Tagged Questions
3
votes
3answers
40 views
dimension of space of polynomials
Let $\mathcal P_k^n$ be the space of all polynomials of degree $\leq k$ in $n$ variables. Prove $\dim\mathcal P_k^n = {n+k\choose k}$.
I tried showing this by taking $n\in\mathbb N$ an arbitrary ...
0
votes
3answers
40 views
Newton's binomial problem
It is known that in the development of $(x+y)^n$ there is a term of the form $1330x^{n-3}y^3$ and a term of the form $5985x^{n-4}y^4$.
Calculate $n$.
So, I know that the binomial formula of Newton ...
1
vote
1answer
55 views
Writing a sum as a fraction
Express
$$\sum^{20}_{i=2}f(x)^i$$where $$f(x)=\sum_{i\geq 1}2^{i-1}x^{3i}$$ as a fraction of polynomials $p(x)/q(x)$ and simplify as much as possible.
Hmm. How to do it? Wolfram is really stupid on ...
1
vote
2answers
31 views
express Pochhammer symbol $(x)_n$ as a polynomial of order $n$ in $x$
Define
$$(x)_{n}=x(x-1)(x-2)...(x-n+1)=\prod_{k=1}^{n} (x-k+1)=\sum_{k=0}^n a_k x^k$$
Q: what is the closed-form expression for $a_k$ ?
1
vote
2answers
32 views
Number of coefficients of multivariable polynomial
Let $g \in \mathbb{F}[x_1, \dots, x_n]$ be a polynomial of degree $d$ with $n$ variables. Number of its coefficients is ${n+d \choose d}$
Is there an easy proof? It clearly holds for univariate ...
2
votes
0answers
106 views
Can a linear combination of even Legendre polynomials have common real root(s) with a linear combination of odd Legendre polynomials?
I am using the following definition of Legendre Polynomials: $P_0(x)=1$, $P_1(x)=x$ and
$$P_{k+1}(x)=\left(\frac{2k+1}{k+1}\right)xP_k(x)â\left(\frac{k}{k+1}\right)P_{kâ1}(x)$$
Let
...
1
vote
0answers
38 views
Upper bound on degree of coefficients required to write polynomials as a linear combination of $f_1,…,f_n$
All polynomials will be elements of $\mathbb{Q}[x]$. Suppose $f_1,...,f_n$ are polynomials of degree at most $d$ which are coprime. What is a (hopefully sharp) upper bound on the degree of ...
6
votes
1answer
101 views
What is the correspondence between combinatorial problems and the location of the zeroes of polynomials called?
In the wikipedia article on the Italian-born American mathematician and philosopher Gian-Carlo Rota, it is stated that the one combinatorial idea he would like to be remembered for
"... is the ...
3
votes
2answers
57 views
Generating Function Example from class
Example: Consider the sequence $(h_n)$ where $h_n$ is the number of nonnegative integer solutions to $$a_1+a_2+a_3+a_4+a_5=n.,$$ where $a_1$ is even, $a_2$ is odd, $a_3$ is a multiple of $5$, $a_4$ is ...
13
votes
2answers
399 views
Signed Multinomial Expansion Coefficients?
I've been spending probably an undue amount of time trying to compute the coefficients of polynomials of the form $p_n(x_1, ..., x_n) = \displaystyle\prod_{\sigma \in \{ -1 , 1 \}^{n-1} } (x_1 + ...
3
votes
3answers
64 views
Find the coefficient for a term in an expression
We have the expression:
$$( 1 + x^1 + x^2 + x^3 + \dots + x^{27})(1 + x^1 + x^2 + \dots + x^{14})^2$$
For this expression how do you calculate the coefficient of $x^{28}$?
I know the answer is ...
1
vote
1answer
177 views
Determine the number of irreducible monic polynomials of degree 3 in $\mathbb F_p[x]$
Determine the number of irreducible monic polynomials of degree 3 in $\mathbb F_p[x]$ where $p$ is a prime.
I'd like to start off by acknowledging that I know there are many posts relating to similar ...
2
votes
1answer
73 views
Evaluate a certain derivative
Let $\{x_1,\dots,x_n\}$ be pairwise distinct complex numbers and $\{l_1,\dots,l_n\}$ a vector of natural numbers such that $l_1+l_2+\dots+l_n=N$. Let
$$ h_j(x)=\prod_{i\neq j,i=1,\dots, n} ...
3
votes
0answers
159 views
Closed-form expression for sum of Vandermonde matrix elements
Given the Vandermonde matrix:
$$\begin{pmatrix}1^0 & 1^1 & 1^2 & ... & 1^n \\
2^0 & 2^1 & 2^2 & ... & 2^n \\
\vdots & \vdots & \vdots & \ddots & ...
