RADICALS OF BINOMIAL IDEALS

In this paper we investigate radical operations on binomial ideals, i. e. ideals generated by sums of at most two terms, especially the L-rad ical, alpha-radical and tau-radical for an arbitrary extension field L of the base field K resp. an arbitrary ordering alpha resp. preorderi ng tau on K. This is the vanishing ideal of the set of L-rational poin ts of the ideal resp, the R-radical for an arbitrary real closure R of alpha resp. the intersection of the alpha-radicals for all orders alp ha on K containing tau. We derive necessary and sufficient conditions on L resp. tau for these radicals of arbitrary binomial ideals to be a gain binomial and find several cases (incl. L = K and L a real or sepa rable closure of K) where this is true. There are counterexamples for the ordinary radical. Further we describe algorithms for radical compu tations and root counting which are designed for the special structure of binomial ideals, and we give Bezout-type bounds for the number of L-rational points in the case that their number is finite. (C) 1997 El sevier Science B.V.