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On one-dimensional formal group laws in characteristic zero. (English) Zbl 1315.14060

Summary: Let \(\mathbb K\) be a field of characteristic zero or, more generally, a \(\mathbb Q\)-algebra. A formal power series \(F(x,y)=x+y+\sum_{i,j\geq 1}a_{i,j}x^iy^j\in \mathbb K{[\![x, y]\!]}\) is called a one-dimensional formal group law if \(F(F(x,y),z)=F(x,F(y,z))\). Using some elementary methods, we prove that for every one-dimensional formal group law \(F(x,y)\) there exists a formal power series \(f(x)=x+\sum_{n\geq 2}f_nx^n\in\mathbb K{[\![} x, y{]\!]}\) so that \(F(x,y)=f^{-1}(f(x)+f(y))\).

MSC:

14L05 Formal groups, \(p\)-divisible groups
13F25 Formal power series rings
16Z05 Computational aspects of associative rings (general theory)
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References:

[1] Fripertinger H., Reich L., Schwaiger J., Tomaschek J.: Associative formal power series in two indeterminates. Semigroup Forum 88(3), 529-540 (2014) · Zbl 1317.39026 · doi:10.1007/s00233-013-9533-4
[2] Hazewinkel M.: Formal Groups and Applications. Academic Press, New York (1978) · Zbl 0454.14020
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