Determining a polynomial

[Rafael]

Bob chooses a polynomial (any polynomial) with positive integer coefficients. Alice wants to find out which polynomial Bob has chosen. To do this, she can ask Bob to evaluate the polynomial at any point any number of times. What is the minimum number of questions that she needs to ask?

Note: Alice does not know the degree of the polynomial.

What are the implications (if any) for Shamir's Secret Sharing algorithm?