Generic consistency and nondegeneracy of vertically parametrized systems

2023/04/05 by Elisenda Feliu, Feliu, Elisenda, Oskar Henriksson +3 · 2 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · #13P15 #14A10 #14A25 #14Q30 #Algebraic Geometry (math.AG) #FOS: Biological sciences #FOS: Mathematics #Gene Regulatory Network Analysis #Markov Chains and Monte Carlo Methods #Mathematical Biology Tumor Growth #Molecular Networks (q-bio.MN) #Quantitative Methods (q-bio.QM)

paper · pdf · doi:10.48550/arxiv.2304.02302

openalex publication_date 2023/04/05 · openalex created_date 2023/04/07 · openalex updated_date 2026/07/28

Abstract

We determine the generic consistency, dimension and nondegeneracy of the zero locus over ℂ^*, ℝ^* and ℝ_>0 of vertically parametrized systems: parametric polynomial systems consisting of linear combinations of monomials scaled by free parameters. These systems generalize sparse systems with fixed monomial support and freely varying parametric coefficients. As our main result, we establish the equivalence among three key properties: the existence of nondegenerate zeros, the zero set having generically the expected dimension, and the system being generically consistent. Importantly, we prove that checking whether a vertically parametrized system has these properties amounts to an easily computed matrix rank condition.

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• An effective criterion for multiple positive zeros of vertically parametrized polynomial systems
• Copositivity, discriminants and nonseparable signed supports

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