Comprehensive Characteristic Decomposition of Parametric Polynomial Systems

Abstract

This paper presents an algorithm that decomposes an arbitrary set F of multivariate polynomials involving parameters into finitely many sets Γ i of (lexicographical) Gröbner bases G i j such that associated with each Γ i there is a system A i of parametric constraints, all the A i 's partition the parameter space, all the G i j 's in each Γ i remain Gröbner bases under specialization of the parameters satisfying the constraints in A i , and for each i the Gröbner bases G i j together with their corresponding W-characteristic sets form a normal characteristic decomposition of F . The sets of Gröbner bases computed by the algorithm provide a comprehensive characteristic decomposition of F that is structure-invariant under the associated constraints and possesses many algebraic and geometric properties, including most of the notable properties on comprehensive Gröbner systems and comprehensive triangular decomposition. Some of these properties are highlighted in the paper and advantages of the proposed algorithm are discussed briefly from the aspects of methodological simplicity and computational performance using illustrative examples and preliminary experiments. CCS CONCEPTS• Computing methodologies → Symbolic and algebraic algorithms; Algebraic algorithms.