Algebraic analysis uncovers conditions for cubic-quadratic compositions to form sums of two polynomial squares, highlighting counterexamples to an integer-coefficient conjecture.
In this paper, for any given monic cubic irreducible polynomial f ∈ ℚ[x], we describe the conditions on rational numbers a ≠ 0 and b, c under which there exist g, h ∈ ℚ[x] such that f (ax2 + bx + c) = g(x)2 + h(x)2 . In particular, our results imply that for any such f there is a suitable triplet (a, b, c), where a ∈ N, b = 0 and c ∈ ℚ, with such a representation. However, this is not always the case with integers a ≠ 0 and b, c. In this direction, we show that, unlike it has been previously conjectured, there exist cubic monic irreducible polynomials in ℤ[x], say, f (x) = x3 − 2 or f (x) = x3 + 3, such that for any integers a ≠ 0 and b, c the polynomial f (ax2 + bx + c) is not expressible as a sum of squares of two polynomials in ℚ[x].
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Artūras Dubickas (2026) studied this question.