Diophantine Equations for Polynomial Recursive Sequences
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915692655149056 |
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| author | N, Darsana Rout, Sudhansu Sekhar |
| author_facet | N, Darsana Rout, Sudhansu Sekhar |
| contents | We study the Diophantine equation of type $U_n(x)=V_m(y)$, where $(U_n)_{n\geq 0}$ and $(V_m)_{m\geq 0}$ are polynomial power sums defined over a number field $K$. By applying the finiteness criterion of Bilu and Tichy, we show under appropriate assumptions that equation $U_n(x)=V_m(y)$ has infinitely many solutions with bounded $\mathcal{O}_S$-denominator. We also study decomposable polynomials in third and second order linear recurrence sequences. In particular, we show that if $W_n(x)=g(h(x))$ for a simple third order linear recurrence sequence $(W_n(x))_{n\geq 0}$ of complex polynomials, then deg $g$ is bounded. Furthermore, we show that if $(u_{n_1}+u_{n_2})(x)=g(h(x))$ for a binary recurrence sequence $(u_n(x))_{n\geq 0}$ then deg $g$ is bounded. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_20384 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Diophantine Equations for Polynomial Recursive Sequences N, Darsana Rout, Sudhansu Sekhar Number Theory We study the Diophantine equation of type $U_n(x)=V_m(y)$, where $(U_n)_{n\geq 0}$ and $(V_m)_{m\geq 0}$ are polynomial power sums defined over a number field $K$. By applying the finiteness criterion of Bilu and Tichy, we show under appropriate assumptions that equation $U_n(x)=V_m(y)$ has infinitely many solutions with bounded $\mathcal{O}_S$-denominator. We also study decomposable polynomials in third and second order linear recurrence sequences. In particular, we show that if $W_n(x)=g(h(x))$ for a simple third order linear recurrence sequence $(W_n(x))_{n\geq 0}$ of complex polynomials, then deg $g$ is bounded. Furthermore, we show that if $(u_{n_1}+u_{n_2})(x)=g(h(x))$ for a binary recurrence sequence $(u_n(x))_{n\geq 0}$ then deg $g$ is bounded. |
| title | Diophantine Equations for Polynomial Recursive Sequences |
| topic | Number Theory |
| url | https://arxiv.org/abs/2512.20384 |