Four squares from three real polynomials

Abstract

We construct three nonconstant polynomials $a,b,c\in \mathbb{R}[X]$ such that \[ab+1,\qquad ac+1,\qquad bc+1,\qquad abc+1 \] are all squares in $\mathbb{R}[X]$. The entries in the construction are quadratic, so the construction has the smallest possible positive degrees. By composition with nonconstant polynomials, this gives infinitely many examples over $\mathbb{R}[X]$ with all entries nonconstant.

Disclosure

“ted by the Croatian Science Foundation Grant No. IP-2022- 10-5008. A. J. was also supported by the European Union – NextGenerationEU, project number uniri-iz-25-62-ALGEBRA. During the exploratory phase of this research, the author utilized GPT-5.5 Pro to investigate examples and candidate constructions. All mathematical results were performed and validated by the au- thor. The author takes full responsibility for the mathematical correctness of these results. The author would like t”

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Classification

Suggesting mathematical examples or conjectures

Multiplier

6

Verified

2026-08-24T00:34:05.733Z

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