You at entitled to claim a child as your dependent if that child lived primarily with you throughout 2016. If you and your wife had an equal amount of overnights with the children, then the parent with the higher adjusted gross income would be entitled to claim the children as your dependent.
If the child did not live with you for the greater number of nights in 2016, or if you do not have the higher adjusted gross income if the overnights are equal, then you are not entitled to claim a child as a dependent in 2016 unless your wife signs the Release. There is no other recourse.
Note that claiming children as dependents is an issue that is often settled in permanent child support agreement or court order so that there is no discrepancy about claiming children on future years’ tax returns.
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