Desingularization of binomial varieties in arbitrary characteristic. Part I. A new resolution function and their properties

摘要

This paper is devoted to give all the technical constructions and definitionsthat will lead to the construction of an algorithm of resolution ofsingularities for binomial ideals. We construct a resolution function that will provide a resolution ofsingularities for binomial ideals, over a field of arbitrary characteristic.For us, a binomial ideal means an ideal generated by binomial equations withoutany restriction, including monomials and $p$-th powers, where $p$ is thecharacteristic of the base field. This resolution function is based in a modified order function, called$E$-order. The $E$-order of a binomial ideal is the order of the ideal along anormal crossing divisor $E$. The resolution function allows us to construct an algorithm of$E$-\emph{resolution of binomial basic objects}, that will be a subroutine ofthe main resolution algorithm.