On Polynomial Representations of Boolean Functions Related to Some Number Theoretic Problems
نویسندگان
چکیده
We say a polynomial P over ZZM strongly M -represents a Boolean function F if F (x) ≡ P (x) (mod M) for all x ∈ {0, 1} . Similarly, P one-sidedly M -represents F if F (x) = 0 ⇐⇒ P (x) ≡ 0 (mod M) for all x ∈ {0, 1} . Lower bounds are obtained on the degree and the number of monomials of polynomials over ZZM , which strongly or one-sidedly M -represent the Boolean function deciding if a given nbit integer is square-free. Similar lower bounds are also obtained for polynomials over the reals which provide a threshold representation of the above Boolean function.
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