Can the particle "no" sometimes be omitted?

Question

It seems to me that sometimes the "no" that should be between two nouns is omitted. if so, can this be done in any case or are there special cases?

For example, a case where I think there should be a "no" particle is this

Birusu-sei de shugyō kaishi da!

I translate this as "It's the training's start in Birusu's planet" (correct me if I'm wrong), and if I would have to write this back for me I would add a "no" to read

Birusu-sei de shugyō no kaishi da!

A similar phrase it's seen to use the "no"

Gokū wa shugyō no tabi he!?"

Why? Is it optional?

Answer 1

Some nouns tend to directly follow arbitrary nouns without any particle and form longer noun phrases. 開始 (kaishi) is one of such nouns, while 旅 is not (although there are several fixed noun + 旅 compounds). See my previous answer for other examples: the omission of an implied "の" creates the appearance of a 四字熟語{よじじゅくご}?

"Birusu-sei de shugyō no kaishi da!" is grammatically fine, but I feel this no should be omitted. That is partly because this is a subtitle (which should generally be short) and partly because this is a vigorous colloquial sentence where particles tend to be omitted.

Answer 2

No, you basically can't omit it.

Birusu-sei doesn't stand for "Birusu's planet" but "planet Birusu", though it happens to be Birusu's planet (Birusu-no hoshi) too in the story. (The globe is ware-ware no hoshi while it can't be ware-ware-boshi.)

shugyō-kaishi is one compound word and in this case it's almost the same as omission from shugyo-no kaishi but not necessarily interpreted so in every context. For example, it doesn't really work when your mentor says it at the beginning of training.