Certifying solutions to polynomial systems over Q
Başlık çevirisi mevcut değil.
- Tez No: 502137
- Danışmanlar: Dr. AGNES SZANTO, Dr. JONATHAN HAUENSTEIN
- Tez Türü: Doktora
- Konular: Matematik, Mathematics
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 2016
- Dil: İngilizce
- Üniversite: North Carolina State University
- Enstitü: Yurtdışı Enstitü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 93
Özet
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Özet (Çeviri)
This dissertation is concerned with certifying that a given point is near an exact root of a polynomial system with rational coefficients. In Chapter 1, we provide prerequisite background material from algebraic geometry, number theory and matrix theory.Most importantlywe introduce the problem of certification and its classical solution with -theory on well-constrained systems. In Chapter 2, we establish a method to certify approximate solutions of an overdetermined system with rational coefficients.The difficulty lies in the fact that consistency of overdetermined systems is not a continuous property. Our certification is based on hybrid symbolic-numeric methods to compute an exact rational univariate representation (RUR) of a component of the input system from approximate roots. For overdetermined polynomial systems with simple roots, we compute an initial RUR from approximate roots. The accuracy of the RUR is increased via Newton iterations until the exact RUR is found, which we certify using exact arithmetic. Since the RUR is well-constrained, we can use it to certify the given approximate roots using -theory. We prove that our algorithms have complexity that are polynomial in the input plus the output size upon successful convergence, and we use worst case upper bounds for termination when our iteration does not converge to an exact RUR. In Chapter 3, we focus on certifying isolated singular roots.We use a determinantal formof the isosingular deflation, which adds new polynomials to the original system without introducing new variables. The resulting polynomial system is overdetermined, but the roots are now simple, thereby reducing the problem to the overdetermined case. Finally, in Chapter 4 we propose a method to certify approximate real solutions of polynomial systems using the signature ofHermite matrices.We use approximate roots to construct theHermite matrices, then rationalize the entries with a preset bound on denominators. Once we ensure that the rationalized Hermite matrices in fact correspond to the given system, one can use the Hermite's theorem to certify a real approximate solutions of the given polynomial system.
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