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B.2.1 Introduction to orderings

SINGULAR offers a great variety of monomial orderings which provide an enormous functionality, if used diligently. However, this flexibility might also be confusing for the novice user. Therefore, we recommend to those not familiar with monomial orderings to generally use the ordering dp for computations in the polynomial ring $K[x_1,\ldots,x_n]$,resp. ds for computations in the localization $\hbox{Loc}_{(x)}K[x_1,\ldots,x_n]$.

For inhomogenous input ideals, standard (resp. groebner) bases computations are generally faster with the orderings $\hbox{Wp}(w_1, \ldots, w_n)$(resp. $\hbox{Ws}(w_1, \ldots, w_n)$)if the input is quasihomogenous w.r.t. the weights $w_1$, $\ldots$, $w_n$ of $x_1$, $\ldots$, $x_n$.

If the output needs to be "triangular" (resp. "block-triangular"), the lexicographical ordering lp (resp. lexicographical block-orderings) need to be used. However, these orderings usually result in much less efficient computations.