Interpolation in Valiant's theory

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 VP = VNP 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...

• Type: Journal
• Year: 2007
• Where: CORR
• Authors: Pascal Koiran, Sylvain Perifel
• Keywords: Algebraic Complexity, Algebraic Complexity Classes, Complexity Classes, CORR 2007, Education