Polinomska Pellova enačba

diplomsko delo

Anita Detela (Author), Daniel Eremita (Mentor)

Abstract

Polinomska Pellova enačba je enačba oblike P^2 - D Q^2 = 1, kjer je D dani polinom, P in Q pa sta neznana polinoma istih spremenljivk kot D in tudi njuni koeficienti so iz istega polja ali kolobarja kot koeficienti polinoma D. Glavni problem pri reševanju polinomske Pellove enačbe je ugotoviti, ali obstajajo netrivialne rešitve ali ne. Bistvo tega diplomskega dela je pokazati, da lahko opišemo rešitve polinomske Pellove enačbe v Z[X], če je znana ena rešitev iste enačbe (z istim D iz Z[X]) v kolobarju C[X]. Ko imamo enkrat rešitev (P,Q), kjer sta P, Q iz C[X], so vse rešitve v kolobarju Z[X] neke potence minimalne kompleksne rešitve. Prvo poglavje je namenjeno definiranju osnovnih pojmov, ki so pogosto uporabljeni v diplomskem delu. Razvita je tudi teorija, ki je potrebna kasneje za dokaz Masonovega izreka. V drugem poglavju je na kratko predstavljena Pellova enačba za števila in z njo povezane ugotovitve, ki so navdih pri raziskovanju polinomske Pellove enačbe, saj obstaja podobnost pri nekaterih sklepih. Glavna tema diplomskega dela je opisana v tretjem poglavju. S pomočjo Masonovega izreka zapišemo potreben pogoj za rešljivost polinomske Pellove enačbe in izkaže se, da je ta pogoj tudi zadosten, če je polinom D kvadraten polinom. Nato je podana popolna karakterizacija rešitev polinomske Pellove enačbe, v primeru, ko le-ta ima netrivialno rešitev. Zapisan je tudi dokaz posplošenega Nathansonovega rezultata. Na koncu je podanih nekaj primerov za polinom D četrte stopnje.

Keywords

matematika;polinomi;enačbe;kolobarji;Masonov izrek;Pellova enačba;diplomska dela;

Data

Language:	Slovenian
Year of publishing:	2010
Source:	Maribor
Typology:	2.11 - Undergraduate Thesis
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
Publisher:	[A. Detela]
UDC:	51(043.2)
COBISS:	17681416
Views:	2660
Downloads:	183
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	POLYNOMIAL PELL'S EQUATION
Secondary abstract:	Polynomial Pell's equation is an equation which is written in a form P^2 - D Q^2 = 1 where D is a given poynomial, P and Q are unknown polynomials in the same variables as D and with coefficients in the same field or ring as those of D. The main difficulty in solving polynomial Pell's equations is to determine whether non-trivial solutions exist or not. The aim of this graduation thesis is to show that we can describe solutions of polynomial Pell's equation in Z[X] if one solution of the same equation (with the same D in Z[X]) in the ring C[X] is known. Once we have a solution (P,Q) where P, Q in C[X], all solutions in Z[X] are certain powers of the minimal complex solution. In the first chapter we define fundamental notions which are frequently used through graduation thesis. There is also theory developed needed later for the proof of Mason's theorem. Pell's equation for integers is shortly introduced in the second chapter. There are some statements that inspire us by researching polynomial Pell's equation. We shall see that certain similar results can be obtained. The main theme of this graduation thesis is described in the third chapter. Using Mason's theorem we give the necessary condition for the solvability of the polynomial Pell's equation which turns out to be also a suficient condition if D is quadratic. We also obtain complete characterization of the solutions of the polynomial Pell's equation in case it has non-trivial solutions. The proof of generalized Nathanson's result is also written. At the end there are some examples for a given quartic polynomial D.
Secondary keywords:	ring;polynomial;Mason's theorem;Pell's equation;polynomial Pell's equation;
URN:	URN:SI:UM:
Type (COBISS):	Undergraduate thesis
Thesis comment:	Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Pages:	56 f.
Keywords (UDC):	mathematics;natural sciences;naravoslovne vede;matematika;mathematics;matematika;
ID:	18530