Az Egri Ho Si Minh Tanárképző Főiskola Tud. Közleményei. 1987. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 18/11)
Bogdan Tropak: Some algebraic properties of linear recurrences
- 77 Without, loos of the generality we can assume that the roots of gCx) are » ct , . . . , ct a , f3 , . . . , and that cx^ are the roots of uCx) and ß. of v(x). ' J Now by the definition of F we have 9 a r p g( xo"-" xk-J - n n * ß « i=i tj=i j = Fu[ Z0> - > Z 8-J Fv( Y0> - > Yr-J what ends the proof. Theorem 3. If gCx) = g 4Cx) ...g^Cx) is a decomposition of g(x) on irreducible factors then <93 F s(x o X^,]» 1 1 r v r where X. c j 3 are linear forms in . . . , X. and F are t O* ' k-1 g ^ forms associated to g^ Cx) , irreducible over the rational field and conversely if Fg( X0>--Xk.l] = Fl[ X0>- Xk-J - Fr[ X0'-- ' X Ic-J is a decomposition of F^ on irreducible factors then gCx) is decomposable on r irreducible factors g^Cx) ,...,g^Cx) , say and F has the form C9). Proo f: By Lemma 2 it is enough to prove that if Fg [ Xo> • * ' > Xk-J= Fi [ Xo> • • ' > Xk-i]' Fa [ Xo> • ' ' > Xk -i] with not constant F , F 2 then gCx) is reducible.
