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Parametric equations in free groups

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Abstract

We introduce the notion of a parametric equation in a free group; this is an equation containing natural parameters as exponents and a system of linear Diophantine equations relating these exponents. For these equations, we introduce elementary transformations that are necessary for the description of general solutions of ordinary equations in a free group. We prove that it is possible to linearize any relation among parameters that appears in the course of transformations of the given equation.

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References

  1. R. C. Lyndon, “Equations in free groups,”Trans. Amer. Math. Soc.,96, 445–457 (1960).

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  2. Yu. M. Ozhigov, “Equations with two unknowns in a free group,”Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.],268, No. 4, 808–814 (1983).

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  3. A. A. Razborov,On Systems of Equations in a Free Group [in Russian], Thesis, Steklov Mathematical Institute, Moscow (1987).

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Authors and Affiliations

  1. Moscow State Technical University “STANKIN”, USSR

    Yu. I. Ozhigov

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  1. Yu. I. Ozhigov
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Translated fromMatematicheskie Zametki, Vol. 58, No. 4, pp. 569–581, October, 1995.

The author is grateful to the corresponding member of the Russian Academy of Sciences S. I. Adyan for attention to the paper and Doctor A. A. Razborov for valuable critical comments.

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Ozhigov, Y.I. Parametric equations in free groups. Math Notes 58, 1074–1083 (1995). https://doi.org/10.1007/BF02305096

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