On the finiteness of solutions for polynomial-factorial Diophantine equations
On the finiteness of solutions for polynomial-factorial Diophantine equations
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DOI：
10.1515/forum-2020-0138
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发表时间：
2019-03
影响因子：
0.8
通讯作者：
Wataru Takeda
中科院分区：
文献类型：
--
作者：
Wataru Takeda
Abstract We study the Diophantine equations obtained by equating a polynomial and the factorial function, and prove the finiteness of integer solutions under certain conditions. For example, we show that there exist only finitely many l such that l!{l!} is represented by NA(x){N_{A}(x)}, where NA{N_{A}} is a norm form constructed from the field norm of a field extension K/𝐐{K/\mathbf{Q}}. We also deal with the equation NA(x)=l!S{N_{A}(x)=l!_{S}}, where l!S{l!_{S}} is the Bhargava factorial. In this paper, we also show that the Oesterlé–Masser conjecture implies that for any infinite subset S of 𝐙{\mathbf{Z}} and for any polynomial P(x)∈𝐙[x]{P(x)\in\mathbf{Z}[x]} of degree 2 or more the equation P(x)=l!S{P(x)=l!_{S}} has only finitely many solutions (x,l){(x,l)}. For some special infinite subsets S of 𝐙{\mathbf{Z}}, we can show the finiteness of solutions for the equation P(x)=l!S{P(x)=l!_{S}} unconditionally.
Abstract We study the Diophantine equations obtained by equating a polynomial and the factorial function, and prove the finiteness of integer solutions under certain conditions. For example, we show that there exist only finitely many l such that l!{l!} is represented by NA(x){N_{A}(x)}, where NA{N_{A}} is a norm form constructed from the field norm of a field extension K/𝐐{K/\mathbf{Q}}. We also deal with the equation NA(x)=l!S{N_{A}(x)=l!_{S}}, where l!S{l!_{S}} is the Bhargava factorial. In this paper, we also show that the Oesterlé–Masser conjecture implies that for any infinite subset S of 𝐙{\mathbf{Z}} and for any polynomial P(x)∈𝐙[x]{P(x)\in\mathbf{Z}[x]} of degree 2 or more the equation P(x)=l!S{P(x)=l!_{S}} has only finitely many solutions (x,l){(x,l)}. For some special infinite subsets S of 𝐙{\mathbf{Z}}, we can show the finiteness of solutions for the equation P(x)=l!S{P(x)=l!_{S}} unconditionally.