Triangular Gröbner basis structure for CRN steady-state systems

Prove that for any system of steady-state polynomial equations arising from a chemical reaction network governed by mass-action kinetics with conservation laws included, the reduced Gröbner basis computed with respect to the lexicographic order x_n > x_{n-1} > ... > x_2 > x_1 has triangular form: the first basis element g_1 is a univariate polynomial in x_1, and for each j = 2, ..., n, the corresponding basis element equals x_j − g_j(x_1) for some polynomial g_j in x_1.

Background

The paper studies steady states of polynomial systems derived from chemical reaction networks (CRNs) under mass-action kinetics, focusing on algebraic methods to find all real positive solutions. Numerical solvers can miss solutions or require good initial guesses, motivating an algebraic approach via Gröbner bases.

A central algebraic tool is a shape theorem for zero-dimensional radical ideals whose reduced Gröbner basis under lex order has a triangular structure. Motivated by consistent computational evidence in CRN examples (including a gene regulatory network and the Wnt signaling pathway), the authors formulate a conjecture asserting that, under mass-action kinetics with conservation laws, the reduced Gröbner basis generically attains this triangular form.

If true, the conjecture would justify a simple algorithm: solve a single univariate polynomial g_1(x_1) = 0 and then recover remaining variables by back-substitution using x_j = g_j(x_1). Establishing this structural result would provide a rigorous foundation for the observed efficiency and completeness of Gröbner-basis-based steady-state computations in CRNs.

References

Given the set of polynomial equations arising from a CRN under the above assumptions, when computing the reduced Gr\"obner basis with respect to the lexicographic ordering $x_n > x_{n-1} > \dots > x_2 > x_1$, the first element $g_1$ of the basis is a univariate polynomial in $x_1$. Each subsequent element $g_j$ of the basis, for $j = 2, \dots, n$, is a polynomial of the form $x_j-g_j(x_1)$. This conjecture implies that the Gr\"obner basis transforms the system into a triangular form, facilitating a sequential solution process starting from $x_1$.

— When algebra twinks system biology: a conjecture on the structure of Gröbner bases in complex chemical reaction networks  (2501.12233 - Ferrari et al., 21 Jan 2025) in Conjecture 1, Subsection 2.2 (The conjecture)