On the number of term orders

Abstract

By an admissible order on a finite subsetQ of ℚⁿ we mean the restriction toQ of a linear order on ℚⁿ compatible with the group structure of ℚⁿ and such that ℕⁿ is contained in the positive cone of the order. We first derive upper and lower bounds on the number of admissible orders on a given setQ in terms of the dimensionn and the cardinality ofQ. Better estimates are possible if the setQ enjoys symmetry properties and in the case whereQ is a discrete hyperbox of the form$\mathop \Pi \limits_{k = 1}^n [1,d_k ].$ In the latter case, we also give asymptotic results as$\mathop {\min }\limits_{1 \leqq k \leqq n} d_k \to \infty$ d _k → ∞ for fixedn. We finally present algorithms which compute the set of admissible orders that extend a given binary relation onQ and their number. The algorithms are useful in connection with the construction of universal Gröbner bases.