While out walking on a sunny, rainy day you meet a leprechaun (who also happens to love math). He says that he will give you his pot of gold if you can give the most efficient solution to the following question:

"I have a favorite polynomial F. Since I hate those pesky negative numbers, F has only non-negative coefficients. You may ask me as many questions about F as you would like. The only questions that I am willing to answer, however, are those of the form "what is F(a)?" for specific values of a. What are the coefficients of my favorite polynomial?"

What is the minimum number of questions that you must ask the leprechaun in order to determine F, and what are the questions that you plan to ask?

"I have a favorite polynomial F. Since I hate those pesky negative numbers, F has only non-negative coefficients. You may ask me as many questions about F as you would like. The only questions that I am willing to answer, however, are those of the form "what is F(a)?" for specific values of a. What are the coefficients of my favorite polynomial?"

What is the minimum number of questions that you must ask the leprechaun in order to determine F, and what are the questions that you plan to ask?