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Keywords:
field; rational function; restricted sum; restricted product
field; rational function; restricted sum; restricted product
Summary:
We study sums and products in a field. Let $F$ be a field with ${\rm ch}(F)\not =2$, where ${\rm {\rm ch} } (F)$ is the characteristic of $F$. For any integer $k\geq 4$, we show that any $x\in F$ can be written as $a_1+\dots +a_k$ with $a_1,\dots ,a_k\in F$ and $a_1\dots a_k=1$, and that for any $\alpha \in F \setminus \{0\}$ we can write every $x\in F$ as $a_1\dots a_k$ with $a_1,\dots ,a_k\in F$ and $a_1+\dots +a_k=\alpha $. We also prove that for any $x\in F$ and $k\in \{2,3,\dots \}$ there are $a_1,\dots ,a_{2k}\in F$ such that $a_1+\dots +a_{2k}=x=a_1\dots a_{2k}$.
References:
[1] Elkies, N. D.: On the areas of rational triangles or how did Euler (and how can we) solve $xyz(x+y+z)=a$?. Available at \let \relax\brokenlink{ http://www.math.harvard.edu/ elkies/{euler_14t.pdf}} (2014), 50 pages.
[2] Klyachko, A. A., Mazhuga, A. M., Ponfilenko, A. N.: Balanced factorisations in some algebras. Available at https://arxiv.org/abs/1607.01957 (2016), 4 pages.
[3] Klyachko, A. A., Vassilyev, A. N.: Balanced factorisations. Available at https://arxiv.org/abs/1506.01571 (2015), 8 pages. MR 3593641
[4] Zypen, D. van der: Question on a generalisation of a theorem by Euler. Question 302933 at MathOverflow, June 16, 2018. Available at http://mathoverflow.net/questions/302933
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