# On Parametric Linear System Solving

## TL;DR

This paper presents a polynomial-time method for solving parametric linear systems with up to three parameters, leveraging Hermite and Smith normal forms to efficiently analyze solution regimes and singularities.

## Contribution

It introduces a novel approach that reduces the complexity of solving parametric linear systems by exploiting algebraic normal forms, improving over previous exponential methods.

## Key findings

- Polynomial-time solution method for systems with up to three parameters.
- Effective identification of singularities and structural changes in the system.
- Reduction of regimes needed from exponential to polynomial in system size.

## Abstract

Parametric linear systems are linear systems of equations in which some symbolic parameters, that is, symbols that are not considered to be candidates for elimination or solution in the course of analyzing the problem, appear in the coefficients of the system. In this work we assume that the symbolic parameters appear polynomially in the coefficients and that the only variables to be solved for are those of the linear system. The consistency of the system and expression of the solutions may vary depending on the values of the parameters. It is well-known that it is possible to specify a covering set of regimes, each of which is a semi-algebraic condition on the parameters together with a solution description valid under that condition. We provide a method of solution that requires time polynomial in the matrix dimension and the degrees of the polynomials when there are up to three parameters. In previous methods the number of regimes needed is exponential in the system dimension and polynomial degree of the parameters. Our approach exploits the Hermite and Smith normal forms that may be computed when the system coefficient domain is mapped to the univariate polynomial domain over suitably constructed fields. Our approach effectively identifies intrinsic singularities and ramification points where the algebraic and geometric structure of the matrix changes.