All Questions
7
questions
1
vote
1
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72
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If $p$ is a prime and $m,n\in\mathbb{N}$, prove that $\binom{pm}{pn}\equiv\binom{m}{n}$ mod $p$.
Hint: Compare the binomial expansions of $(1+x)^{pm}$ and of $(1+x^m)^p$ in $\mathbb{F}_p[x]$.
If $R$ is a commutative ring, then the set of all polynomials with coefficients in $R$ is denoted by $R[x]...
- 366
1
vote
0
answers
71
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Why do two ways of expanding the same formal polynomial lead to matching coefficients?
There are a few proofs in which the technique is to expand the product of some formal polynomials in $\mathbb{R}[x_1,x_2,\ldots,x_k]$ in more than one distinct way and then we can match up the ...
- 3,415
1
vote
1
answer
34
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how to reduce $(1-\alpha)^{T-i}$ into a sum
I'm given the following proof:
\begin{align}
& \sum^T_{i=k} \alpha^i(1-\alpha)^{T-i} \binom{T}{i}
\\&=\sum^T_{i=k} \alpha^i \binom{T}{i} \sum_{j=0}^{T-i}(-\alpha)^j\binom{T-i}{j}
\\&=\...
- 877
1
vote
1
answer
49
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Binomial expansion of $(1+X)^b$ of the form $\sum\limits_k b^kf_k(X)$ with $f_k(X)$ polynomial
For each $b \in \mathbb{N}, (1+X)^b=\sum\limits_{n=0}^b{{b}\choose {n}}X^{n}$ is a polynomial function of $X$. How to write $(1+X)^b$ in terms of $b$ as a closed expression of the form $$(1+X)^b=\sum\...
- 1,816
3
votes
1
answer
463
views
Gaussian polynomial identities
I'd appreciate any hints for showing that these identities are true for Gaussian polynomials. I've tried to approach the problem using basic algebra but it gets messy very quickly and I've gotten ...
- 371
0
votes
0
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44
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How to calculate the $k$-dimension of a subspace of a polynomial ring? [duplicate]
Let $k$ be an infinite field and $R:=k[x_1,...,x_n]$ the polynomial ring in $n$ indeterminates.
Why is the $k$-dimension of $U$ given by $\begin{pmatrix} n+m-1 \\ m\end{pmatrix}$, when $U$ is the ...
- 540
10
votes
1
answer
1k
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Vandermonde identity in a ring
Let $R$ be a commutative $\mathbb{Q}$-algebra. For $r \in R$ and $n \in \mathbb{N}$ we can define the binomial coefficient $\binom{r}{n}$ as usual by $\binom{r}{0}=1$ and $\binom{r}{n+1}=\frac{r-n}{n+...
- 165k