Detecting binomiality

Abstract

Abstract. Binomial ideals are special polynomial ideals with many algorithmically and theoretically nice properties. We discuss the problem of deciding if a given polynomial ideal is binomial. While the methods are general, our main motivation and source of examples is the simplification of steady state equations of chemical reaction networks. For homogeneous ideals we give an efficient, Gröbner-free algorithm for binomiality detection, based on linear algebra only. On inhomogeneous input the algorithm can only give a sufficient condition for binomiality. As a remedy we construct a heuristic toolbox that can lead to simplifications even if the given ideal is not binomial.

Gröbner bases of reaction networks with intermediate species

“…Therefore the steady state ideal is generated by 29 polynomials in 29 variables and 46 parameters. Using Singular [7] and monomial order grevlex with x 1 > · · · > x 29 (the same monomial order that is used in [4]), it took between 110 and 115 seconds 1 to compute the reduced Gröbner basis. This basis consists of 169 polynomials.…”

“…Remark 3.14. In [4], a method for determining whether a homogeneous ideal is binomial is introduced. The method avoids the computation of Gröbner bases and is regarded as a fast method.…”

“…Example 5.3. When a binomial basis of the steady state ideal is obtained from linear combinations of the steady state polynomials (see [4]), then the steady state ideal of the core network is binomial if and only if that of the extended network is.…”

“…In typical networks, this extra condition can be readily checked from the network structure alone. When the core network has a homogeneous steady state ideal (which happens frequently for realistic reaction networks), then one can employ the linear algebra-based method introduced in [4] to detect whether the steady state ideal of the core network is binomial. Then, combined with our result, we obtain a faster method to address whether the steady state ideal of the original network is binomial, which does not rely on the computation of a Gröbner basis.…”

“…Intermediates are introduced in Section 2. In Section 3 we address Gröbner bases of networks with intermediates, discuss binomial steady state ideals and relate our work to [4]. In Section 4 a technical condition of algebraic independence of a set of rational functions, which is assumed in the former sections, is discussed.…”