Irreducibility and locus of complex roots of polynomials related to Fermat's Last Theorem
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arXiv
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| Format: | Preprint |
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2025
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| author | Karapetyan, Hayk Hambardzumyan, Ruben |
| author_facet | Karapetyan, Hayk Hambardzumyan, Ruben |
| contents | We study the polynomials $x^n + (1-x)^n + a^n, a \in\mathbb{Q}$, whose rational roots would yield counterexamples to Fermat's Last Theorem. We investigate their factorization over $\mathbb{Q}$. In the case $a \notin \{0, \pm 1\}$, we ask whether they are irreducible over $\mathbb{Q}$, prove the irreducibility for several infinite families, and investigate the location of the roots of these polynomials on the complex plane. For $a=\pm1$, the factorization of $K_{a,n}$ is intimately related to that of the Cauchy--Mirimanoff polynomials $E_n$ and the polynomials $T_n$ and $S_n$ introduced by P. Nanninga. After removing the trivial factors $x$, $x-1$, and $x^2-x+1$, the remaining components agree (up to change of variable) with $E_n$, $S_n$, or $T_n$. We prove several new irreducibility results for these factors. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_00020 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Irreducibility and locus of complex roots of polynomials related to Fermat's Last Theorem Karapetyan, Hayk Hambardzumyan, Ruben Number Theory 11R09, 12D10 We study the polynomials $x^n + (1-x)^n + a^n, a \in\mathbb{Q}$, whose rational roots would yield counterexamples to Fermat's Last Theorem. We investigate their factorization over $\mathbb{Q}$. In the case $a \notin \{0, \pm 1\}$, we ask whether they are irreducible over $\mathbb{Q}$, prove the irreducibility for several infinite families, and investigate the location of the roots of these polynomials on the complex plane. For $a=\pm1$, the factorization of $K_{a,n}$ is intimately related to that of the Cauchy--Mirimanoff polynomials $E_n$ and the polynomials $T_n$ and $S_n$ introduced by P. Nanninga. After removing the trivial factors $x$, $x-1$, and $x^2-x+1$, the remaining components agree (up to change of variable) with $E_n$, $S_n$, or $T_n$. We prove several new irreducibility results for these factors. |
| title | Irreducibility and locus of complex roots of polynomials related to Fermat's Last Theorem |
| topic | Number Theory 11R09, 12D10 |
| url | https://arxiv.org/abs/2510.00020 |