1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Prove two polynomials are equal in R^n

  1. Nov 6, 2016 #1
    1. The problem statement, all variables and given/known data
    The task is to prove that $$\lim_{x\rightarrow0}\frac{Q_1(x)-Q_2(x)}{\|x\|^k}=0 \implies Q_1=Q_2,$$ where ##Q_1,Q_2## are polynomials of degree ##k## in ##\mathbb{R}^n##.

    2. Relevant equations

    $$
    \lim_{x\to 0} \frac{a x^\alpha}{\|x\|^n}=\left\{\begin{array}{c}
    0 \textrm{ if } |\alpha|>n \\
    a \textrm { if } |\alpha|=n \\
    \infty \textrm { if } |\alpha|<n \textrm{ and } a\neq 0 \\
    0 \textrm{ if } a=0
    \end{array}\right.
    $$

    $$|\alpha|=k=\alpha_1!\alpha_2!\cdot...\cdot\alpha_n!$$

    3. The attempt at a solution Proof by contradiction. Assume that ##Q_{1}\neq{Q_2}## and let's denote ##Q_1(x)-Q_2(x)=F(x)+G(x)## where ##F## is lowest degree (##l##) polynomial and and ##G## contains the rest. Then let's consider the limit $$\lim_{t\rightarrow0}\frac{F(tx)+G(tx)}{\|tx\|^l},$$ where ##b\neq{0}## and ##F(b)\neq{0}##.

    $$\lim_{t\rightarrow0}\frac{F(tb)+G(tb)}{\|tb\|^l}=\lim_{t\rightarrow0}\frac{G(tb)}{\|tb\|^l}+\lim_{t\rightarrow0}\frac{F(tb)}{\|tb\|^l}=...\neq{0}$$ which is contradiction. Therefore it must hold that ##Q_1=Q_2##.


    I have problem expanding the limit expression.
     
    Last edited: Nov 6, 2016
  2. jcsd
  3. Nov 6, 2016 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    Are you sure you can split up the limit that way?
    If yes (which is not trivial), you can just split it into k+1 limits and show that every term of the polynomial has to be zero.

    I would look for all polynomials of degree <= k which satisfy the given limit. The difference between two polynomials of degree k has to be such a polynomial.
     
  4. Nov 6, 2016 #3

    Stephen Tashi

    User Avatar
    Science Advisor

    What does "in ##\mathbb{R}^n##" mean in this context? Are we talking about polynomials in n-variables ?
     
  5. Nov 6, 2016 #4
    Yes.
    ##F(tb)=t^{\alpha}F(b)##, but how to use that?
    Is ##||tb||^l=|t|^l||b||##?

    $$\lim_{t\rightarrow0}\frac{F(tb)+G(tb)}{\|tb\|^l}=\lim_{t\rightarrow0}\frac{t^lF(b)+t{^\alpha}G(b)}{|t|^l\|b\|^l}=\lim_{t\rightarrow0}(\frac{t^lF(b)}{|t|^l\|b\|^l}+\frac{t{^\alpha}G(b)}{|t|^l\|b\|^l})=...?$$, where ##|\alpha|>l##
    Or
    $$\lim_{t\rightarrow0}\frac{F(tb)+G(tb)}{\|tb\|^l}=(\lim_{t\rightarrow0}\frac{a(tb)^l}{\|tb\|^l}+\frac{a(tb)^{\alpha}}{\|tb\|^l})=a+\lim_{t\rightarrow0}\frac{a(tb)^{\alpha}}{\|tb\|^l}=a+0\neq{0}$$ for some ##a_i##?
     
    Last edited: Nov 6, 2016
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted