20.500.12556/DKUM-14066 POLINOMSKA PELLOVA ENAČBA POLYNOMIAL PELL'S EQUATION 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. 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. kolobar polinom Masonov izrek Pellova enačba polinomska Pellova enačba ring polynomial Mason's theorem Pell's equation polynomial Pell's equation true true false [A. Detela] Slovenski jezik Angleški jezik Diplomsko delo 2010-05-24 09:34:15 2010-06-17 15:40:48 2022-04-12 11:42:27 2023-12-17 03:13:41 2010 0 Maribor 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 51(043.2) 17681416 URN:SI:UM:DK:U1DDFCJK UNI_Detela_Anita_2010.pdf UNI_Detela_Anita_2010.pdf 1 F20055E71FC92BF640BFA73D264CC909 c6290f24395c71aae5e98f78dee77bf670bcd1707360d514d5e3d3b51b406e1d 105bb6a4-7c04-11eb-bb7a-00155d0001ca 20.500.12556/dkum/d69bb7f1-d69d-4843-9401-6dc8e5477741 https://dk.um.si/Dokument.php?lang=slv&id=14728 Fakulteta za naravoslovje in matematiko 0 0 0