Irreducibility of lacunary polynomials with 0,1 coefficients
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917801929736192 |
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| author | Kalogirou, Alexandros |
| author_facet | Kalogirou, Alexandros |
| contents | We show that $0,1$-polynomials of high degree and few terms are irreducible with high probability. Formally, let $k\in\mathbb{N}$ and $F(x)=1+\sum_{i=1}^kx^{n_i}$, where $ 0<n_1<\cdots<n_k\leq N. $ Then we show that $\lim_{k\rightarrow\infty}\limsup_{N\rightarrow\infty}\mathbb{P}(\text{$F(x)$ is reducible})=0.$ The probability in this context is derived from the uniform count of polynomials $F(x)$ of the above form. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_10035 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Irreducibility of lacunary polynomials with 0,1 coefficients Kalogirou, Alexandros Number Theory We show that $0,1$-polynomials of high degree and few terms are irreducible with high probability. Formally, let $k\in\mathbb{N}$ and $F(x)=1+\sum_{i=1}^kx^{n_i}$, where $ 0<n_1<\cdots<n_k\leq N. $ Then we show that $\lim_{k\rightarrow\infty}\limsup_{N\rightarrow\infty}\mathbb{P}(\text{$F(x)$ is reducible})=0.$ The probability in this context is derived from the uniform count of polynomials $F(x)$ of the above form. |
| title | Irreducibility of lacunary polynomials with 0,1 coefficients |
| topic | Number Theory |
| url | https://arxiv.org/abs/2410.10035 |