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LSpice
LSpice
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zero Zero of a power series in a local field

Let $f(z)=\sum_{n\ge1}a_nz^n$ be a power series of $\mathbb C_p[[z]]$ where the $a_n$ are such that $|a_n|=1$ for every positive integer $n$. Consider $z_0\in\mathbb C_p$ such that $|z_0|<1$. Can one assert that there exist $x_0\in \mathbb C_p$ with $|x_0|<1$ such that $f(x_0)=z_0$.

Thanks in advance for any answer.

zero of a power series in a local field

Let $f(z)=\sum_{n\ge1}a_nz^n$ be a power series of $\mathbb C_p[[z]]$ where the $a_n$ are such that $|a_n|=1$ for every positive integer $n$. Consider $z_0\in\mathbb C_p$ such that $|z_0|<1$. Can one assert that there exist $x_0\in \mathbb C_p$ with $|x_0|<1$ such that $f(x_0)=z_0$.

Thanks in advance for any answer.

Zero of a power series in a local field

Let $f(z)=\sum_{n\ge1}a_nz^n$ be a power series of $\mathbb C_p[[z]]$ where the $a_n$ are such that $|a_n|=1$ for every positive integer $n$. Consider $z_0\in\mathbb C_p$ such that $|z_0|<1$. Can one assert that there exist $x_0\in \mathbb C_p$ with $|x_0|<1$ such that $f(x_0)=z_0$.

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joaopa
joaopa
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zero of a power series in a local field

Let $f(z)=\sum_{n\ge1}a_nz^n$ be a power series of $\mathbb C_p[[z]]$ where the $a_n$ are such that $|a_n|=1$ for every positive integer $n$. Consider $z_0\in\mathbb C_p$ such that $|z_0|<1$. Can one assert that there exist $x_0\in \mathbb C_p$ with $|x_0|<1$ such that $f(x_0)=z_0$.

Thanks in advance for any answer.