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When naming some number of objects, there's only one plural form in English (2 dogs, 3 dogs, 10 dogs). But in Slavic languages like Polish the form changes depending on the number (2 psy, 4 psy, 5 psów, 12 psów, 22 psy, 100 psów).
Does the form vary in Turkish? If so, what are the rules behind that?
In Turkish, a plural declension is not needed if the noun is modified by a numeral.
(If it is used, it is emphatic.)
In fact, it is not needed with any modifier that already implies plural.
arkadaş (friend)
arkadaşlar (friends)
bir arkadaş (one friend)
iki arkadaş (two friends)
birçok arkadaş (many friends, a lot of friends)
çok fazla arkadaş (too many friends)
sayıda arkadaş (numerous friends)
Note all the situations in which -lar is not required.
The same is true of languages from other language families like Persian, Armenian, Georgian and Kurdish that existed in the region before Turkish, and of Turkic languages generally.
There is only one plural form in Turkish* but it's usage is quite different from English or the Slavic examples you give.
So far so good. That's the only plural form there is, but the trick is you only ever use it when you need a stand alone noun. Any time you precede the noun with a number or any other indication of quantity you actually use the singular form.
Note only in the last example is the plural form used.
* The actual suffix could be -lar or -ler depending on vowel harmony rules but either way it's the same form.
Wikipedia says "If a numeral is used with a noun, then the plural suffix is usually not used". (Unreferenced, unfortunately, but it agrees with my memory from when I studied Turkish.) So, no, neither the case nor the number of nouns changes with different numerals.
I believe the use of the genitive singular for (numbers ending in) 2, 3 or 4, as opposed to the genitive plural for larger numbers in Slavonic languages is generally regarded as a paucal form, a generalisation of the dual.
Turkish has only one suffix for plural (* ler) and with the number of objects It doesn't change.
But as it is written many times above, Turkish has tendency to drop the plurals in most cases. This feature can be observed through the many proposed Altaic languages (Japanese, Mongolian etc.)
