On Goldbach's Conjecture for Integer Polynomials

Linear Forms in Monic Integer Polynomials

“…have been given in [1,3,5,7,[11][12][13]. In particular, for a monic polynomial f in the ring Z[x], there is an asymptotical formula for the number of representations of f by the sum of several irreducible monic polynomials, each of height at most T (see [2,6,9]).…”

“…To give a more complete treatment of the subject, we shall investigate the representations of a monic polynomial f ∈ Z [2,6,9]. It is shown that then there are asymptotically c k,d T (k−1)(d −1) of such representations as T → ∞, where d = deg f 2 and c k,d > 0 is a constant independent of T. Obviously, for some collections u 1 , .…”

An asymptotic formula for Goldbach’s conjecture with monic polynomials in \(\mathbb {Z}[\theta ][x]\)

“…In a recent note, Saidak [15], improving on a result of Hayes, gave Chebyshevtype estimates for the number R(y) = R f (y) of representations of the monic polynomial f (x) ∈ Z[x] of degree d > 1 as a sum of two irreducible monics g(x) and h(x) in Z[x], with the coefficients of g(x) and h(x) bounded in absolute value by y.…”

Irreducibility of a Polynomial Shifted by a Power of Another Polynomial

“…e same happens with Goldbach-type problems in polynomials with integer coefficients when much more is known compared to classical Goldbach problems for integers. ere is a considerable literature concerning this, see, for instance, [13][14][15][16][17][18][19][20][21][22].…”