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Explicit bounds for primes in residue classes
 Math. Comp
, 1996
"... Abstract. Let E/K be an abelian extension of number fields, with E ̸ = Q. Let ∆ and n denote the absolute discriminant and degree of E. Letσdenote an element of the Galois group of E/K. Weprovethefollowingtheorems, assuming the Extended Riemann Hypothesis: () (1) There is a degree1 prime p of K su ..."
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Abstract. Let E/K be an abelian extension of number fields, with E ̸ = Q. Let ∆ and n denote the absolute discriminant and degree of E. Letσdenote an element of the Galois group of E/K. Weprovethefollowingtheorems, assuming the Extended Riemann Hypothesis: () (1) There is a degree1 prime p of K such that p = σ, satis
Interpolating Between Quantum and Classical Complexity Classes
, 2008
"... We reveal a natural algebraic problem whose complexity appears to interpolate between the wellknown complexity classes BQP and NP: ⋆ Decide whether a univariate polynomial with exactly m monomial terms has a padic rational root. In particular, we show that while (⋆) is doable in quantum randomized ..."
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We reveal a natural algebraic problem whose complexity appears to interpolate between the wellknown complexity classes BQP and NP: ⋆ Decide whether a univariate polynomial with exactly m monomial terms has a padic rational root. In particular, we show that while (⋆) is doable in quantum randomized polynomial time when m = 2 (and no classical randomized polynomial time algorithm is known), (⋆) is nearly NPhard for general m: Under a plausible hypothesis involving primes in arithmetic progression (implied by the Generalized Riemann Hypothesis for certain cyclotomic fields), a randomized polynomial time algorithm for (⋆) would imply the widely disbelieved inclusion NP⊆BPP. This type of quantum/classical interpolation phenomenon appears to new. 1 Introduction and Main Results Thanks to quantum computation, we now have exponential speedups for important practical problems such as Integer Factoring and Discrete Logarithm [Sho97]. However, a fundamental