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 > 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
 A: The problem as stated is not well formulated; there are different ways it could be given a precise mathematical meaning, but then the main content of the question becomes the way in which it is made precise.  
One thing missing is a specification of the probability measure, or a class of probability measures, for "random integer" and "random polynomial".  Also implicitly missing, although not explicitly mentioned, is either a mathematical or a practical model of computation.
If you choose any particular probability distribution on polynomials with integer coefficients, then for any $\epsilon$ there are particular integers $m$ so that the probability of two of the polynomials agreeing at $m$ is less than $\epsilon$.  As an example, if you consider any set of polynomials whose coefficients are chosen with any probability measure on integers between -1,000,000 to +1,000,000, then the value for $n = 2,000,001$ is definitive.  This may not help you with your actual problem, because computing the value when $n = 2,000,000$ may involve computation with integers of greater than than machine precision, depending on the degree and the machine. If it's a polynomial of degree 10, it's probably better to compute its value on all integers between -5 and 5, rather than one integer of size 2000000; but there are many possible strategies for computation, and this gets into a different set of issues.
For a probability measure that does not have finite support, you can't usually determine identity of the polynomial with absolute certainty from value at a particular integer, but it can still be made as nearly certain as you like.
Similarly, the value on any real number chosen from a distribution with no atoms, or the value on a single known-to-be transcendental real, is definitive --- but computation up to machine precision might or might not tell you equality.
A: Sorry can't add comments yet so I have to post this way. To comment on David's last post,
one doesn't even have to go to transcendental numbers.
If $P-Q$ is not zero, it only has finitelly many roots, thus the probability for a random chosen real number to be a root of $P-Q$ is zero. 
A: Use probabilistic identity testing. This is noteworthy for being one of the few problems known to be in BPP but not in P.
A: Verifying if two black-box polynomials are identical is easy if you know that they have the same degree $n$; simply compute values at $n+1$ points, and if those $n+1$ values agree, they're identical.
A: If the coefficients are non-negative then you can always do it with at most two integer evaluations.
That is, $P$ and $Q$ are equal if and only if


*

*$P(1)=Q(1)$, and

*$P(P(1)+1)=Q(Q(1)+1)$.


Update. If we allow for negative coefficients then this won't work.  However, if in addition we are told that all coefficients $c$ satisfy $|c| \leq b$, then I believe we can do it with one integer evaluation.  Namely, choose $n$ satisfying $n \geq 2b+1$.  Then I think $P$ and $Q$ are equal if and only if


*

*$P(n)=Q(n)$.  


See my comments below for an explanation.
A: If $t$ is chosen randomly in the reals, then with probability one $t$ is transcendental, so $P(t) = Q(t)$ iff $P = Q$. (This is just a generalization of the answer from Gerry Myerson)
A: If $P$ and $Q$ are polynomials with integer coefficients, and $P(\pi)=Q(\pi)$, then $P$ and $Q$ are guaranteed identical. 
