Is your goal to actually hide your identity? Like you're advocating the violent overthrow of the government and you don't want the police to track you down? Or maybe more realistically, you're afraid your writing might interfere with business relationships, like you don't want co-workers to know that you're writing sex novels? Or is it that you think a pseudonym would help sell your book? Perhaps you think your real name doesn't sound cool enough, or your name sounds inconsistent with the type of book you're writing. Like, you want to publish a Chinese cook book and your real name is Patrick O'Malley?
What do you see as the problem? I can only think of two likely issues: 1. Establishing copyright. 2. Being able to receive and cash the royalty checks.
RE copyright: In the U.S., when you register your copyright there is a place on the form to say you're using a pseudonym. You can also give your real name if you want to, but you are not required to. If you do give your real name, it will be possible for people to find it by looking up the records at the copyright office. The duration of the copyright is different if you don't give your real name, because they then can't base it on when you die. See this link for more info.
RE royalty checks: Again, speaking for the U.S.: If you use a pseudonym, you can endorse a check with a pseudonym and then endorse it with your real name. I think that would be hard for an outsider to track down. Lots of places today use direct deposit, in which case I don't know exactly what information is sent to the bank. Anybody here know? Does the payer tell the bank, "this deposit is for Fred Smith, account number 12345"? Or do they just say "for account number 12345"? If they do give a name, does the bank compare it to the name on the account? In any case, if this was an issue, you could simply open a business account and have the deposits made in the name of the business. That's what I did for my second book anyway. If there was a crime involved I'm sure the police could track down the true owner of the account, but such information is generally confidential. Someone curious about the real author couldn't call the bank and ask who is authorized to withdraw money from the account. Well, they could ask, but the bank won't tell them.