2 corrected the condition k >deg(P+Q) to k > max(deg(P),deg(Q))

Let t be a positive real number. Let P(x) and Q(x) be two random polynomials with integer coefficients. If P(t) = Q(t), then what is the probability that P(x) is not identical to Q(x)?

Will it make a difference if I restrict t to be an integer?

Suppose I had a set T ={t0,t1,…tk}, can we answer a similar question --- If P(ti) = Q(ti) for all ti in the set T, what is the probability that P is identical to Q? If k > deg(P +Q), max(deg(P), deg(Q)), the probability is 1. But can we say something about how many points we need to check before we can be fairly certain that the polynomials are identical?

Thanks

1

# How to tell if two random polynomials are identical

Let t be a positive real number. Let P(x) and Q(x) be two random polynomials with integer coefficients. If P(t) = Q(t), then what is the probability that P(x) is not identical to Q(x)?

Will it make a difference if I restrict t to be an integer?

Suppose I had a set T ={t0,t1,…tk}, can we answer a similar question --- If P(ti) = Q(ti) for all ti in the set T, what is the probability that P is identical to Q? If k > deg(P +Q), the probability is 1. But can we say something about how many points we need to check before we can be fairly certain that the polynomials are identical?

Thanks