A Majorization Order on Monomials and Termination of a Successive Difference Substitution Algorithm

Abstract When the ordering of variables is fixed, e.g., x1 ≥ x2 ≥ · · · ≥ xn, the monomial X α = x α1 1 · · ·xn n majorizing the monomial X β = x1 1 · · ·xn n (|α| = |β|) means that ∑k i=1 αi ≥ ∑k i=1 βi (k = 1, · · · , n − 1). In this paper, a necessary condition of positively terminating of a general successive difference substitution algorithm (KSDS) for an input f is obtained by using a majorization order on monomials. That is, every single term with negative coefficients in the form f is majorized at least by a single term with positive coefficients of f in an arbitrary ordering of variables.