2007/10/01 by Pascal Koiran, Koiran, Pascal, Sylvain Perifel +1
Computer Science · #Complexity and Algorithms in Graphs #Computational Complexity (cs.CC) #Computational Geometry and Mesh Generation #Data Management and Algorithms #FOS: Computer and information sciences #cs.CC
paper · pdf · doi:10.48550/arxiv.0710.0360
13 pages
arxiv created 2007/10/01 · openalex publication_date 2007/10/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the following question: if a polynomial can be evaluated at rational points by a polynomial-time boolean algorithm, does it have a polynomial-size arithmetic circuit? We argue that this question is certainly difficult. Answering it negatively would indeed imply that the constant-free versions of the algebraic complexity classes VP and VNP defined by Valiant are different. Answering this question positively would imply a transfer theorem from boolean to algebraic complexity. Our proof method relies on Lagrange interpolation and on recent results connecting the (boolean) counting hierarchy to algebraic complexity classes. As a byproduct we obtain two additional results: (i) The constant-free, degree-unbounded version of Valiant's hypothesis that VP and VNP differ implies the degree-bounded version. This result was previously known to hold for fields of positive characteristic only. (ii) If exponential sums of easy to compute polynomials can be computed efficiently, then the same is true of exponential products. We point out an application of this result to the P=NP problem in the Blum-Shub-Smale model of computation over the field of complex numbers.