INA §212(a)(10)(A), 8 U.S.C. §1182(a)(10)(A)
Polygamy occurs when an individual is simultaneously married to more than one person. U.S. immigration last has prohibited polygamy almost from the first days of the republic.
Now, you might say that your fiancée or spouse has no intention of marrying more than one person. But here’s where the issue comes up – and it does come up surprisingly often. The problem arises when there was a previously married and a problem with the divorce paperwork. We see this on both sides of the equation – both U.S. petitioners and foreign nationals. A divorce decree that wasn’t properly filed, or even not properly certified by the court, can lead to problems.
The same issue arises if a former spouse died. In such cases, a formal death certificate is required. Otherwise, the consulate will have concerns that the former spouse is still alive, which would lead to polygamy.
The minimum you need to do here is to ensure that you understand what official divorce decrees and/or death certificates look like for the country involved. The State Department keeps a partial list of these in its “reciprocity tables,” which are available online. Within the United States, you can check with the local clerk of court (for divorce decrees) or records department (for death certificates).