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Results tagged with polynomials
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user 4213
Questions in which polynomials (single or several variables) play a key role. It is typically important that this tag is combined with other tags; polynomials appear in very different contexts. Please, use at least one of the top-level tags, such as nt.number-theory, co.combinatorics, ac.commutative-algebra, in addition to it. Also, note the more specific tags for some special types of polynomials, e.g., orthogonal-polynomials, symmetric-polynomials.
8
votes
Accepted
Entire function interpolation with control over multiplicities/derivatives
If I read you right,
you want an entire function that takes the values $0$ and $1$ at only
finitely many (specified) points. This implies that the function must be a polynomial,
by Picard's great theo …
- 20.8k
11
votes
The resultant and the ideal generated by two polynomials in $\mathbb{Z}[x]$
An interesting example where the resultant and the "reduced resultant"
differ comes from the theory of elliptic curves. Take an elliptic curve
$$E:\qquad y^2=x^3+ax+b$$
where $a$ and $b$ are integers. …
- 20.8k
15
votes
Is it true that all the "irrational power" functions are almost polynomial ?
There is an "iterated" version of the mean value theorem which states that for
a smooth enough function $f$ on an interval that $\Delta^k f(x)=f^{(k)}(\xi)$
where $\xi$ is between $x$ and $x+k$. This …
- 20.8k
11
votes
Accepted
Positivity of a finite sum
These are Stirling numbers of the second kind. More precisely
your sum is S(k,i-1) where $S$ denotes Stirling number of the second kind.
- 20.8k
1
vote
Given an integer n and a finite extension K of Q , find a polynomial of degree n that is irr...
Your "$x^n-\alpha$" approach is the "ramified" way to go:
the extension you get localizes to a totally ramified extension
of the local field $K_{\mathfrak{p}}$, which has degree $n$,
so the global ext …
- 20.8k
3
votes
Expressing power sum symmetric polynomials in terms of lower degree power sums
If $e_1,\ldots,e_N$ are the elementary symmetric functions of $x_1,\ldots,x_N$
then for $k\ge N$ one has
$$p_k=\sum_{j=1}^N(-1)^{j-1}e_j p_{k-j}.$$
This formula uses the elementary symmetric functions …
- 20.8k
12
votes
Accepted
Ehrhart polynomial
If you mean the polytope with vertices $(0,\ldots,0,\pm1,0,\ldots,0)$
then it is easily seen to be
$$\sum_{k=0}^d 2^k{d\choose k}{x\choose k}.$$
- 20.8k
9
votes
Accepted
A special integral polynomial
An easy way to ensure that a polynomial $g$ of degree $m$ over $\mathbf{Z}$ has
Galois group $S_m$ is to take primes $p_1$, $p_2$ and $p_3$
with $g$ irreducible modulo $p_1$, a linear times an irreduc …
- 20.8k
17
votes
Accepted
Maximal Ideals in the ring k[x1,...,xn ]
We can write the coefficients of $m_2$ as polynomials
in $a_1$ over $k$. Doing this, and replacing $a_1$ by $x_1$ and the free
variable by $x_2$ gives a polynomial $f_2$ in $x_1$ and $x_2$. … We get a sequence of polynomials $f_i$ in $x_1,\ldots,x_i$
and it's not hard to prove these generate $I$. …
- 20.8k
9
votes
Accepted
Irreducibility of polynomials related to quadratic residues
These are known as Fekete polynomials:
http://en.wikipedia.org/wiki/Fekete_polynomial .
I don't know of any results on their Galois groups. …
- 20.8k