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We study the algebraic varieties defined by the conditional independence statements of Bayesian networks. A complete algebraic classification is given for Bayesian networks on at most five random variables. Hidden variables are related to the geometry of higher secant varieties.
We obtain local equations for the toric Hilbert scheme, which parametrizes all ideals with the same multigraded Hilbert function as a given toric ideal. We also prove a conjecture of Sturmfels' providing a criterion for an ideal to have such a Hilbert function.
Multivariate polynomial dynamical systems over finite fields have been studied in several contexts, including engineering and mathematical biology. An important problem is to construct models of such systems from a partial specification of dynamic properties, e.g., from a collection of state transition measurements. Here, we consider static models, which… (More)
In the present paper we study algorithms based on the theory of Gröbner bases for computing free resolutions of modules over polynomial rings. We propose a technique which consists in the application of special selection strategies to the Schreyer algorithm. The resulting algorithm is efficient and, in the graded case, allows a straightforward… (More)
We show that the Hilbert scheme, that parametrizes all ideals with the same Hilbert function over an exterior algebra, is connected. We give a new proof of Hartshorne's Theorem that the classical Hilbert scheme is connected. More precisely: if Q is either a polynomial ring or an exterior algebra, we prove that every two strongly stable ideals in Q with the… (More)
We report on our experiences exploring state of the art Gröbner basis computation. We investigate signature based algorithms in detail. We also introduce new practical data structures and computational techniques for use in both signature based Gröbner basis algorithms and more traditional variations of the classic Buchberger algorithm. Our… (More)