In this paper we develop a new technique to compute the Betti table of a monomial ideal. We present a prototype implementation of the resulting algorithm and we perform numerical experiments suggesting a very promising efficiency. On the way of descr
ibing the method, we also prove new constraints on the shape of the possible Betti tables of a monomial ideal.
Let $X$ be a set of points whose coordinates are known with limited accuracy; our aim is to give a characterization of the vanishing ideal $I(X)$ independent of the data uncertainty. We present a method to compute a polynomial basis $B$ of $I(X)$ whi
ch exhibits structural stability, that is, if $widetilde X$ is any set of points differing only slightly from $X$, there exists a polynomial set $widetilde B$ structurally similar to $B$, which is a basis of the perturbed ideal $ I(widetilde X)$.
Given a set $X$ of empirical points, whose coordinates are perturbed by errors, we analyze whether it contains redundant information, that is whether some of its elements could be represented by a single equivalent point. If this is the case, the emp
irical information associated to $X$ could be described by fewer points, chosen in a suitable way. We present two different methods to reduce the cardinality of $X$ which compute a new set of points equivalent to the original one, that is representing the same empirical information. Though our algorithms use some basic notions of Cluster Analysis they are specifically designed for thinning out redundant data. We include some experimental results which illustrate the practical effectiveness of our methods.
