Questions tagged [weierstrass-preparation]
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8 questions
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Approximating a strictly increasing non-negative function on a non-negative domain by polynomials with non-negative coefficients
Let $f:[0,2]\rightarrow [0,\infty)$ be a strictly increasing smooth function. The Weierstrass approximation theorem says that we can uniformly approximate $f$ by polynomials. But my concern is
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- 11
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Expression for a logarithm of the Weierstrass sigma functions
Lets consider a following function
\begin{equation}
f(z) = \ln{\frac{\sigma (z - v)}{\sigma (z + v)}},
\end{equation}
where $\sigma$ is Weierstrass sigma function, and $v$ is a constant. Is there a ...
- 19
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How to analyze the roots of two variable $p$-adic power series $f(x,y)$ in the maximal ideal $m$?
Let $K \subseteq \mathbb{Q}_p$ be the $p$-adic field, $O_K$ be the ring of integer and $m$ be the maximal ideal.
Consider the single variable $p$-adic power series $f(x)=\sum_{n=0}^{\infty}a_n x^n \...
- 930
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Multivariate Weierstrass preparation Theorem?
Let $(K,|\cdot|)$ be a complete local field and $\mathcal{O}$ be its ring of integers. Let $C$ be a complete algebraic closure of $K$ and let $\mathfrak{m}:=\{x\in \mathcal{O}_{C}~|~|x|<1\}$ where $...
- 1,479
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Can a squarefree polynomial in K[x,y] not be squarefree in K[[x]][y]?
In a UFD, as usual one says that $f$ is square-free if it is not divisible by the square of any irreducible element, i.e., if it has no multiple factor.
An polynomial $f\in k[x,y]$ can have more ...
- 81
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Roots of a partially holomorphic function
Let $\Omega$ be an open subset of $\mathbb R^d$, let $U$ be an open subset of $\mathbb C$ and let $f:\Omega\times U\rightarrow\mathbb C$ be a $C^\infty$ function which is holomorphic with respect to $\...
- 16.2k
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Is there an algorithm to find out the number of small solutions to a polynomial equation, when we vary all the coefficients?
Let $\Phi (z,t)$ be a polynomial given by
$$ \Phi(z,t) := z^n + A_{n-1}(t) z^{n-1} + \ldots + A_1(t) z + A_0(t).$$
Assume that $\Phi(0,0) =0$. It is a fact that a solution $z(t)$ of the equation
$$ \...
- 3,245
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What is the closure of space of polynomials in a dense subspace along with a marked point equal to?
EDIT
Let $\mathbb{C}^{m*}$ be the space of non zero polynomials of degree at most
$d$ in two variables. So an element of this space is essentially
$$ f:=f_{00} + f_{10} x + f_{01} y + \ldots f_{0d}...
- 3,245