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239
Invariance of Conjunctions of Polynomial Equalities for Algebraic Differential Equations?
"... Abstract In this paper we seek to provide greater automation for formal deductive verification tools working with continuous and hybrid dynamical systems. We present an efficient procedure to check invariance of conjunctions of polynomial equalities under the flow of polynomial ordinary differenti ..."
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differential equations. The procedure is based on a necessary and sufficient condition that characterizes invariant conjunctions of polynomial equalities. We contrast this approach to an alternative one which combines fast and sufficient (but not necessary) conditions using differential cuts for soundly
Automatic Generation of Polynomial Loop Invariants: Algebraic Foundations
 In International Symposium on Symbolic and Algebraic Computation 2004 (ISSAC04
, 2004
"... This paper presents the algebraic foundation for an approach for generating polynomial loop invariants in imperative programs. It is first shown that the set of polynomials serving as loop invariants has the algebraic structure of an ideal. Using this connection, a procedure for finding loop invaria ..."
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Cited by 34 (6 self)
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. This yields a correct and complete algorithm for inferring conjunctions of polynomial equations as invariants. The method has been implemented in Maple using the Groebner package. The implementation has been used to automatically discover nontrivial invariants for several examples to illustrate the power
On polynomial solutions of differential equations
, 1991
"... A general method of obtaining linear differential equations having polynomial solutions is proposed. The method is based on an equivalence of the spectral problem for an element of the universal enveloping algebra of some Lie algebra in the ”projectivized” representation possessing an invariant subs ..."
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A general method of obtaining linear differential equations having polynomial solutions is proposed. The method is based on an equivalence of the spectral problem for an element of the universal enveloping algebra of some Lie algebra in the ”projectivized” representation possessing an invariant
Characterizing Algebraic Invariants by Differential Radical Invariants ⋆
"... Abstract We prove that any invariant algebraic set of a given polynomial vector field can be algebraically represented by one polynomial and a finite set of its successive Lie derivatives. This socalled differential radical characterization relies on a sound abstraction of the reachable set of solu ..."
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Cited by 4 (2 self)
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Abstract We prove that any invariant algebraic set of a given polynomial vector field can be algebraically represented by one polynomial and a finite set of its successive Lie derivatives. This socalled differential radical characterization relies on a sound abstraction of the reachable set
Generating Polynomial Invariants with DISCOVERER and QEPCAD
"... This paper investigates how to apply the techniques on solving semialgebraic systems to invariant generation of polynomial programs. By our approach, the generated invariants represented as a semialgebraic system are more expressive than those generated with the wellestablished approaches in the ..."
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Cited by 5 (2 self)
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in the literature, which are normally represented as a conjunction of polynomial equations. We implement this approach with the computer algebra tools DISCOVERER and QEPCAD 1. We also explain, through the complexity analysis, why our approach is more efficient and practical than the one of [17] which directly
Integrodifferential polynomials and operators
, 2008
"... We propose two algebraic structures for treating integral operators in conjunction with derivations: The algebra of integrodifferential polynomials describes nonlinear integral and differential operators together with initial values. The algebra of integrodifferential operators can be used to solv ..."
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Cited by 9 (7 self)
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We propose two algebraic structures for treating integral operators in conjunction with derivations: The algebra of integrodifferential polynomials describes nonlinear integral and differential operators together with initial values. The algebra of integrodifferential operators can be used
Generating All Polynomial Invariants in Simple Loops
, 2007
"... This paper presents a method for automatically generating all polynomial invariants in simple loops. It is rst shown that the set of polynomials serving as loop invariants has the algebraic structure of an ideal. Based on this connection, a xpoint procedure using operations on ideals and Grobner bas ..."
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Cited by 25 (2 self)
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with the ideals generated by the procedure either remain the same or increase their dimension at every iteration of the xpoint procedure. This yields a correct and complete algorithm for inferring conjunctions of polynomial equalities as invariants. The method has been implemented in Maple using the Groebner
Constructing Invariants for Hybrid Systems
 IN HYBRID SYSTEMS: COMPUTATION AND CONTROL, LNCS 2993
, 2004
"... An invariant of a system is a predicate that holds for every reachable state. In this paper, we present techniques to generate invariants for hybrid systems. This is achieved by reducing the invariant generation problem to a constraint solving problem using methods from the theory of ideals over p ..."
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Cited by 59 (7 self)
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polynomial rings. We extend our previous work on the generation of algebraic invariants for discrete transition systems in order to generate algebraic invariants for hybrid systems. In doing so, we present a new technique to handle consecution across continuous differential equations. The techniques we
Solving Second Order Linear Differential Equations via Algebraic Invariant Curves
"... Abstract—The idea to find first integrals for polynomial vector fields via algebraic invariant curves can be traced back to Darboux in the 19th century. In 1983, this approach was further developed by Prelle and Singer to become a semidecision algorithm for finding elementary first integrals. In th ..."
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. In this paper, we describe how to extend the PrelleSinger method to deal with second order linear differential equations via Kovacic’s results on algebraic solutions of Riccati equations. Some illustrative examples on using this approach are provided. Index Terms—Algebraic invariant curves, the Prelle
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