A p-adic approach to the computation ofGröbner bases
Journal of Symbolic Computation
A new method for solving algebraic systems of positive dimension
Discrete Applied Mathematics - Special volume on applied algebra, algebraic algorithms, and error-correcting codes
Solving zero-dimensional algebraic systems
Journal of Symbolic Computation
Representation for the radical of a finitely generated differential ideal
ISSAC '95 Proceedings of the 1995 international symposium on Symbolic and algebraic computation
Zeros, multiplicities, and idempotents for zero-dimensional systems
Algorithms in algebraic geometry and applications
Modern computer algebra
On the theories of triangular sets
Journal of Symbolic Computation - Special issue on polynomial elimination—algorithms and applications
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
The differential ideal [P]: M∞
Journal of Symbolic Computation - Special issue on differential algebra and differential equations
Deformation techniques for efficient polynomial equation solving
Journal of Complexity
A Gröbner free alternative for polynomial system solving
Journal of Complexity
Resultants and moving surfaces
Journal of Symbolic Computation
Relations between roots and coefficients, interpolation and application to system solving
Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
Algebraic Solution of Systems of Polynomial Equations Using Groebner Bases
AAECC-5 Proceedings of the 5th International Conference on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
How Lower and Upper Complexity Bounds Meet in Elimination Theory
AAECC-11 Proceedings of the 11th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
When Polynomial Equation Systems Can Be "Solved" Fast?
AAECC-11 Proceedings of the 11th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
On the Invariants of the Quotients of the Jacobian of a Curve of Genus 2
AAECC-14 Proceedings of the 14th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Probabilistic algorithms for sparse polynomials
EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
On Improving Approximate Results of Buchberger's Algorithm by Newton's Method
EUROCAL '85 Research Contributions from the European Conference on Computer Algebra-Volume 2
Degree bounds and lifting techniques for triangular sets
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Computation with polynomial systems
Computation with polynomial systems
Notes on triangular sets and triangulation-decomposition algorithms I: polynomial systems
SNSC'01 Proceedings of the 2nd international conference on Symbolic and numerical scientific computation
Sharp estimates for triangular sets
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Lifting techniques for triangular decompositions
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
On approximate triangular decompositions in dimension zero
Journal of Symbolic Computation
Change of order for regular chains in positive dimension
Theoretical Computer Science
Almost linear time operations with triangular sets
ACM Communications in Computer Algebra
Optimization techniques for small matrix multiplication
Theoretical Computer Science
Bit-size estimates for triangular sets in positive dimension
Journal of Complexity
On the complexity of computing with zero-dimensional triangular sets
Journal of Symbolic Computation
Modular Composition Modulo Triangular Sets and Applications
Computational Complexity
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We study the representation of the solutions of a polynomial system by triangular sets, and concentrate on the positive-dimensional case. We reduce to dimension zero by placing the free variables in the base field, so the solutions can be represented by triangular sets with coefficients in a rational function field.We give intrinsic-type bounds on the degree of the coefficients in such a triangular set, and on the degree of an associated degeneracy hypersurface. Then we show how to apply lifting techniques in this context, and point out the role played by the evaluation properties of the input system.Our algorithms are implemented in Magma; we present three applications, relevant to geometry and number theory.