Linear estimates for solutions of quadratic equations in free groups. (Q2882397)

A quadratic equation \(E\) with variables \(\{x_i,y_i,z_i\}\) and non-trivial coefficients \(\{C_j,C\}\) over a free group \(F(A)\) is said to be in `standard form' if its coefficients are expressed as freely and cyclically reduced words in \(A^{\pm 1}\) and \(E\) has either the form \((\prod_{i=1}^g[x_i,y_i])(\prod_{j=1}^{m-1}z_j^{-1}C_jz_j)C=1\) or \((\prod_{i=1}^g[x_i,y_i])C=1\), in which case it is called `orientable' or it has the form \((\prod_{i=1}^gx_i^2)(\prod_{j=1}^{m-1}z_j^{-1}C_jz_j)C=1\) or \((\prod_{i=1}^gx_i^2)C=1\), in which case it is called `non-orientable'. -- The length of a quadratic equation \(E\) is defined as \(\text{length}(E)=\sum_j|C_j|+C+2(\)number of variables).NEWLINENEWLINE It is well-known fact that an arbitrary quadratic equation over a free group can be brought to standard form in time polynomial in its length.NEWLINENEWLINE Let \(E\) be a standard consistent quadratic equation in a free group \(F\). Let \(s\) be the sum of the lengths of the coefficients in \(E\). Then there is a solution such that every value of a variable has length \(\leq 2s\) in the orientable case, and \(\leq 12s^4\) in the non-orientable case.