In Hamlet, why wouldn't Claudius provide a state funeral for Polonius?
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.
Claudius fears the outcome of such a funeral. As Hamlet is the guilty party, and being much loved by the multitude, he cannot risk the attention it would bring to Hamlet's supposed madness and conspicuous absence. Even under the guise of obtaining help for Hamlet's madness in England, he would risk calling attention to the murder and the people of Denmark forgiving him considering the sudden death and hasty marriage of Claudius and Gertrude. Also, Denmark is in political turmoil and Claudius holds a tenuous position. He has been approved by vote, and can be ousted just as easily, as is seen when Laertes returns to avenge his father's death with a mob calling for him to be king.