If someone legally changes their name, is that name the one that needs to be written in the Death Note?

Heh. That's a pretty good question.

I would say.. Yeah, Once the names been changed legally, आप would need to write that new name in the deathnote in order to kill someone.

Though... your birth name could still be used against you. Since आप can't exactly change your borth certificates details, because आप can only be born once. So द्वारा changing your name, your just giving someone anouther name to be able to use to kill you. Since both names would be legal, in a way.

या maybe not. I don't know lol. I'd need to know और about the laws regarding changing your name legally. Laws might even vary depending on your country. Plus theres the dethnotes rules to consider aswel, and no one but the maker of the Death note can make the पुस्तकें rules लोल

Only one way to find out..........

.................ಠ_ಠ

I wouldn't know my friend...
why don't आप give Tsugumi Ohba a call-

One of the "Unanswerable Death Note Questions".
आप could ask that here:
link

Highly doubtful. Shinigami couldn't care less about our law and many characters in the ऐनीमे lived under aliases but couldn't die from them being written down.

I think what would be most in teresting is if आप were never दिया a name in the first place.

या giving twins the same name since the rule saying that if 2 individuals having the same name are unaffected. But if one dies would that still work?

Many interesting possibilities when आप consider how limited the deathnote is.

It's what the person considers to be their name; the birth certificate has nothing to do with it.
For example: If a very young girl was found on the सड़क, स्ट्रीट (and wasn't aware of her name) and was taken in द्वारा a family who called her "Sue", and yet on her birth certificate she was named "Lindsay", Sue would be the name written in the Death Note.
So, returning to the original question, it is most likely their new name would be the one that would be written in the Death Note if they considered it to be THEM, if that makes any sense. If it was simply to avoid the power of the Death Note that they changed their name, though, then it's basically the same as an alias and won't work in the Death Note.
It is not a matter of legal issues, but personal belief/attachment to the name.

Well, in a lawful state, when आप change your name, आप sign to relinquish all ownership and hold over the old name. आप may not legally use that name anymore. Your name आप chose to accept would be the name that would be written down. Shinigami eyes however, would probably still see old name as the real one. आप might get away with it... who knows? try it and see ;)