Arturas Dubickas: Polynomials expressible by sums of monic integer irreducible polynomials, p.65-81

Abstract:

We prove an asymptotical formula for the number of representations of a given monic polynomial ƒ ∈ Ζ[x] by the sum of $k \geq 2$ monic irreducible polynomials in Ζ[x] whose heights are bounded by $T$. The main term turns out to be $c_{d,k}T^{(d-1)(k-1)}$, where $d=\deg f$ and $c_{d,k}$ is some positive rational number. The binary case $k=2$ was first considered by Hayes in 1965 as a version of a binary Goldbach problem for polynomials. In this case, we improve the error term in a recent asymptotical formula (due to Kozek) and show that our error term is best possible for each $d \geq 2$.

Key Words: Irreducible polynomial, height, Goldbach's problem for polynomials..

2000 Mathematics Subject Classification: Primary: 11R09;
Secondary: 11C08.

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