Not being around when something happened is not a valid excuse. If your TV was physically damaged while you owned it, it's your problem to deal with.
Your only hope is to be able to explain that the physical gouge is not the cause of the TV's picture problem. You don't have to prove it without a doubt, as long as there is a reasonable possibility that the TV problem is not the result of physical trauma that should be enough. I would try to show this in one of two ways: 1) go look around in stores and try to find another set of the same model of TV as yours with the same or similar problem, 2) if they give you problems, ask the repairman or warranty company to send you a technical explanation of what they think the problem is, and ask THEM to explain their theory on how the display-related problem you have could be caused by physical trauma to the TV.
Unfortunately I think that physical trauma on the TV looks real bad on paper and in court, so I think your only solid avenue lies in what I wrote in 1) above.
Also, find your warranty papers and read them carefully. If they say something like "Warranty service does not apply to sets that have been damaged by the user" then you are probably in big trouble regardless. If they say something more like "Warranty service does not apply to issues that are the result of user misuse or damage" then you still have hope if you can argue your case successfully.
Also, be aware that what your warranty papers say is not the end-all of a company's liability to ensure that your set functions properly. That is why there is all this legal fine-print gibberish at the bottom of any warranty card. So that can be a possibility for you as well, but generally only useful if you want to take them to court over your TV, which probably isn't going to happen.
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