According to my grammar books, it reads that you can say "Discount on your next purchase" but not "Discount on next purchase" because you have to put "your" or "the" before "next".
Could anyone can explain why?
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Purchase needs a determiner, but next is an adjective.
yourdeterminer nextadjective purchase
Next can't fill the determiner slot here, so you can't leave out your.
*discount on next purchase ← ungrammatical in Standard English
This doesn't work because the noun phrase next purchase is incomplete without a determiner. And because the meaning of next is definite (it refers to something specific), we want to use some sort of definite determiner, like your or the.
It's true that you might run across this sort of example in an elliptical style – for example in a headline or on a coupon – where words that are usually necessary are left out. But in Standard English, a determiner is required.
Although it's usually not a determiner, we might want to consider next a determiner in a few specific phrases, like next Wednesday or next week:
In this week, it's clear that this is a determiner. We might want to consider last and next determiners in these examples by analogy. But most of the time, as in your example, next is an adjective and cannot be a determiner.
In this answer, the * symbol marks a phrase as ungrammatical.
If you are referring to THE next purchase, then you need to use the definite article. It doesn't sound correct to say "discount on next purchase", because you are talking about THE NEXT purchase. It is about a particular object, therefore you need the article.
If you were mentioning whose is the next purchase (like in "discount on your next purchase"), then you should not use article, because possessive pronouns do not require articles in English.
When you have contact with native speakers and immerse yourself in English, you are able to perceive what sounds natural and what does not. Sometimes you don't find such things on grammar books, you need to live with English.
In addition, I have an observation about your comment. You said "next" is a superlative and it is not. Superlative is the highest degree of a quality, the most highlighted object or person of a group. In the context of distance, the superlatives are: "closest" and "nearest". Perceive superlatives either end with "-est" or have "most" before the adjective.
"Next" cannot be used as superlative. You can read a publication from Cambridge Dictionary about "nearest" and "next": http://dictionary.cambridge.org/pt/gramatica/gramatica-britanica/nearest-or-next
The word "next" may have been used as superlative in the past (probably for "nigh", as dictionary.com says), but not today anymore.
Tip: I think you are reading grammar and forgetting the naturalness of the language. If you want to learn speaking English, you should commit yourself to learn the natural language with people and less with grammar books. Unless you want to be a grammarian or understand the historical dynamics of English, you don't need to attach yourself to archaic terms, such as "nigh", because it helps little for English learning as second language.
All the best
There are several words that, in common contemporary English, count as determiners -- a, the, some, this, that, my, your, their, whatever, and so on.
A singular and countable noun (such as "purchase") requires some sort of (singular and countable) determiner to form a complete and coherent phrase.
The difference between "next purchase" and phrasings like "your next purchase", "the next purchase", "some next purchase", "whatever next purchase" and so on is that the latter phrasings include a determiner where one is required. On its own, "next" is a reasonable adjective, but it is not a sufficient determiner.
The most common determiners are articles -- the definite and indefinite articles specifically.
"Your" and "the" are not the only sensible options, but they are the most obvious options in this specific context for a supplied determiner. In the absense of such, the phrasing is either not coherent or not countable.