Geodesic
A polynomial bound on solutions of quadratic equations in free groups
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algorithmic aspects of algebra and logic, Tome 274 (2011), pp. 148-190

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We provide polynomial upper bounds on the size of a shortest solution for quadratic equations in a free group. A similar bound is given for parametric solutions in the description of solution sets of quadratic equations in a free group. 
@article{TM_2011_274_a9,
     author = {Igor G. Lysenok and Alexei G. Myasnikov},
     title = {A polynomial bound on solutions of quadratic equations in free groups},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {148--190},
     publisher = {mathdoc},
     volume = {274},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2011_274_a9/}
}
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Igor G. Lysenok; Alexei G. Myasnikov. A polynomial bound on solutions of quadratic equations in free groups. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algorithmic aspects of algebra and logic, Tome 274 (2011), pp. 148-190. http://geodesic.mathdoc.fr/item/TM_2011_274_a9/