Computation of Pommaret Bases Using Syzygies

We investigate the application of syzygies for efficiently computing (finite) Pommaret bases. For this purpose, we first describe a non-trivial variant of Gerdt's algorithm to construct an involutive basis for the input ideal as well as an involutive basis for the syzygy module of the output basis. Then we apply this new algorithm in the context of Seiler's method to transform a given ideal into quasi stable position to ensure the existence of a finite Pommaret basis. This new approach allows us to avoid superfluous reductions in the iterative computation of Janet bases required by this method. We conclude the paper by proposing an involutive variant of the signature based algorithm of Gao et al. to compute simultaneously a Grobner basis for a given ideal and for the syzygy module of the input basis. All the presented algorithms have been implemented in Maple and their performance is evaluated via a set of benchmark ideals.Comment: Computer Algebra in Scientific Computing (CASC 2018), Lille, France, 201

Computer Algebra in Scientific Computing [electronic resource] : 20th International Workshop, CASC 2018, Lille, France, September 17–21, 2018, Proceedings /

Chapter “Positive Solutions of Systems of Signed Parametric Polynomial Inequalities” is available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.Proof-of-Work Certificates that Can Be Efficiently Computed in the Cloud (Invited Talk) -- On Unimodular Matrices of Difference Operators -- Sparse Polynomial Arithmetic with the BPAS Library -- Computation of Pommaret Bases Using Syzygies -- A Strongly Consistent Finite Difference Scheme for Steady Stokes Flow and its Modified Equations -- Symbolic-Numeric Methods for Nonlinear Integro-Differential Modeling -- A Continuation Method for Visualizing Planar Real Algebraic Curves with Singularities -- From Exponential Analysis to Padé Approximation and Tensor Decomposition, in One and More Dimensions -- Symbolic Algorithm for Generating the Orthonormal Bargmann-Moshinsky Basis for SU(3) Group -- About Some Drinfel'd Associators -- On a Polytime Factorization Algorithm for Multilinear Polynomials over F2 -- Tropical Newton-Puiseux Polynomials -- Orthogonal Tropical Linear Prevarieties -- Symbolic-Numerical Algorithms for Solving Elliptic Boundary-Value Problems Using Multivariate Simplex Lagrange Elements -- Symbolic-Numeric Simulation of Satellite Dynamics with Aerodynamic Attitude Control System -- Finding Multiple Solutions in Nonlinear Integer Programming with Algebraic Test-Sets -- Positive Solutions of Systems of Signed Parametric Polynomial Inequalities -- Qualitative Analysis of a Dynamical System with Irrational First Integrals -- Effective Localization Using Double Ideal Quotient and Its Implementation -- A Purely Functional Computer Algebra System Embedded in Haskell -- Splitting Permutation Representations of Finite Groups by Polynomial Algebra Methods -- Factoring Multivariate Polynomials with Many Factors and Huge Coefficients -- Beyond the First Class of Analytic Complexity -- A Theory and an Algorithm for Computing Sparse Multivariate Polynomial Remainder Sequence -- A Blackbox Polynomial System Solver on Parallel Shared Memory Computers.Chapter “Positive Solutions of Systems of Signed Parametric Polynomial Inequalities” is available open access under a Creative Commons Attribution 4.0 International License via link.springer.com