Translation:I would not have published her book.
Thanks a lot for the example. It is a pity that the translation exercises of conditional forms of verbs are mostly (all of them?) partial phrases without the conditional parts of full sentences. These partial phrases are not as effective for learning as full sentences. You are one of the contributers, could you replace at least some of current exercises?
"ihr Buch" is in the accusative form, it is the direct object of the sentence and the basic phrase is "etwas (acc) veröffentlichen". There is no noun is the whole sentence which might be in the dative form (there is no indirect object corresponding to the English "to him/to her").
The context suggest whether the book would be published or not, not to be shown to the public. Context is important in translation, so it gives meaning to what you say. Be careful on that. I have seen that Duolingo does not pay attention to the context very often, and when it does,it does not make sense.
I wrote, "I wold not have published your book," and it was wrong. But the possessive pronoun "ihr" means "your" in the plural formal you - form. And it also means "her" in the singular familiar - form. So shouldn't my answer have been accepted?
The formal you(r) is always capitalised.
Ich hätte ihr Buch nicht veröffentlicht = I wouldn't have published her/their book
Ich hätte Ihr Buch nicht veröffentlicht = I wouldn't have published your book (formal)
By the way, the formal doesn't distinguish between singular and plural.