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Results tagged with polynomials
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user 21907
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.
4
votes
Accepted
Closed formula for Hermite polynomials
Up to some normalization, the harmonic oscillator $H$ is self-adjoint such that
$$
\langle Hu, u\rangle=\sum_{k\ge 0}(\frac12+k) \vert u_k\vert^2,
$$
and thus defining a self-adjoint $A$ by the equali …
- 16.2k
7
votes
Symmetric version of Hilbert's seventeenth problem?
he proves that
Every symmetric smooth fonction on $\mathbb R^n$ is equal to
$g(\sigma_1,⋯,\sigma_n)$ for a $g$ smooth on $\mathbb R^n$,
where
$\sigma_1, \dots, \sigma_n$
are the elementary symmetric polynomials …
- 16.2k
2
votes
Division theorem for vector-valued distributions
I think I have an answer to my own question : let us consider $Q$ the transposed of the comatrix of $P$. The determinant of $P$ is a polynomial and by the Lojasiewicz-Hörmander theorem, we can find a …
- 16.2k
3
votes
1
answer
88
views
Division theorem for vector-valued distributions
The classical division theorem for scalar distributions on $\mathbb R^n$ can be formulated as follows. Let $T$ be a tempered distribution on $\mathbb R^n$ and let $P$ be a non-zero polynomial of $n$ …
- 16.2k
6
votes
Accepted
Multivariate Hermite Polynomials
In one dimension, you have
$$
h_k(t)e^{-π t^2}=e^{π t^2}(\frac{d}{dt})^k\bigl(e^{-2π t^2}\bigr),
$$
and $h_k$ is easily proven to be with degree $k$.
The completeness question amounts to proving
$L^2= …
- 16.2k