[throwaway125]

You are not required to make your application public as it is not considered a derivative of the linux kernel. If you distribute the device you are required to provide your customers with the source code to the running linux kernel.

The exact details depend on multiple factors (are you linking to libraries? Are they considered system libraries?) and if you are actually thinking of doing this you should invest some time in investigating this, and probably in consulting a lawyer.

Note that this is all based on my layman's understanding of the law and the license, and I'm definitely not a lawyer.