Abstract
We consider the problem of testing whether the points in a complex or real variety with non-zero coordinates form a multiplicative group or, more generally, a coset of a multiplicative group. For the coset case, we study the notion of shifted toric varieties which generalizes the notion of toric varieties. This requires a geometric view on the varieties rather than an algebraic view on the ideals. We present algorithms and computations on 129 models from the BioModels repository testing for group and coset structures over both the complex numbers and the real numbers. Our methods over the complex numbers are based on Gröbner basis techniques and binomiality tests. Over the real numbers we use first-order characterizations and employ real quantifier elimination. In combination with suitable prime decompositions and restrictions to subspaces it turns out that almost all models show coset structure. Beyond our practical computations, we give upper bounds on the asymptotic worst-case complexity of the corresponding problems by proposing single exponential algorithms that test complex or real varieties for toricity or shifted toricity. In the positive case, these algorithms produce generating binomials. In addition, we propose an asymptotically fast algorithm for testing membership in a binomial variety over the algebraic closure of the rational numbers.
Highlights
We are interested in situations where the points with non-zero coordinates in a given complex or real variety form a multiplicative group or, more generally, a coset
Following the definition of shifted toric varieties and using Proposition 1, we show in the following proposition that ideals of shifted toric varieties have Gröbner bases of a specific form
We have taken a geometric approach to studying steady state varieties, which—besides significant theoretical results—generated comprehensive empirical data from computations on 129 networks from BioModels repository
Summary
We are interested in situations where the points with non-zero coordinates in a given complex or real variety form a multiplicative group or, more generally, a coset. V is a toric variety if and only I (V ) is prime and the reduced Gröbner basis (with respect to any term order) of I (V ) contains only binomials of the form X α − X β where α, β ∈ Nn. By definition, V is toric if and only if V is irreducible and there exists a torus T such that V = T. Proposition 1 gives the form of the polynomials in a Gröbner basis of the ideal describing the toric component of a binomial variety. From a computational point of view, a variety V = V (I ) is usually given by a set of generators of I and we would like to derive information about toricity, binomiality or coset property of V by computations over the generators of I This can be done via Gröbner bases.
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