Real solutions to systems of polynomial equations and parameter continuation
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Zachary A. Griffin
Abstract
Given a parameterized family of polynomial equations, a fundamental question is to determine upper and lower bounds on the number of real solutions a member of this family can have and, if possible, compute where the bounds are sharp. A computational approach to this problem was developed by Dietmaier in 1998 who used a local linearization procedure to move in the parameter space to change the number of real solutions. He used this approach to show that there exists a Stewart-Gough platform that attains the maximum of forty real assembly modes. Due to the necessary ill-conditioning near the discriminant locus, we propose replacing the local linearization near the discriminant locus with a homotopy-based method derived from the method of gradient descent arising in optimization. This new hybrid approach is then used to develop a new result in real enumerative geometry.
© 2015 by Walter de Gruyter Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Division algebras and transitivity of group actions on buildings
- Projections of del Pezzo surfaces and Calabi–Yau threefolds
- Almost soliton duality
- Remarks on Kähler–Ricci solitons
- Real solutions to systems of polynomial equations and parameter continuation
- Division pairs: a new approach to Moufang sets
- Very special divisors on 4-gonal real algebraic curves
- On the intersection of a Hermitian surface with an elliptic quadric
- Geometric properties of semitube domains
- Threefolds in ℙ6 of degree 12
Artikel in diesem Heft
- Frontmatter
- Division algebras and transitivity of group actions on buildings
- Projections of del Pezzo surfaces and Calabi–Yau threefolds
- Almost soliton duality
- Remarks on Kähler–Ricci solitons
- Real solutions to systems of polynomial equations and parameter continuation
- Division pairs: a new approach to Moufang sets
- Very special divisors on 4-gonal real algebraic curves
- On the intersection of a Hermitian surface with an elliptic quadric
- Geometric properties of semitube domains
- Threefolds in ℙ6 of degree 12
