How Many Even Coefficients?

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Let $$f(x)$$ be a $$2014$$ degree polynomial whose coefficients are all even integers. Suppose $$f(x)$$ can be written as $$f(x)= g(x) h(x),$$ where $$g(x)$$ and $$h(x)$$ again are non-constant polynomials with integer coefficients, both having degree $$\dfrac{2014}{2}$$. Let $$N_g$$ denote the number of even coefficients of $$g(x),$$ and let $$N_h$$ denote the number of even coefficients of $$h(x).$$ As $$f(x)$$ ranges over all such irreducible $$2014$$ degree polynomials in $$\mathbb{Z}[x],$$ find the minimum possible value of $$N_g + N_h-1.$$

Details and assumptions

• As an explicit example, the polynomial $$x^2+2x+1$$ has $$1$$ even coefficient (coefficient of $$x$$).

• Note that in a polynomial with degree $$n,$$ the coefficients of $$x^k$$ for all $$k>n$$ are zero. These coefficients aren't counted in the count of even coefficients.

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