Elimination Methods

Algebraic Approaches to Stability Analysis of Biological Systems

“…The triangular sets T i and systems [T i , U i ] may satisfy some additional requirements for being regular, simple, or irreducible [3,4,43].…”

“…They provide a simple and effective way to eliminate one or several variables simultaneously from the given set P of polynomials, allowing one to triangularize P or to establish conditions for P to have zeros. See, e.g., [43,Section 5.4], [14], and references therein for more information about the classical theory and modern developments of resultants.…”

“…The main algebraic tools used in the approach described in [46] are the methods of triangular decomposition [43,49] and real solution classification [52]. It turns out that other methods based on Gröbner bases [8], resultants [14,43], cylindrical algebraic decomposition (CAD) [11,12], quadratic quantifier elimination [48], and discriminant varieties [25] for variable elimination and real solving may also be applied to the same problem of stability analysis.…”

“…It turns out that other methods based on Gröbner bases [8], resultants [14,43], cylindrical algebraic decomposition (CAD) [11,12], quadratic quantifier elimination [48], and discriminant varieties [25] for variable elimination and real solving may also be applied to the same problem of stability analysis. Moreover, the approach may be generalized to analyze the bifurcation of limit cycles, hysteresis, oscillation, and other phenomena of biological systems.…”

Algebraic Analysis of Bifurcation and Limit Cycles for Biological Systems

“…In this paper, we analyze bifurcation and construct limit cycles for a concrete two-dimensional (planar) system, the self-assembling micelle system with chemical sinks [2], by using an algebraic approach based on triangular decomposition [21,25], Gröbner bases [3,9], discriminant varieties [11], real solution classification [27], and quantifier elimination by partial CAD [6]. The stability and bifurcation for this biological system have been analyzed initially in [16], where our general algebraic approach is described in generality.…”