Generic finiteness of solutions to parametrized systems of generalized polynomial inequalities

Source abstract

We study parametrized systems of $\mathit{generalized}$ polynomial inequalities, $(c \circ x^B) \in C$, for a positive variable vector $x \in \mathbb{R}^n_>$, a $\mathit{real}$ exponent matrix $B \in \mathbb{R}^{n \times m}$, a positive parameter vector $c \in \mathbb{R}^m_>$, and a positive, blunt coefficient cone $C \subset \mathbb{R}^m_>$. This setting includes (i) generalized polynomial equations, $A \, (c \circ x^B) = 0$, arising e.g. from power-law dynamical systems with real exponents (and involving a real coefficient matrix $A \in \mathbb{R}^{n' \times m}$), and (ii) generalized semi-algebraic sets. Our main result guarantees the generic finiteness of solutions modulo unavoidable exponential fibers. Building on a framework for parametrized $\mathit{polynomial}$ systems developed by Müller and Regensburger, we reformulate the problem as a system of $\mathit{binomial}$ equations on the coefficient set $P = C \cap Δ\subseteq \mathbb{R}^m_>$, where $Δ$ is a product of (standard) simplices. Solvability is governed by the monomial-dependency subspace $D$. Using elementary arguments from analysis and topology, including a logarithmic boundary analysis and Sard's theorem, we control the accumulation of solutions on the boundary and in the interior of $P$. We show that, if $\dim P \le \dim D$, then generically in both the measure-theoretic and topological sense, there are finitely many solutions on $P$ (and hence finitely many exponential fibers of solutions in $\mathbb{R}^n_>$). As an application, we prove generic finiteness of steady states (within forward-invariant sets) for parametrized power-law dynamical systems with real exponents.