Laurent polynomial劳伦特多项式

Laurent polynomial劳伦特多项式的系数 p k p_k pk p k ∈ F p_k\in F pkF,F为域, k k k为整数(可为正数和负数),具体可表示为:
p = ∑ k p k X k = p − k X − k + p − ( k − 1 ) X − ( k − 1 ) + . . . + p 0 + p 1 X + . . . + p k X k p=\sum_{k} p_kX^k=p_{-k}X^{-k}+p_{-(k-1)}X^{-(k-1)}+...+p_0+p_1X+...+p_kX^k p=kpkXk=pkXk+p(k1)X(k1)+...+p0+p1X+...+pkXk

Laurent polynomial劳伦特多项式具有如下加法和乘法特性:

  • ( ∑ i a i X i ) + ( ∑ i b i X i ) = ∑ i ( a i + b i ) X i (\sum_{i}a_iX^i)+(\sum_{i}b_iX^i)=\sum_{i}(a_i+b_i)X^i (iaiXi)+(ibiXi)=i(ai+bi)Xi
  • ( ∑ i a i X i ) ⋅ ( ∑ j b j X j ) = ∑ k ( ∑ i < = k , j ; j = k − i a i b j ) X k (\sum_{i}a_iX^i)\cdot (\sum_{j}b_jX^j)=\sum_{k}(\sum_{i<=k,j;j=k-i}a_ib_j)X^k (iaiXi)(jbjXj)=k(i<=k,j;j=kiaibj)Xk

参考资料:
[1] http://mathworld.wolfram.com/LaurentPolynomial.html
[2] https://en.wikipedia.org/wiki/Laurent_polynomial

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