A state funeral would draw too much attention to Polonius' death and risk opening up questions about the circumstances surrounding his demise. One of the first questions might be, What was Polonius doing in the queen's bedroom? All kinds of rumors could have been raised. The truth was also a little awkward. He was there to spy on the conversation between Gertrude and her son. Why would he need to do that? What might Gertrude and/or Hamlet be hiding? And, if Gertrude was hiding something, would that mean that Claudius also had something to hide. In addition, the heir to the throne was the one who killed Claudius when he discovered Polonius in his mother's bedroom. That could have lead to all kinds of rumors and innuendo. So, Polonius was buried quickly and without honor, much to his children's dismay.