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I am given a quotient ring $R=\frac{\mathbb{Z}[x]}{\left< x^n +t\right>}$ for $t\in\mathbb{Z}$, and two polynomials from $R$, $A$,$B$ and let $C$ to be there product.

Defining the norms $$\Vert A\Vert_{2}= \sqrt{\sum_{i=0}^{n-1}\vert a_i\vert^2} \text{ and } \Vert A\Vert_{\infty}=\max_i \vert a_i\vert $$,

what is the probability of $\Vert C\Vert_{\infty}\leq \beta$ for a positive $\beta$ such that the $a_i$s and $b_i$s are selected uniformely random from $\{-1,0,1\}$.

the same question to the norm $\Vert \cdot\Vert_2$.

I have tried to do some numerical experiments but failed ...

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  • $\begingroup$ Before asking for a probability, you should define a probability space. Also you seem to use $B$ for two different things. $\endgroup$ Nov 3, 2022 at 1:19
  • $\begingroup$ thanks @BrendanMcKay $\endgroup$
    – Don Freecs
    Nov 3, 2022 at 1:20

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