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I am not a native english speaker. I have learned about defining and non-defining relative clauses from english grammar books. The following example from p.12 of Hungerford's Algebra illustrates what is confusing me:

Theorem 6.7. (Fundamental Theorem of Arithmetic) Any positive integer $n \gt 1$ may be written uniquely in the form $n = p_1^{t_1}p_2^{t_2} \cdots p_k^{t_k}$, where $p_1 \lt p_2 \lt \cdots \lt p_k$ are primes and $t_i \gt 0$ for all $i$.

I think the clause "where $p_1 \lt p_2 \lt \cdots \lt p_k$ are primes and $t_i \gt 0$ for all $i$" is a defining relative clause since it gives essential information about the form $n = p_1^{t_1}p_2^{t_2} \cdots p_k^{t_k}$. Grammar books tell me not to use commas in defining relative clauses, so I can't understand why there is a comma preceding "where". I looked up the word "where" in three mathematical textbooks and all of them use commas between formulae and the words "where" in similar situations. Here is another example extracted from Theorem 4.59.(Sylow Theorems) of Anthony W. Knapp's Basic Algebra, Digital Second Edition:

Let $G$ be a finite group of order $p^mr$, where $p$ is prime and $p$ does not divide $r$ .

So, is it a convention in mathematical writing? I would appreciate your help with this situation.

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    This isn't exclusive to math. See http://ell.stackexchange.com/questions/32382/comma-and-where . As a native English speaker I find lots of commas in mathematical writing helpful to break up the long sentences. – 2017-01-10
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    For me, this is a clear, simple *grammar* convention. – 2017-01-10
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    I find it quite praiseworthy that a non-native speaker shows interest in such fine grammatical points, while some native speakers here dare write about a polynomal that "it's roots are distinct" ! – 2017-01-10
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    In another direction, Halmos (_How to Write Mathematics_, though I don't own the book and can't give a page reference) advised against the "where" construction, arguing that notation should be explained _before_ its first use. For instance, Halmos might have suggested, "For every integer $n > 1$, there exist unique primes $p_1 < \cdots < p_k$ and positive integers $t_i$ such that $n = p_1^{t_1} p_2^{t_2} \cdots p_k^{t_k}$" for the fundamental theorem, or "Let $r$ be a positive integer, $p$ a prime not dividing $r$, and let $G$ be a finite group of order $p^{m}r$" for the theorem in Knapp. – 2017-01-10
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    I'm no English expert, but it's not clear to me that this is a defining relative clause. It seems that the thing being defined is "form" and the thing that defines it is "$n = p_1^{t_1}\cdots p_k^{t_k}$", not the following "where" clause. Maybe someone with better expertise can run with that and find some references or something. – 2017-01-10
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    Halmos is right, of course, if you can give the information at the start. If not, considerations of grammar and logic aside, _custom_ demands the comma. – 2017-01-21

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Consider an example:

(1)$\qquad$"We can find $x\in A$ such that $f(x,y)=0$, where $g(y)\in B$"

(2)$\qquad$"We can find $x\in A$ such that $f(x,y)=0$ where $g(y)\in B$".

The first statement could be rewritten, with a change of emphasis, as

$\qquad$"We can find $(x,y)\in A\times \{y:g(y)\in B\}$ such that $f(x,y)=0$".

The second statement has the same structure as "There is fire where there is smoke" and might be interpreted as

$\qquad$"We can find $x\in A$ such that $f(x,y)=0$ whenever $g(y)\in B$".

Usually in mathematics, this latter type of interpretation is unintended, and the comma is needed.

Ideally, all notation should be defined before it is used. However, it often happens that the defining condition—for example, "where $c$ is some positive constant"—is not the focus of interest of the statement; so we may not wish to preface our statement as "There is some positive constant $c$ such that ... ". This is especially the case when the notation and condition are routine and conventional. In such cases, the where clause (preceded, of course, by a comma!) is unobjectionable.