My ex wife has our 3 children more than half the time and is considered the custodial parent. She comes from a very wealthy family and is supported by her parents but she lives in her own home. She claims no income and does not file taxes. I do not pay alimony or child support per our legal agreement.
I have our children a third of the year and support them in all of their activities during that time. I maintain a 4 bedroom home to accommodate them and pay out of pocket for family health insurance.
I want to claim the child tax credit but my ex wants to prevent it. Can she claim the credit if she doesn't file a tax return? Can her parents claim the credit if our kids don't live with them? Can anybody else claim them?

If nobody else can claim the credit, can I file for it even though I am the Non-custodial parent and my x-wife won't sign the tax form relinquishing that claim?