Defect and local uniformization (Q2331760)

The goal of the authors is to describe a detail procedure for the reduction of the multiplicity of a hypersurface singularity over an algebraically closed ground field of positive characteristic along a valuation under additional assumptions. One of them is a requirement that there exists a finite linear projection of the hypersurface singularity which is defectless. The authors underline that their main result (Theorem 7.1) follows also from the paper [\textit{J.-C. San Saturnino}, J. Algebra 481, 91--119 (2017; Zbl 1370.13005)]. It should be remarked that the paper under review contains an interesting brief historical background concerning the development of the classical theories of local uniformization and resolution of singularities originated by O. Zariski, a series of fruitful ideas, comments and useful references related to these topics.