Certifying Reality of Projections
| Autor: | Samantha N. Sherman, Avinash Kulkarni, Jonathan D. Hauenstein, Emre Can Sertöz |
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| Rok vydání: | 2018 |
| Předmět: |
Quadratic growth
Polynomial 010102 general mathematics System of polynomial equations 010103 numerical & computational mathematics 01 natural sciences Square (algebra) law.invention symbols.namesake Invertible matrix Rate of convergence law Genus (mathematics) ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION symbols Applied mathematics 0101 mathematics Newton's method Mathematics |
| Zdroj: | Mathematical Software – ICMS 2018 ISBN: 9783319964171 ICMS |
| Popis: | Computational tools in numerical algebraic geometry can be used to numerically approximate solutions to a system of polynomial equations. If the system is well-constrained (i.e., square), Newton’s method is locally quadratically convergent near each nonsingular solution. In such cases, Smale’s alpha theory can be used to certify that a given point is in the quadratic convergence basin of some solution. This was extended to certifiably determine the reality of the corresponding solution when the polynomial system is real. Using the theory of Newton-invariant sets, we certifiably decide the reality of projections of solutions. We apply this method to certifiably count the number of real and totally real tritangent planes for instances of curves of genus 4. |
| Databáze: | OpenAIRE |
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