| | SLO | ENG | Cookies and privacy

Bigger font | Smaller font
First page > Show document

Show document Help

Title:POLINOMSKA PELLOVA ENAČBA
Authors:ID Detela, Anita (Author)
ID Eremita, Daniel (Mentor) More about this mentor... New window
Files:.pdf UNI_Detela_Anita_2010.pdf (444,07 KB)
MD5: F20055E71FC92BF640BFA73D264CC909
PID: 20.500.12556/dkum/d69bb7f1-d69d-4843-9401-6dc8e5477741
 
Language:Slovenian
Work type:Undergraduate thesis
Organization:FNM - Faculty of Natural Sciences and Mathematics
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 skozi diplomsko delo. 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. Zatem 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:kolobar, polinom, Masonov izrek, Pellova enačba, polinomska Pellova enačba
Place of publishing:Maribor
Publisher:[A. Detela]
Year of publishing:2010
PID:20.500.12556/DKUM-14066 New window
UDC:51(043.2)
COBISS.SI-ID:17681416 New window
NUK URN:URN:SI:UM:DK:U1DDFCJK
Publication date in DKUM:17.06.2010
Views:3523
Downloads:260
Metadata:XML DC-XML DC-RDF
Categories:FNM
:
DETELA, Anita, 2010, POLINOMSKA PELLOVA ENAČBA [online]. Bachelor’s thesis. Maribor : A. Detela. [Accessed 13 April 2025]. Retrieved from: https://dk.um.si/IzpisGradiva.php?lang=eng&id=14066
Copy citation
  
Average score:
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
(0 votes)
Your score:Voting is allowed only for logged in users.
Share:Bookmark and Share


Hover the mouse pointer over a document title to show the abstract or click on the title to get all document metadata.

Secondary language

Language:English
Title:POLYNOMIAL PELL'S EQUATION
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.
Keywords:ring, polynomial, Mason's theorem, Pell's equation, polynomial Pell's equation


Comments

Leave comment

You must log in to leave a comment.

Comments (0)
0 - 0 / 0
 
There are no comments!

Back
Logos of partners University of Maribor University of Ljubljana University of Primorska University of Nova Gorica