Computer Science – Computational Complexity
Scientific paper
2009-07-31
Computer Science
Computational Complexity
14 pages
Scientific paper
A polynomial identity testing algorithm must determine whether a given input polynomial is identically equal to 0. We give a deterministic black-box identity testing algorithm for univariate polynomials of the form $\sum_{j=0}^t c_j X^{\alpha_j} (a + b X)^{\beta_j}$. From our algorithm we derive an exponential lower bound for representations of polynomials such as $\prod_{i=1}^{2^n} (X^i-1)$ under this form. It has been conjectured that these polynomials are hard to compute by general arithmetic circuits. Our result shows that the "hardness from derandomization" approach to lower bounds is feasible for a restricted class of arithmetic circuits. The proof is based on techniques from algebraic number theory, and more precisely on properties of the height function of algebraic numbers.
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