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Polynomials? That sounds familiar!

Another CMI problem: (and I could solve this)

Define polynomials \( P_n(x) \) in this way:

  1. \( \displaystyle P_0(x) = 1 \)

  2. \( \displaystyle P_n(x) = \dfrac{x(x-1)(x-2)\ldots(x-n)} {n!} \) where \( n \) is a positive integer.

Prove that for any polynomial \( f(x) \), there always exist unique real numbers \( b_0, b_1, \ldots b_n \) such that:

\( \displaystyle f(x) = \sum_{0 \le i \le n} b_i p_i (x) \)

Further prove that the \( b_i \)'s are all integers, if it is given that \( f(m) \) is an integer for each integer \( m \).

Note by Parth Thakkar
3 years, 4 months ago

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Note that the polynomials \(P_i(x), 0\leq i \leq n\) have increasing degrees and hence they are linearly independent in the vector space of polynomials of degree \(n\). Hence they form a basis and the proof for the first part follows. Abhishek Sinha · 3 years, 1 month ago

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Part (a): Proving the existence and uniqueness of \({b_i}\):

Let \( f(x)= c_0 +c_1x +c_2x^2+...+c_nx^n\) and take note of the fact that:

\(p_n(x)= \dbinom{x}{n}(x-n)\). Since \(p_0(x)=1\), we may write:

\(\sum_{0 \le i \le n}(b_i)(p_i(x)) = (b_0)+ \sum_{1 \le i \le n}(b_i)(p_i(x))\)

Expanding \(\sum_{1 \le i \le n}(b_i)(p_i(x))\): (Without loss of generality, assume \(\sum_{1 \le i \le n}(b_i)(p_i(x))\)= \(\sum_{i=1}^{n}(b_i)(p_i(x))\))

\(\sum_{i=1}^{n}(b_i)(p_i(x))\))= \(b_1(x^2-x)+\frac{b_2}{2}(x^3-3x^2+2x)+\frac{b_3}{6}(x^4-6x^3+11x^2-6x)+\frac{b_4}{24}(x^5-10x^4+35x^3-50x^2+24x)+... +\frac{b_nx!(x-n)}{n!(x-n)!}\)

Equating coefficients of f(x), \(c_i\), with those of \(\sum_{i=1}^{n}(b_i)(p_i(x))\)):

\(c_0=b_0\)

\(c_1=-b_1+b_2-b_3+b_4+....=\sum_{i=1}^{n}(-1)^ib_i\)

\(c_2=b_1-\frac{3}{2}b_2+\frac{11}{6}b_3-\frac{50}{24}b_4+....\)

\(c_3= \frac{b_2}{2} - b_3 +\frac{35}{24}b_4-....\)

\(c_4= \frac{b_3}{6} - \frac{10}{24}b_4+.....\)

\(c_5= \frac{b_4}{24}+....\)

This is a linear system of n equations (excluding \(c_0=b_0\)) in n unknowns. Therefore, we may construct an n x n matrix A such that: \(A\vec{b}=\vec{c}\), where \(\vec{b}\) contains all \(b_i\) and \(\vec{c}\) contains all \(c_i\). \(A=\)

\[ \begin{pmatrix} -1 & 1 & -1&...&\alpha_1 \\ 1 & \frac{-3}{2} & \frac{11}{6}&...&\alpha_2 \\ 0 & \frac{1}{2} & -1&...&\alpha_3\\ 0&0&\frac{1}{6}&...&\alpha_4\\ .&.&.&.&.\\ .&.&.&.&.\\ .&.&.&.&.\\ 0&0&0&....&\alpha_n\\ \end{pmatrix} \]

\(A\vec{b}=\vec{c}\) has a unique solution if and only if A is an invertible matrix. A is invertible if and only if \(det(A)\neq 0\). Then \(A\vec{b}=\vec{c}\) has a unique solution if and only if \(det(A)\neq 0\). If we use Gaussian Elimination to reduce A, that is, if we add line (1) to line (2), then line (2) to line (3), then line (3) to line (4), etc. we see that A will be reduced to an upper triangular matrix with every term on the main diagonal of A of the form: \(A_{ji}=\frac{-1}{i!}\), where i corresponds to the column number of \(A_{ji}\). Hence \(det(A)=\prod_{i=1}^n\frac{-1}{i!}\).

Since \(det(A)=\prod_{i=1}^n\frac{-1}{i!}\) and \(i\ge1\), \(det(A)\neq 0\). Then A is invertible and \(A\vec{b}=\vec{c}\) has a unique solution, hence \(\vec{b}=A^{-1}\vec{c}\), which of course exists. This proves the existence of \(b_i\) but does not prove the uniqueness of all \(b_i\). We will now proceed to prove this.

Let \(A_{ji}^{-1}\) denote the element of \(A^{-1}\) in the \(j^{th}\) row and \(i^{th}\) column. Since \(\vec{b}=A^{-1}\vec{c}\), we have:

\(b_n=A_{n1}^{-1}c_1 + A_{n2}^{-1}c_2+A_{n3}^{-1}c_3+...+A_{nn}^{-1}c_n\)

Suppose \(\exists\) \(b_i\) such that \(b_i=b_n\) but \(i \neq n\), i.e. suppose all \(b_i\) are not unique. Then, since \(b_n\) is described by the above equation:

\( A_{i1}^{-1}c_1=A_{n1}^{-1}c_1\)

\( A_{i2}^{-1}c_2=A_{n2}^{-1}c_2\)

\( A_{i3}^{-1}c_3=A_{n3}^{-1}c_3\)

And in general: \( A_{in}^{-1}c_n=A_{nn}^{-1}c_n\) \(\Rightarrow\) \( A_{in}^{-1}=A_{nn}^{-1}\)

If \( A_{in}^{-1}=A_{nn}^{-1}\) and \(i \neq n\), then \(A^{-1}\) contains at least two identical rows, which implies \(det(A^{-1})=0\). Then \(A^{-1}\) is not invertible, which is impossible because \((A^{-1})^{-1}=A\). This is a contradiction, hence we are forced to conclude that all \(b_i\) are unique. Part (b) coming soon.

QED Ethan Robinett · 3 years, 1 month ago

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did any one get a seat in CMI (i got one wihtout interview) i solved nearly 6 of the ques.... Priyadarshi Anubhav · 3 years, 2 months ago

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@Priyadarshi Anubhav I did (without interview). Didn't expect it though! Pretty much set on joining there. Parth Thakkar · 3 years, 2 months ago

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Solved this one. Nice question. Vijay Raghavan · 3 years, 4 months ago

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@Vijay Raghavan How do you solve this question? Can you give the solution? Or atleast an hint Mridul Sachdeva · 3 years, 4 months ago

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@Mridul Sachdeva Hint: Notice that \( P_k (x) \) vanishes at \( x = k \), and also do the "higher" \( P \)'s. That is, \( P_{k+1}, P_{k+2} \) and so on. But the "lower" \( P \)'s don't. If you use this properly, you're done! Parth Thakkar · 3 years, 4 months ago

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@Parth Thakkar Nice :) Mridul Sachdeva · 3 years, 4 months ago

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