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Interpolation in Valiant’s theory
, 2007
"... Abstract. We investigate the following question: if a polynomial can be evaluated at rational points by a polynomialtime boolean algorithm, does it have a polynomialsize arithmetic circuit? We argue that this question is certainly difficult. Answering it negatively would indeed imply that the cons ..."
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Abstract. We investigate the following question: if a polynomial can be evaluated at rational points by a polynomialtime boolean algorithm, does it have a polynomialsize arithmetic circuit? We argue that this question is certainly difficult. Answering it negatively would indeed imply that the constantfree 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 constantfree, degreeunbounded version of Valiant’s hypothesis VP 6 = VNP implies the degreebounded 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 BlumShubSmale model of computation over the field of complex numbers.
Finding a vector orthogonal to roughly half a collection of vectors. Available from http://perso.enslyon.fr/pascal.koiran/publications.html. Accepted for publication in
 Journal of Complexity
, 2006
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There Are No Sparse NP_WHard Sets
"... . In this paper we prove that, in the context of weak machines over IR, there are no sparse NPhard sets. 1 Introduction In [1977] Berman and Hartmanis conjectured that all NPcomplete sets are polynomially isomorphic. That is, that for all NPcomplete sets A and B, there exists a bijection &apos ..."
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. In this paper we prove that, in the context of weak machines over IR, there are no sparse NPhard sets. 1 Introduction In [1977] Berman and Hartmanis conjectured that all NPcomplete sets are polynomially isomorphic. That is, that for all NPcomplete sets A and B, there exists a bijection ' : \Sigma ! \Sigma such that x 2 A if and only if '(x) 2 B. In addition both ' and its inverse are computable in polynomial time. Here \Sigma denotes the set f0; 1g and \Sigma the set of all finite sequences of elements in \Sigma. Should this conjecture be proved, we would have as a consequence that no "small" NPcomplete set exists in a precise sense of the word "small". A set S ` \Sigma is said to be sparse when there is a polynomial p such that for all n 2 IN the subset S n of all elements in S having size n has cardinality at most p(n). If the BermanHartmanis conjecture is true, then there are no sparse NPcomplete sets. Partially supported by CERG grant 9040393. 1 In 1982 Mahaney (...