PIT and Reconstruction for Sums of Powers
We study deterministic black-box polynomial identity testing (PIT) and reconstruction for polynomials of the form \[ f(x)=\sum_{i=1}^r α_i f_i(x)^d, \] where each $f_i$ belongs to some underlying polynomial class $\mathcal{C}$. This representation generalizes Waring decomposition, several natural small-depth circuit models, and polynomial decompositions arising in moment-based learning problems. Our main results are three lifting theorems showing that, when the outer power is sufficiently large, PIT and reconstruction for the inner class can be lifted to sums of powers. When the power degree is small, we can perform these tasks directly in some of the cases using closure properties of the underlying models. In particular we obtain the following results: \begin{itemize} \item When the inner polynomials have bounded-individual-degrees and are sparse, we obtain deterministic reconstruction for every d: the original power-sum decomposition is recovered at high degree, while at low degree the polynomial is reconstructed nonproperly by sparse interpolation. \item For sums of powers of same-order and any-order read-once oblivious algebraic branching programs, the same proper/nonproper dichotomy gives deterministic reconstruction for every powering degree. \item Explicit quasipolynomial-size hitting sets for every powering degree for several depth-5 and depth-6 circuit families, as well for the models mentioned above. Our results for high powering degree rely on the polynomial ABC theorem, which to the best of our knowledge is the first application of this theorem in the context of PIT.