Mathematics Colloquia and Seminars

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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 in G as fixed coefficients, and X in G as the unknown. Certain word equations, such as XAXAX=B, have solutions in terms of radicals:

X = A^{-1/2}(A^{1/2}BA^{1/2})^{1/3}A^{-1/2},

while others such as X^2AX = 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 new combinatorial object P_w in Z[x,y], called the word polynomial, 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. (joint work with Lionel Levine and Darren Rhea).