Linear estimates for solutions of quadratic equations in free groups
  We prove that in a free group the length of the value of each variable in a minimal solution of a standard quadratic equation is bounded by $2s$ for orientable equation and by $12s^4$ for non-orientable equation, where $s$ is the sum of the lengths of the coefficients 
Olga Kharlampovich, Alina Vdovina
math.GR