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

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

2011-09-04

arxiv.org/abs/1109.0686v1

Computer Science

Symbolic Computation

6 pages

Scientific paper

When the ordering of variables is fixed, e.g., $x_1\geq x_2 \geq...\geq x_n$, the monomial $X^{\alpha}=x_1^{\alpha_1}...x_n^{\alpha_n}$ majorizing the monomial $X^{\beta}=x_1^{\beta_1}...x_n^{\beta_n}\ (|\alpha|=|\beta|)$ means that $\sum^k_{i=1}\alpha_i\geq \sum^k_{i=1}\beta_i\ (k=1,...,n-1)$. 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.

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