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Word equations in a uniquely divisible group

Abstract : We study equations in groups $G$ with unique $m$-th roots for each positive integer $m$. A word equation in two letters is an expression of the form$w(X,A) = B$, where $w$ is a finite word in the alphabet ${X,A}$. We think of $A,B ∈G$ as fixed coefficients, and $X ∈G$ as the unknown. Certain word equations, such as $XAXAX=B$, have solutions in terms of radicals: $X = A^-1/2(A^1/2BA^1/2)^1/3A^-1/2$, while others such as $X^2 A X = B$ do not. We obtain the first known infinite families of word equations not solvable by radicals, and conjecture a complete classification. To a word w we associate a polynomial $P_w ∈ℤ[x,y]$ in two commuting variables, which factors whenever $w$ is a composition of smaller words. We prove that if $P_w(x^2,y^2)$ has an absolutely irreducible factor in $ℤ[x,y]$, then the equation $w(X,A)=B$ is not solvable in terms of radicals.
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Christopher J. Hillar, Lionel Levine, Darren Rhea. Word equations in a uniquely divisible group. 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 2010, San Francisco, United States. pp.749-760, ⟨10.46298/dmtcs.2807⟩. ⟨hal-01186233⟩

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