Generic finiteness of solutions to parametrized systems of generalized polynomial inequalities

Abstract

We study parametrized systems of generalized polynomial inequalities, (c ∘ x^B) ∈ C, for a positive variable vector x ∈ ℝⁿ_>, a real exponent matrix B ∈ ℝ^{n × m}, a positive parameter vector c ∈ ℝ^m_>, and a positive, blunt coefficient cone C ⊂ ℝ^m_>. This setting includes (i) generalized polynomial equations, A (c ∘ x^B) = 0, arising e.g. from power-law dynamical systems with real exponents (and involving a real coefficient matrix A ∈ ℝ^{n' × 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 polynomial systems developed by Müller and Regensburger, we reformulate the problem as a system of binomial equations on the coefficient set P = C ∩ Δ⊆ 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 ≤ 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 ℝⁿ_>). As an application, we prove generic finiteness of steady states (within forward-invariant sets) for parametrized power-law dynamical systems with real exponents.

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BibTeX RIS

BibTeX

@misc{muller2026generic,
  title = {{Generic finiteness of solutions to parametrized systems of generalized polynomial inequalities}},
  author = {Stefan Müller},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.28588},
  eprint = {2610.28588},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.28588},
}

RIS

TY  - UNPB
TI  - Generic finiteness of solutions to parametrized systems of generalized polynomial inequalities
AU  - Stefan Müller
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.28588
UR  - https://arxiv.org/abs/2610.28588
AB  - We study parametrized systems of generalized polynomial inequalities, (c ∘ x^B) ∈ C, for a positive variable vector x ∈ ℝ^n_>, a real exponent matrix B ∈ ℝ^n × m, a positive parameter vector c ∈ ℝ^m_>, and a positive, blunt coefficient cone C ⊂ ℝ^m_>. This setting includes (i) generalized polynomial equations, A (c ∘ x^B) = 0, arising e.g. from power-law dynamical systems with real exponents (and involving a real coefficient matrix A ∈ ℝ^n' × 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 polynomial systems developed by Müller and Regensburger, we reformulate the problem as a system of binomial equations on the coefficient set P = C ∩ Δ⊆ 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 ≤ 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 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.
ER  -

Abstract

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