1
vote
1answer
69 views
Polynomial Formula like Infinite Sum with non-natural index
By polynomial formula $$(\sum_{i\in m} x_i)^n=\sum_{\substack{j_i \in \mathbb{Z}^+ \\ \sum j_i=n}}\left(\begin{array}{c}
n\\
j_{0},\ldots , j_{m-1}
\end{array}\right)\prod_{i \in m} x_i^{j_i}$$ where ...
5
votes
3answers
77 views
$(1-t)^{-d}= \Sigma_{k=0}^\infty {d+k-1 \choose d-1} t^k$?
I'm trying to see why the equation $(1-t)^{-d}= \Sigma_{k=0}^\infty {d+k-1 \choose d-1} t^k$ holds in the power series ring $\mathbb{Z}[[t]]$. I assume it's a counting argument about the number of ...
2
votes
1answer
88 views
Polynomials and partitions
There is a question I have based on the fact:
If you take a quadratic polynomial with integer coefficients, and take the set (1,2,3,4,5,6,7,8), and make a partition A=(1,4,6,7), and B=(2,3,5,8), and ...
1
vote
0answers
218 views
Using the Multinomial Theorem
I know it's a simple question, but I keep getting different general formulas for the coefficients when I am trying to use the multinomial theorem for the following:
$$
...
5
votes
3answers
248 views
The coefficient of $x^{18}$ in $(1+x^5+x^7)^{20}$
I was asked about a simple question that is: "What is the coefficient of $x^{18}$ in $(1+x^5+x^7)^{20}$? Generally, we know that; $$(x+y+z)^n= ...
3
votes
0answers
86 views
The polynomial where only the terms in the multinomial series where each variable's exponent is $>0$ are kept?
I'm wondering if there's a special polynomial with a name out there with $x_1,x_2,\ldots,x_k$ as variables that's defined like this:
$$
\sum_{\substack{i_1>0,i_2>0, \ldots,i_k>0 \\ i_1 ...
6
votes
3answers
244 views
Intuitive explanation for a polynomial expansion?
Is there an ituitive explanation for the formula:
$$
\frac{1}{\left(1-x\right)^{k+1}}=\sum_{n=0}^{\infty}\left(\begin{array}{c}
n+k\\
n
\end{array}\right)x^{n}
$$
?
Taylor expansion around x=0
...
4
votes
0answers
54 views
Necessary and sufficient condition for $f(q^n)$ to be in $\mathbb{Z}[q,q^{-1}]$ when $f\in\mathbb{Q}(q)[x]$?
In this question, user bgins shows that for each $k$ there is a unique polynomial $P_k(x)$ of degree $k$ whose coefficients are in $\mathbb{Q}(q)$, the field of rational functions, such that ...
3
votes
0answers
62 views
About $t$-analogue of the Euler polynomials.
A certain way to define the $t$-analogue of the Euler polynomials $C_n(x)$ is by
$$
C_n(x,t)=\sum_{\pi\in S_n}x^{\text{des}(\pi)+1}t^{\text{maj}(\pi)}
$$
where $des(\pi)$ is the descents in $\pi$, ...
8
votes
3answers
269 views
Polynomial in $\mathbb{Q}[x]$ sending integers to integers?
We can view the binomial coefficient $\binom{x}{k}$ has a polynomial in $x$ with degree $k$. So taking some $f\in\mathbb{Q}[x]$, why is $f(n)\in\mathbb{Z}$ for all $n\in\mathbb{Z}$, precisely when the ...
2
votes
2answers
274 views
Product of all irreducibles with degree divisible by $n$ in $\mathbb{F}_{q^n}$?
In the finite field of $q^n$ elements, the product of all monic irreducible polynomials with degree dividing $n$ is known to simply be $X^{q^n}-X$. Why is this?
I understand that $q^n=\sum_{d\mid ...
5
votes
1answer
196 views
A nicer recurrence for the Eulerian polynomials.
I was perusing the subject of Eulerian polynomials. I'm assuming the definition that the Eulerian polynomial is defined by $C_n(t)=\sum_{\pi\in S_n}t^{1+d(\pi)}$, where $d(\pi)$ is the number of ...
1
vote
1answer
54 views
Recurrence $C_n(t)=t(1-t)C'_{n-1}(t)+ntC_{n-1}(t)$ for Eulerian Polynomials?
I was reading about Eulerian polynomials on OEIS, and there is a recurrence given for them, namely:
$$
C_0(t)=1
$$
and
$$
C_n(t)=t(1-t)C'_{n-1}(t)+ntC_{n-1}(t)\qquad (n\geq 1).
$$
How can ...
