Some geometric properties of successive difference substitutions
This paper provides a new, geometric perspective to study successive difference substitutions, and proves that the sequence of the successive difference substitution sets is not convergent. An interesting result that a given k-dimensional rational hyperplane can be transformed to a k-dimensional coordinate hyperplane of new variables by finite steps of successive difference substitutions is presented. Moreover, a sufficient condition for the sequence of the successive difference substitution sets of a form being not terminating is obtained. That is, a class of polynomials which cannot be proved to be positive semi-definite by the successive difference substitution method are obtained.
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