Can someone please explain this terminology? I'm a computer science guy, about half way through a book on abstract algebra.
I'm reading a a paper titled "Integer multiplication in time O(n log n)", which is on Cornell's arXiv site, by David Harvey and Joris van der Hoeven.
In it, they mention "multivariate polynomial rings" in one of their overview sections.  They set $r=2^k$ for some $k$, and then have $d \ge 2$.  They talk about the ring 
$$R[x_1,\ldots,x_{d-1}]/(x_1^{t_1}-1,\dots,x_{d-1}^{t_{d-1}}-1), \quad R:=\mathbb{C}[y]/(y^r+1)$$
"where $t_i \mid 2r$ for all $i$"
I'm wondering what they intend by this statement.  Could someone please explain this to me, in as much detail as possible?  
 A: Let's unravel what $R$ is first. It starts as the polynomial ring $\mathbb C[y]$, i.e. the ring of polynomials in the variable $y$ with coefficients in $\mathbb C$.
From this ring, a new ring, one in which all polynomials which have $y^r+1$ as a factor are "turned to zero", is constructed by taking the quoteint with respect to the ideal $(y^r + 1)$ generated by $y^r + 1$, i.e. the set of polynomials
$$(y^r+1)=\{(y^r+1)\cdot g\mid g\in\mathbb{C}[y]\}.$$
The elements of this ring are equivalence classes of polynomials in $\mathbb C[y]$, i.e. if $[f]$ denotes the equivalence class represented by $f$, then for $f,g\in\mathbb C[y]$ we have 
$$[f]=[g] \text{ if and only if } f-g\in(y^r-1).$$ The "all polynomials which have $y^r+1$ as a factor are turned to zero" is now explicit: the equivalence class $[f]$ represented by $f$ is the equivalence class $[0]$ represented by $0$ if and only if $f-0\in(y^r+1)$, i.e. if and only if $f=g(y^r+1)$ for some $g\in\mathbb C[y]$.
Addition and multiplication in this ring are defined as you would expect:
$$[f]+[g]=[f+g],\quad [f]\cdot[g]=[f\cdot g].$$
The construction and algebra of quotients of rings is a standard topic and should be thoroughly explained in any abstract algebra textbook.

Next we have the ring $R[x_1,\dots,x_{d-1}]$. This is the ring of all polynomials in variables $x_1,\dots,x_{d-1}$ with coefficients in the ring $R$. For $d=2$ this is just $R[x_1]$, so its elements are polynomials in $x_1$ with coefficients in $R$. For $d=3$ the elements of this ring are finite sums
$$\sum a_{i,j}x_1^{i}x_2^{j},$$
where $a_{i,j}\in R$ and $(i,j)$ range over all pairs of non-negative integers, but only finitely many $a_{i,j}\neq 0$. For $d\geq 3$ the polynomials are similarly described.
The ideal $(x_1^{t_1}-1,\dots,x_{d-1}^{t_{d-1}}-1)$ is the ideal of all polynomials in $R[x_1,\dots,x_{d-1}]$ generated by the polynomials $x_1^{t_1}-1,\dots,x_{d-1}^{t_{d-1}}-1$, so it is the set
$$\{f_1(x_1^{t_1}-1)+\cdots+f_{d-1}(x_{d-1}^{t_{d-1}}-1)\mid f_1,\dots,f_{d-1}\in R[x_1,\dots,x_{d-1}]\}$$
The elements of the quotient ring $R[x_1,\dots,x_{d-1}]/(x_1^{t_1}-1,\dots,x_{d-1}^{t_{d-1}}-1)$ are again equivalence classes such that two equivalence classes $[f]$ and $[g]$ represented by $f,g\in R[x_1,\dots,x_{d-1}]$ are the same equivalence class $[f]=[g]$ if and only if $f-g\in(x_1^{t_1}-1,\dots,x_{d-1}^{t_{d-1}}-1)$.

The condition $t_i | 2r$ is not relevant for the definition of these rings, and why the authors chose to work with precisely this ring and how it relates to integer multiplication algorithms is very far from my area of expertise.

To get more experience and intuition about these rings, I would encourage you to take $d=2$, $k=1$ and $t_1=1$, then $t_1=2$, and after that $t_1=4$ and work out what the elements of these rings are by following the definitions.
