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Numerical Algebraic Geometry and Differential Equations [PDF]

, 2014
In this paper we review applications of numerical algebraic geometry to differential equations. The techniques we address are direct solution, bootstrapping by filtering, and continuation and bifurcation. We review differential equations systems with multiple solutions and bifurcations.
Bei Hu, Wenrui Hao, Andrew J. Sommese
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Differential forms in computational algebraic geometry [PDF]

Proceedings of the 2007 international symposium on Symbolic and algebraic computation, 2007
We give a uniform method for the two problems #CCC and #ICC of counting connected and irreducible components of complex algebraic varieties, respectively. Our algorithms are purely algebraic, i.e., they use only the field structure of C. They work efficiently in parallel and can be implemented by algebraic circuits of polynomial depth, i.e., in ...
Peter Scheiblechner, Peter Bürgisser
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Differential forms in algebraic geometry

2011
Before considering more general spaces we shall first discuss (1) the r-dimensional projective space Π r . In this space we shall consider a homogeneous coordinate system (Z0, Z1, ... , Z r ). Let U α be that part of Π r in which Z α ≠ 0. In U α we may then introduce non-homogeneous coordinates zαi = Zι/Zα (ι≠α).
W. Hodge
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Embedding Method by Real Numerical Algebraic Geometry for Structurally Unamenable Differential-Algebraic Equations

Journal of Systems Science and Complexity, 2021
Existing structural analysis methods may fail to find all hidden constraints for a system of differential-algebraic equations with parameters if the system is structurally unamenable for certain values of the parameters.
Wenqiang Yang, Wenyuan Wu, G. Reid
semanticscholar   +1 more source

An algebraic geometry approach for the computation of all linear feedback Nash equilibria in LQ differential games

IEEE Conference on Decision and Control, 2015
In this paper, Linear-Quadratic (LQ) differential games are studied, focusing on the notion of solution provided by linear feedback Nash equilibria. It is well-known that such strategies are related to the solution of coupled algebraic Riccati equations,
C. Possieri, M. Sassano
semanticscholar   +1 more source

Noncommutative differential geometry of matrix algebras

Journal of Mathematical Physics, 1990
The noncommutative differential geometry of the algebra Mn (C) of complex n×n matrices is investigated. The role of the algebra of differential forms is played by the graded differential algebra C(sl(n,C),Mn (C))=Mn (C)⊗Λsl(n,C)*,sl(n,C) acting by inner derivations on Mn (C).
Dubois-Violette, Michel, Kerner, Richard, Madore, John   +2 more
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Differential geometry on Grassmann algebras

Letters in Mathematical Physics, 1976
H. C. Lee [1] developed the analogue of Riemannian geometry on a real symplectic manifold — the fundamental skew two-form taking the place of the symmetric tensor. The usual Riemannian concepts do not adapt themselves very well, thus ‘curvature’ is represented by a tensor of the third rank and ‘Killing's equations’ now involve this ‘curvature tensor ...
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Algebraic geometry of Abel differential equation

Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 2012
A solution $$y(x)$$ of an Abel differential equation $$(1) \ y^{\prime }=p(x)y^2 + q(x) y^3$$ is called “closed” on
Clara Shikhelman, Y. Yomdin, Y. Shelah, Sh. Giat   +3 more
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Rational Real Algebraic Models of Compact Differential Surfaces with Circle Actions

Polynomial Rings and Affine Algebraic Geometry, 2018
We give an algebro-geometric classification of smooth real affine algebraic surfaces endowed with an effective action of the real algebraic circle group $\mathbb{S}^1$ up to equivariant isomorphisms.
A. Dubouloz, Charlie Petitjean
semanticscholar   +1 more source