First, let's look at the words of Jesus:
"You have heard that it was said, 'You shall not commit adultery.' But I say to you that everyone who looks at a woman with lustful intent has already committed adultery with her in his heart. If your right eye causes you to sin, tear it out and throw it away. For it is better that you lose one of your members than that your whole body be thrown into hell. And if your right hand causes you to sin, cut it off and throw it away. For it is better that you lose one of your members than that your whole body go into hell. "It was also said, 'Whoever divorces his wife, let him give her a certificate of divorce.' But I say to you that everyone who divorces his wife, except on the ground of sexual immorality, makes her commit adultery, and whoever marries a divorced woman commits adultery." - Matthew 5:27-32
And Pharisees came up to him and tested him by asking, "Is it lawful to divorce one's wife for any cause?" He answered, "Have you not read that he who created them from the beginning made them male and female, and said, 'Therefore a man shall leave his father and his mother and hold fast to his wife, and the two shall become one flesh'? So they are no longer two but one flesh. What therefore God has joined together, let not man separate." They said to him, "Why then did Moses command one to give a certificate of divorce and to send her away?" He said to them, "Because of your hardness of heart Moses allowed you to divorce your wives, but from the beginning it was not so. And I say to you: whoever divorces his wife, except for sexual immorality, and marries another, commits adultery." - Matthew 19:9
So, if anyone looks at a woman with lustful intent, that's adultery of the heart. Now, for adultery to be adultery, then either the man doing the looking, or the woman being looked at (or both of them) needs to be married, and they can't be married to each other. With that in mind, it's impossible for the man to seek out a second wife without committing adultery of the heart. Therefore, if he's committing adultery of the heart, any resulting flirtation, hookup, or romantic relationship, whether formalized or not, would also be adultery. Furthermore, consider that the man who divorces his wife and marries another woman commits adultery. Interesting that it's adultery, and not simple abandonment here. This tells us that despite the man's dissolving the legal union, he is not free to marry another blamelessly. If it is adultery to get together with another woman after divorcing your first wife, how much more is it adultery when you keep the first wife around and try to marry another! The only difference is he's not abandoning the first wife. Remember, if the script is flipped, and a married woman gets together with other men while remaining married to her husband, even with no intent to divorce him, that is still adultery. Adultery isn't A, B, and C for women but only A and B for men. Adultery is adultery. Jesus was using the term adultery to demonstrate the seriousness of his standards, which were quite a bit stricter than many Jewish leaders'.
So with that in mind, there is no room for polygamy for the Christian. Any special dispensations that might have been given in the Old Testament are over.
