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On Goldbach’s conjecture for integer polynomials. (English) Zbl 1192.11074

Introduction: We give a short proof of the fact that every monic polynomial \(f(x)\) in \(\mathbb Z[x]\) can be written in the form \(f(x)=g(x)+h(x)\), where \(g(x)\) and \( h(x)\) are both irreducible monic polynomials in \(\mathbb Z[x]\). For the number \(\mathfrak R(f;y)\) of representations of \(f(x)\) as a sum of two irreducible monics \(g(x)\) and \(h(x)\) in which all coefficients of \(g(x)\) and \(h(x)\) are bounded in absolute value by \(y\), we prove that \[ y^ d\ll_ f\mathfrak R(f;y)\ll_ fy^ d \] as \(y\to\infty\), where the notation ‘\(\ll_ f\)’ means that the constant could depend on (the degree and coefficients of) the polynomial \(f(x)\).

MSC:

11P99 Additive number theory; partitions
13B25 Polynomials over commutative rings
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