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

This paper is devoted to give all the technical constructions and definitions that will lead to the construction of an algorithm of resolution of singularities for binomial ideals. We construct a resolution function that will provide a resolution of singularities for binomial ideals, over a field of arbitrary characteristic. For us, a binomial ideal means an ideal generated by binomial equations without any restriction, including monomials and $p$-th powers, where $p$ is the characteristic 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 a normal 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 of the main resolution algorithm.