Abstract
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 by-product, we obtain two additional results:
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(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.
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(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.
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Koiran, P., Perifel, S. Interpolation in Valiant’s Theory. comput. complex. 20, 1–20 (2011). https://doi.org/10.1007/s00037-011-0002-8
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DOI: https://doi.org/10.1007/s00037-011-0002-8
Keywords
- Computational complexity
- algebraic complexity
- Valiant’s model
- polynomials
- interpolation
- Blum–Shub–Smale model