ISSAC 2025 Short Communications: An Effective Criterion for Multiple Positive Solutions to Vertically Parametrized Polynomial Systems

Polynomial systems that arise in applications often have fixed support and their coefficients depend on a set of parameters via some function. An example is the class of vertically parametrized systems [ 13, 10, 11 ], which appear naturally, for example, in the study of chemical reaction networks [ 6 ], polynomial optimization [ 3 ], or geometric modelling [ 5 ]. These are systems defined by vectors of polynomials of the form

[EQUATION]

where C,M ∈ R ^n×m are matrices and C has full rank, x ^M refers to the vector whose i -th entry is the monomial in variables x ₁ , ..., x _n with exponents given by the i -th column of M , and κ o x ^M is componentwise multiplication with the vector of parameters κ := ( κ ₁ , ..., κ _m ). To understand the structure of these systems, it is relevant to notice that each parameter is associated with a single monomial, that is, it only appears as a linear term of the coefficient of the same monomial. Sparse (or freely parametrized) polynomial systems , in which each monomial in each polynomial contributes with a different parameter, are a particular instance of vertically parametrized systems.