Elimination for generic sparse polynomial systems

We present a new probabilistic symbolic algorithm that, given a variety defined in an n-dimensional affine space by a generic sparse system with fixed supports, computes the Zariski closure of its projection to an l-dimensional coordinate affine space with l < n. The complexity of the algorithm depends polynomially on combinatorial invariants associated to the supports.Comment: 22 page

Invariant Form of BK-factorization and its Applications

Invariant form of BK-factorization is presented, it is used for factorization of the LPDOs equivalent under gauge transformation and for construction of approximate factorization simplifying numerical simulsations with corresponding LPDEs of higher orderComment: 11 pages, 7 figure

Efficiently and Effectively Recognizing Toricity of Steady State Varieties

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 Gr\"obner 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

A Galois connection between classical and intuitionistic logics. II: Semantics

Three classes of models of QHC, the joint logic of problems and propositions, are constructed, including a class of subset/sheaf-valued models that is related to solutions of some actual problems (such as solutions of algebraic equations) and combines the familiar Leibniz-Euler-Venn semantics of classical logic with a BHK-type semantics of intuitionistic logic. To test the models, we consider a number of principles and rules, which empirically appear to cover all "sufficiently simple" natural conjectures about the behaviour of the operators ! and ?, and include two hypotheses put forward by Hilbert and Kolmogorov, as formalized in the language of QHC. Each of these turns out to be either derivable in QHC or equivalent to one of only 13 principles and 1 rule, of which 10 principles and 1 rule are conservative over classical and intuitionistic logics. The three classes of models together suffice to confirm the independence of these 10 principles and 1 rule, and to determine the full lattice of implications between them, apart from one potential implication.Comment: 35 pages. v4: Section 4.6 "Summary" is added at the end of the paper. v3: Major revision of a half of v2. The results are improved and rewritten in terms of the meta-logic. The other half of v2 (Euclid's Elements as a theory over QHC) is expected to make part III after a revisio

Identifying the parametric occurrence of multiple steady states for some biological networks

We consider a problem from biological network analysis of determining regions in a parameter space over which there are multiple steady states for positive real values of variables and parameters. We describe multiple approaches to address the problem using tools from Symbolic Computation. We describe how progress was made to achieve semi-algebraic descriptions of the multistationarity regions of parameter space, and compare symbolic results to numerical methods. The biological networks studied are models of the mitogen-activated protein kinases (MAPK) network which has already consumed considerable effort using special insights into its structure of corresponding models. Our main example is a model with 11 equations in 11 variables and 19 parameters, 3 of which are of interest for symbolic treatment. The model also imposes positivity conditions on all variables and parameters. We apply combinations of symbolic computation methods designed for mixed equality/inequality systems, specifically virtual substitution, lazy real triangularization and cylindrical algebraic decomposition, as well as a simplification technique adapted from Gaussian elimination and graph theory. We are able to determine multistationarity of our main example over a 2-dimensional parameter space. We also study a second MAPK model and a symbolic grid sampling technique which can locate such regions in 3-dimensional parameter space.Comment: 60 pages - author preprint. Accepted in the Journal of Symbolic Computatio