arXiv subjects

AmirHosein Sadeghimanesh

Publications and source records attributed to AmirHosein Sadeghimanesh.

8 recordsLinked to original sources

arXiv2026

Resultant Tools for Parametric Polynomial Systems with Application to Mathematical Biology

To decompose the parameter space of a parametric system of polynomial equations with respect to the number of real solutions that the system attains, first the discriminant variety of the system should be computed. This is an elimination type problem because the discriminant variety is the union of the projections into the parameter space of the intersection of the variety of the original system with one or several new hypersurfaces. Despite the popularity of Gröbner Bases (GBs) for this task, in this work we focus on a lesser used alternative tool, namely resultants. We develop new approaches to build the discriminant variety using different resultant methods such as Dixon resultant and iterated univariate resultants. We discuss how the minimal discriminant variety, the discriminant varieties computed by our methods, and the known border polynomials are related. We demonstrate with numerous examples from population dynamics and chemical reaction network theory that our new approach successfully handles larger size examples than a GB can.

cs.SC↗

arXiv2026

Letting Homogeneity Entropy Select S-Pairs in Buchberger's Algorithm

We present a novel S-pair selection strategy called Homogeneity Entropy, for deciding the sequence of S-polynomials to construct in Buchberger's algorithm to compute a Groebner basis. The strategy uses an information theoretic measure derived from the distribution of degrees among the monomials of the S-polynomial: a very different approach to the classical heuristics such as Degree, Normal and Sugar, or indeed the more recent machine learning approaches to the problem. We implement this strategy and evaluate it on two different datasets: (1) variations of randomly generated polynomial systems with controlled numbers of variables, degrees, densities and number of polynomials per system; and (2) the PHCpack benchmark dataset sourced from real world problems. The Homogeneity Entropy strategy significantly outperforms classical strategies on random polynomial datasets, but on the PHCpack dataset the classical strategies perform better. This suggests the right strategy varies with the shape of the data and we explore this in several experiments. The new strategy offers practically meaningful gains on certain distributions, and represents the first use of such information-theoretic guidance in the optimisation of symbolic computation algorithms.

cs.IT↗

arXiv2026

Optimising Cylindrical Algebraic Coverings for use in SMT by Solving a Set Covering Problem with Reasons

The Conflict-Driven Cylindrical Algebraic Covering algorithm has proven well suited for performing theory validation checks in the satisfiability modulo theories paradigm for non-linear real arithmetic. CDCAC repurposes the theory underpinning classical cylindrical algebraic decomposition for SMT solving and is implemented in the SMT solvers cvc5 and SMT-RAT, as well as the computer algebra system Maple. It was previously observed that when using cylindrical algebraic decomposition for an SMT theory call, the output can be optimised by solving a single set covering problem instance that minimises the conflict clause. In this paper we consider the corresponding optimisation for CDCAC and observe that CDCAC naturally gives rise to multiple such optimisations within a single call. Each time a covering is generalised in one dimension, the resulting cell in the next dimension is labelled with theory constraints that cannot be satisfied together. We seek the smallest subset of constraints whose union covers all labels from the cells in the current covering. We call this optimisation problem a set covering problem with reasons. To simplify this problem, we introduce a data reduction step that generalises Beasley reduction for the classical set covering problem and show that this step alone solves many of the instances arising from SMT-LIB benchmarks. We then propose an exact solver based on linear programming to efficiently solve the remaining cases. Integrating these techniques into CDCAC has the potential to significantly improve SMT solver performance for non-linear real arithmetic problems.

cs.DS↗

arXiv2025

Understanding Multistationarity of Fully Open Reaction Networks

This work addresses multistationarity of fully open reaction networks equipped with mass action kinetics. We improve upon the existing results relating existence of positive feedback loops in a reaction network and multistationarity; and we provide a novel deterministic operation to generate new non-multistationary networks. This is interesting because while there were many operations to create infinitely many new multistationary networks from a multistationary example, this is the first such operation for the non-multistationary counterpart. Such tools for the generation of example networks have a use-case in the application of data science to reaction network theory. We demonstrate this by using the new data, along with a novel graph representation of reaction networks that is unique up to a permutation on the name of species of the network, to train a graph attention neural network model to predict multistationarity of reaction networks. This is the first time machine learning tools are used for studying classification problems of reaction networks.

q-bio.MN↗

arXiv2022

Polynomial Superlevel Set Representation of the Multistationarity Region of Chemical Reaction Networks

In this paper we introduce a new representation for the multistationarity region of a reaction network, using polynomial superlevel sets. The advantages of using this polynomial superlevel set representation over the already existing representations (cylindrical algebraic decompositions, numeric sampling, rectangular divisions) is discussed, and algorithms to compute this new representation are provided. The results are given for the general mathematical formalism of a parametric system of equations and so may be applied to other application domains.

q-bio.MN↗

arXiv2022

Kac-Rice formulas and the number of solutions of parametrized systems of polynomial equations

Kac-Rice formulas express the expected number of elements a fiber of a random field has in terms of a multivariate integral. We consider here parametrized systems of polynomial equations that are linear in enough parameters, and provide a Kac-Rice formula for the expected number of solutions of the system when the parameters follow continuous distributions. Combined with Monte Carlo integration, we apply the formula to partition the parameter region according to the number of solutions or find a region in parameter space where the system has the maximal number of solutions. The motivation stems from the study of steady states of chemical reaction networks and gives new tools for the open problem of identifying the parameter region where the network has at least two positive steady states. We illustrate with numerous examples that our approach successfully handles a larger number of parameters than exact methods.

math.NA↗

arXiv2019

The multistationarity structure of networks with intermediates and a binomial core network

This work addresses whether a reaction network, taken with mass-action kinetics, is multistationary, that is, admits more than one positive steady state in some stoichiometric compatibility class. We build on previous work on the effect that removing or adding intermediates has on multistationarity, and also on methods to detect multistationarity for networks with a binomial steady state ideal. In particular, we provide a new determinant criterion to decide whether a network is multistationary, which applies when the network obtained by removing intermediates has a binomial steady state ideal. We apply this method to easily characterize which subsets of complexes are responsible for multistationarity; this is what we call the \emph{multistationarity structure} of the network. We use our approach to compute the multistationarity structure of the $n$-site sequential distributive phosphorylation cycle for arbitrary n.

q-bio.MN↗

arXiv2018

Groebner bases of reaction networks with intermediate species

In this work we consider the computation of Groebner bases of the steady state ideal of reaction networks equipped with mass-action kinetics. Specifically, we focus on the role of intermediate species and the relation between the extended network (with intermediate species) and the core network (without intermediate species). We show that a Groebner basis of the steady state ideal of the core network always lifts to a Groebner basis of the steady state ideal of the extended network by means of linear algebra, with a suitable choice of monomial order. As illustrated with examples, this contributes to a substantial reduction of the computation time, due mainly to the reduction in the number of variables and polynomials. We further show that if the steady state ideal of the core network is binomial, then so is the case for the extended network, as long as an extra condition is fulfilled. For standard networks, this extra condition can be visually explored from the network structure alone.

cs.SC↗