Sparse Perfect Powers

We examine the problem of detecting when a given sparse polynomial f is equal to hr for some other polynomial h and integer r ≥ 2. In this case we say f is a perfect power, and h is its r th root. We give randomized algorithms to detect when f is a perfect power, by repeatedly evaluating the polynomial in specially-chosen finite fields and checking whether the evaluations are themselves perfect powers. These detection algorithms are randomized of the Monte Carlo type, meaning that they are always fast but may return an incorrect answer (with small probability). In fact, the only kind of wrong answer ever returned will be a false positive, so we have shown that the perfect power detection problem is in the complexity class coRP. As a by-product, these decision problems also compute an integer r > 2 such that f is an r th perfect power, if such an r exists.