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National Repository of Grey Literature 2 records found  Search took 0.01 seconds. 
Algebraic inequalities over the real numbers
 Raclavský, Marek ; Šťovíček, Jan (advisor) ; Příhoda, Pavel (referee)
This thesis analyses the semialgebraic sets, that is, a finite union of solu- tions to a finite sequence of polynomial inequalities. We introduce a notion of cylindrical algebraic decomposition as a tool for the construction of a semialge- braic stratification and a triangulation of a semialgebraic set. On this basis, we prove several important and well-known results of real algebraic geometry, such as Hardt's semialgebraic triviality or Sard's theorem. Drawing on Morse theory, we finally give a proof of a Thom-Milnor bound for a sum of Betti numbers of a real algebraic set. 1
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Rational points on elliptic curves
 Raclavský, Marek ; Stanovský, David (advisor) ; Šťovíček, Jan (referee)
This thesis concerns with rational points on elliptic curves. By the Mordell theorem we know that the group of rational points on elliptic curve is finitely generated. First, we study torsion subgroup, which turns out to be well described by theorem of Nagell-Lutz. Next, we focus on torsion-free part, which is characterized by the notion of rank. The thesis consists of solved problems and we also provide a summary of theoretical foundations. We find points of finite order on particular elliptic curves and compute their ranks. Powered by TCPDF (www.tcpdf.org)
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1 Raclavský, Martin
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