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Difference between revisions of "YWANG Generalization"

(Created page with "The '''YWANG Generalization for Polynomial Factorization''' (discovered by AoPS user Yufan Wang) is a generalization of polynomial factorization. It differs...")
 
 
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The '''YWANG Generalization for Polynomial Factorization''' (discovered by AoPS user [[User:Yufanwang|Yufan Wang]]) is a generalization of polynomial factorization. It differs from traditional techniques (such as [[Synthetic Division|synthetic division]]) in that the coefficients of each term is explicitly defined, and is applicable to the factoring of more than one element of the form <math>(x-a_i)</math> at the same time.
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The '''YWANG Generalization for Polynomial Factorization''' (discovered by AoPS user [https://artofproblemsolving.com/community/user/593414 Yufanwang]) is a generalization of polynomial factorization. It differs from traditional techniques (such as [[Synthetic Division|synthetic division]]) in that the coefficients of each term is explicitly defined, and is applicable to the factoring of more than one element of the form <math>(x-a_i)</math> at the same time.
  
 
==Theorem==
 
==Theorem==

Latest revision as of 16:24, 23 February 2023

The YWANG Generalization for Polynomial Factorization (discovered by AoPS user Yufanwang) is a generalization of polynomial factorization. It differs from traditional techniques (such as synthetic division) in that the coefficients of each term is explicitly defined, and is applicable to the factoring of more than one element of the form $(x-a_i)$ at the same time.

Theorem

As per traditional techniques, we limit our constants to be within $\mathbb{R}$ (i.e. we will define $c_0, c_1, \cdots, c_n \in \mathbb{R}$, $i\in \mathbb{N}$, and $n\in \mathbb{Z}$). Then, for a given $f(x)=c_0x^n+c_1x^{n-1}+\cdots+c_{n-1}x+c_n=\textstyle\sum^n_{m=0}c_mx^{n-m}$ and its factors $(x-a_1)(x-a_2)\cdots(x-a_i)$ we have \[f(x)=\prod^i_{k=0}(x-a_k)\sum^{n-1}_{m=0}\gamma_{i, m}x^{n-m-1}\] where $\gamma_{i,m}$ designates the $m$th term of the $i$th YWANG gamma sequence. Each term of the gamma sequence is defined as follows: \[\gamma_{i,j}=\sum_{k=0}^j\left(c_k\sum_{\substack{b_1\cdots b_i\in [0,1]\cap\mathbb{Z} \\ \sum_{l=1}^i b_l = k}} \left(\prod_{l=1}^i a_l^{b_l}\right)\right).\] This result can be derived from strong division with synthetic division or by taking the Taylor series of \[\frac{\sum_{m=0}^n c_m x^{n-m}}{\prod_{k=0}^i (x-a_k)}\] at $x=\infty$. A rigorous proof with both methods will be added in due time.

YWANG Gamma Sequences

The $i$th YWANG gamma sequence is given in the form \[\gamma_{i,j}=\sum_{k=0}^j\left(c_k\sum_{\substack{b_1\cdots b_i\in [0,1]\cap\mathbb{Z} \\ \sum_{l=1}^i b_l = k}} \left(\prod_{l=1}^i a_l^{b_l}\right)\right).\] We will consider the $i$th YWANG gamma sequence solved if there are not nested sums/products within the final expression.

Solved YWANG Gamma Sequences

A list of currently solved YWANG gamma sequences follows. \begin{align*} \gamma_{1,j} &= \sum_{k=0}^j c_k a_1^{j-k} \\ \gamma_{2,j} &= \sum_{k=0}^j c_k \frac{a_1^{j-k}-a_2^{j-k}}{a_1-a_2} \\ \gamma_{3,j} &= \sum_{k=0}^j c_k \frac{a_1^{2(j-k)-1}(a_3-a_2)+a_2^{2(j-k)-1}(a_1-a_3)+a_3^{2(j-k)-1}(a_2-a_1)}{(a_1-a_2)(a_2-a_3)(a_3-a_1)} \end{align*} This list will be updated as further gamma sequences are solved.

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