Representation for the radical of a finitely generated differential ideal
ISSAC '95 Proceedings of the 1995 international symposium on Symbolic and algebraic computation
Existence and uniqueness theorems for formal power series solutions of analytic differential systems
ISSAC '99 Proceedings of the 1999 international symposium on Symbolic and algebraic computation
Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation
ACM Transactions on Mathematical Software (TOMS)
Geometric completion of differential systems using numeric-symbolic continuation
ACM SIGSAM Bulletin
A probabilistic algorithm to test local algebraic observability in polynomial time
Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
Rankings of derivatives for elimination algorithms and formal solvability of analytic partial differential equations
Symbolic-numeric completion of differential systems by homotopy continuation
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Algorithms and implementations for differential elimination
Algorithms and implementations for differential elimination
Application of numerical algebraic geometry and numerical linear algebra to PDE
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Overdetermined Elliptic Systems
Foundations of Computational Mathematics
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Notes on triangular sets and triangulation-decomposition algorithms II: differential systems
SNSC'01 Proceedings of the 2nd international conference on Symbolic and numerical scientific computation
Implicit Riquier Bases for PDAE and their semi-discretizations
Journal of Symbolic Computation
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Riquier Bases for systems of analytic pde are, loosely speaking, a differential analogue of Gröbner Bases for polynomial equations. They are determined in the exact case by applying a sequence of prolongations (differentiations) and eliminations to an input system of pde. We present a symbolic-numeric method to determine Riquier Bases in implicit form for systems which are dominated by pure derivatives in one of the independent variables and have the same number of pde and unknowns. The method is successful provided the prolongations with respect to the dominant independent variable have a block structure which is uncovered by Linear Programming and certain Jacobians are non-singular when evaluated at points on the zero sets defined by the functions of the pde. For polynomially nonlinear pde, homotopy continuation methods from Numerical Algebraic Geometry can be used to compute approximations of the points. We give a differential algebraic interpretation of Pryce's method for ode, which generalizes to the pde case. A major aspect of the method's efficiency is that only prolongations with respect to a single (dominant) independent variable are made, possibly after a random change of coordinates. Potentially expensive and numerically unstable eliminations are not made. Examples are given to illustrate theoretical features of the method, including a curtain of Pendula and the control of a crane.