Spherical quadratic equations in free metabelian groups. (Q2790248)

For each positive integer \(m\), let \(C_m\) be the class of spherical quadratic equations in a group \(G\) of the form \(z_1g_1z_1^{-1}\cdots z_mg_mz_m^{-1}=1\), where \(g_1,\ldots,g_m\) are elements of \(G\) (coefficients), and \(z_1,\ldots,z_m\) are unknowns. Let \(C\) denote \(\bigcup_{m=1}^\infty\). In this paper, the authors consider spherical quadratic equations in the free metabelian group \(M_n\) of rank \(n\geq 2\). They prove that: for a fixed \(m\), there exists a polynomial time algorithm solving the Diophantine problem for equations in \(C_m\); the Diophantine problem for \(C\) is \textbf{NP}-complete. Recall that the Diophantine problem for equations of the form \(w(z_1,\ldots,z_m)=g\) in \(M_n\), where \(w(z_1,\ldots,z_m)\) is coefficient-free and \(g\in G\), is undecidable for \(m\geq 2\) [\textit{V. A. Roman'kov}, Sib. Mat. Zh. 20, 671-673 (1979; Zbl 0419.20030); translation in Sib. Math. J. 20, 469-471 (1980)].