For a field , let be an affine variety over . Denote by the function field of , containing all rational functions . My question is, if a rational function has a pole at , is there an expression where are regular functions, and , ?
When , this is clear, since if we have where , we can simply reduce the expression of and and eliminate the factor until we get such that , . But when are multivariate functions, I wonder how to get such a reduced expression?