On a conjecture of Murthy (Q1647383)

An old question of \textit{M. P. Murthy} asks whether and ideal $I\subset k[x_1,\dots, x_r]=R$ can be generated by the same number of elements required to generate $I/I^2$ [in: Conf. commut. Algebra, Kingston196--211 (1975; Zbl 0354.14015)]. If we denote by $\mu(M)$, the minimal number of generators of a module $M$, the question asks whether $\mu(I)=\mu(I/I^2)$. The reviewer had given an affirmative answer when $\mu(I/I^2)\geq\dim R/I +2$ [\textit{N. Mohan Kumar}, Invent. Math. 46, 225--236 (1978; Zbl 0395.13009)]. Recently a complete affirmative answer was announced by \textit{J. Fasel} [Ann. Math. (2) 184, No. 1, 315--331 (2016; Zbl 1372.13014)], but the proof had gaps and thus remains incomplete; and the question remained open in general. In the paper under review, the author improves the earlier result of the reviewer in the case where $k$ is the algebraic closure of a finite field and with the weaker hypothesis, $\mu(I/I^2)\geq\dim R/I+1$.