5
votes
2answers
279 views
Counting Irreducible Polynomials
I'm investigating irreducible polynomials over finite fields at the moment, and I wanted to know if there is a formula for the number of irreducible polynomials of degree n over a fixed finite field ...
2
votes
0answers
96 views
Elementary symmetrical polynomial equations, whose solutions are known to be natural numbers.
Let $n_1,n_2,\dots,n_k$ be natural numbers (excluding 0), and for each $1\leq i\leq k$ let $\sigma_i(n_1,n_2,\dots,n_k)$ be the elementary symmetrical polynomial consisting of the sum of all products ...
3
votes
3answers
152 views
Do these special power functions generate all homogeneous symmetric polynomials?
Over rational numbers, the set of all power functions up to a certain degree generate all symmetric polynomials in that degree.
My question is as follows. To be succinct, let's say we have four ...
7
votes
2answers
525 views
Is there a General Formula for the Transition Matrix from Products of Elementary Symmetric Polynomials to Monomial Symmetric Functions?
Given the elementary symmetric polynomials $e_k(X_1,X_2,...,X_N)$ generated via
$$
\prod_{k=1}^{N} (t+X_k) = e_0t^N + e_1t^{N-1} + \cdots + e_N.
$$
How can one get the monomial symmetric functions ...
0
votes
1answer
65 views
Multiplication of the factors $(x+a_i)$, what are the coefficients? [duplicate]
Possible Duplicate:
Create polynomial coefficients from its roots
When multiplying some factors $(x+a_i)$, is there a neat way to write the coefficients in $x$? I tried up to three factors, ...
13
votes
4answers
452 views
Intriguing polynomials coming from a combinatorial physics problem
For real $0<q<1$, integer $n >0 $ and integer $k\ge 0$, define
$$[k, n]_q \equiv -\sum_{m=1}^{n} q^{m(k+1)} (q^{-n}; q)_m = -\sum_{m=1}^{n} q^{m(k+1)} \prod_{l=0}^{m-1} (1-q^{l-n})$$
...
3
votes
3answers
132 views
The sum of a polynomial over a boolean affine subcube
Let $P:\mathbb{Z}_2^n\to\mathbb{Z}_2$ be a polynomial of degree $k$ over the boolean cube.
An affine subcube inside $\mathbb{Z}_2^n$ is defined by a basis of $k+1$ linearly independent vectors and an ...
1
vote
1answer
60 views
Multinomial coefficient not coming out right
I'm trying to find the coefficient of $ x^{36} $ in the expansion of $ (2 - x + x^2)^{21} $
So I found all possible combinations of $ \displaystyle 2^x, x^y, (x^2)^z $ that yield 36 and $ x + y + z = ...
3
votes
2answers
561 views
Roots of polynomials on the unit circle
Given an integer polynomial, how can I detect if its roots in $\mathbb{C}$ are on the unit circle, that is, of absolute value 1?
I'm not sure what the best formulation for this problem is; that may ...
2
votes
1answer
152 views
Help Identify this Pattern in the coefficients of a polynomial
I have a polynomial that is the related to a solution of a system of 4 differential equations that represent that density matrix evolution of some quantum system (details not particularly important). ...
6
votes
1answer
263 views
A Curious Binomial Sum Identity without Calculus of Finite Differences
Let $f$ be a polynomial of degree $m$ in $t$. The following curious identity holds for $n \geq m$,
\begin{align}
\binom{t}{n+1} \sum_{j = 0}^{n} (-1)^{j} \binom{n}{j} \frac{f(j)}{t - j} = (-1)^{n} ...
2
votes
1answer
92 views
Counting some vanishing polynomials in $\mathbb{Z}_n[X]$
Given the ring of polynomials $\mathbb{Z}_n[X]$, consider $$\mathbb{P}_n = \{a_0 +a_1x+a_2x^2+\cdots+a_{n-1}x^{n-1}| a_i \in \mathbb{Z}_n\},$$ i.e.
$\mathbb{P}_n$ is the set of all polynomials in ...
1
vote
1answer
206 views
The number of symmetric polynomials of n degree
How many symmetric polynomials of n degree with all their coefficients $\ =1 $ are there?Is there a type that computes their number?
7
votes
1answer
113 views
Has this Extension to a Series been Studied Before?
We know from Calculus what a series is, and you might have seen infinite products as well. But the Elementary Symmetric Polynomials give an entire spectrum of operators between a sum and product over ...
19
votes
8answers
940 views
why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$
why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$?
I know this is well-known. But how to prove it rigorously? Even mathematical induction does not seem so straight-forward.
...
6
votes
2answers
188 views
Number of terms in a monomial symmetric polynomial
Is there a closed form expression for the number of terms in a monomial symmetric polynomial in a given number of variables for a particular partition of exponents, in terms of which/how many ...
