on http://eprint.iacr.org/2020/1481.pdf at page 3, it defines this quotient:
$$B = \mathbb{Z}[Y_1, \cdots, Y_k]/(\phi_{m_1}(Y_1), \cdots, \phi_{m_k}(Y_k))$$
where $\phi_m(X)$ is the mth cyclotomic polynomial, and $m = m_1\cdots m_k$, a bunch of prime powers.
I'm trying to understand what this quotient looks like so I can understand later what a basis for it looks like.
Starting with $\mathbb{Z}[Y_1, \cdots, Y_k]$, it looks like a polynomial with multiple variables, like $f(Y_1,Y_2,Y_3) = Y_1 + Y_2 + Y_3$.
What is $(\phi_{m_1}(Y_1), \cdots, \phi_{m_k}(Y_k))$? Is it the polynomial space generated by all the cyclotomic polynomials? How does it looks like?
What the quotient looks like finally?
$(\phi_{m_1}(Y_1),\ldots,\phi_{m_k}(Y_k))$ is the ideal of $\mathbb{Z}[Y_1,\ldots,Y_k]$ generated by the $\phi_{m_i}(Y_i)$'s. Its elements are of the form $$ \sum_{i=1}^k\phi_{m_i}(Y_i)P_i(Y_1,\ldots,Y_k) $$ where $P_i\in\mathbb{Z}[Y_1,\ldots,Y_k]$. Now suppose $i_1,\ldots,i_k\geqslant 0$, what it the class of $Y_1^{i_1}\ldots Y_k^{i_k}$ in $B$ ? Suppose the euclidean division of $Y_j^{i_j}$ by $\phi_{m_j}$ is $Y_j^{i_j}=\phi_{m_j}Q_j+R_j$ where $Q_j,R_j\in\mathbb{Z}[Y_j]$ and $\deg R_j<\deg\phi_{m_j}$. The class of $Y_1^{i_1}\ldots Y_k^{i_k}$ is the same as the class of $R_1\ldots R_k$. This means that all the elements of $B$ can be written as a sum of such products. Or, equivalently, each element of $B$ is a sum of $\alpha Y_1^{i_1}\ldots Y_k^{i_k}$ with $\alpha\in\mathbb{Z}$ and $i_j<\deg\phi_{m_j}$ for all $j$.