This is a self-contained purely algebraic treatment of desingularization of fields of fractions $mathbf{L}:=Q(mathbf{A})$ of $d$-dimensional domains of the form [mathbf{A}:=bar{mathbf{F}}[underline{x}]/langle b(underline{x})rangle] with a purely algebraic objective of uniquely describing $d$-dimensional valuations in terms of $d$ explicit (independent) local parameters and $1$ (dependent) local unit, for arbitrary dimension $d$ and arbitrary characteristic $p$. The desingularization will be given as a rooted tree with nodes labelled by domains $mathbf{A}_k$ (all with field of fractions $Q(mathbf{A}_k)=mathbf{L}$), sets $EQ_k$ and $INEQ_k$ of equality constraints and inequality constraints, and birational change-of-variables maps on $mathbf{L}$. The approach is based on d-dimensional discrete valuations and local monomial orderings to emphasize formal Laurent series expansions in $d$ independent variables. It is non-standard in its notation and perspective.
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