想知道更多請去看 wikipedia - Convex set

Minkowskis First Theorem

Any Lattice L

\lambda_1(L) \le \sqrt{n}\ \text{det}(L)^{\frac{1}{n}}