Radicals of binomial ideals, Original Research Article

In this paper we investigate radical operations on binomial ideals, i.e. ideals generated by sums of at most two terms, especially the L-radical, α-radical and τ-radical for an arbitrary extension field L of the base field K resp. an arbitrary ordering α resp. preordering τ on K. This is the vanishing ideal of the set of L-rational points of the ideal resp. the R-radical for an arbitrary real closure R of α resp. the intersection of the α-radicals for all orders α on K containing τ. We derive necessary and sufficient conditions on L resp. τ for these radicals of arbitrary binomial ideals to be again binomial and find several cases (incl. L = K and L a real or separable closure of K) where this is true. There are counterexamples for the ordinary radical. Further we describe algorithms for radical computations 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.