Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1994. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 22)

HERENDI, T. and PETHŐ, A., Trinomials, which are divisible by quadratic polynomials

64 Tamás Herendi and Attila Pethő From this we get (1) (1 + vT+4t) = e

n

+i • (1 - vT+4Í). , where £

n_|_i is a n + l-th primitive root of unity, and 1 < j < n. Also j (if it is integer), because in this case 1 -f \/\ -f 41 — yj\ + 41 — 1 woula.^iold ei Vti.;. • which is impossible. From équation (1) we obtain t — — . j "

+

1 .

2. The next question is how many différent values t can have. If j = 0 then t = — | and it is easy to see that F

m ^ 0 for any m = 0,1,2,.. .. C í' Further - , = — , ; n+

1 , where 0 < i, j < n + 1 and i j if and only if i+j = n+l. It me ans that t has at most [y] différent values. We know [*

] f e \ that deg {F

n(x)) = [f ] which implies that F

n(x) = ü z + > n-f

l • j = l V ) t} (b) F

n(x) has a rational root if and only if — . j "+

1 2 = E for j G 9 2 {1,2,..., [f]},p,q G Z, ç/0. This is équivalent to0 = p-(^

+

1+ l) + <? • fn+1 = P • (fn+1 ) + (? + 2P)£n+l + P- Henc

e 1 has to be a root of the polynomial pz

2 + (ç + 2p)x + p, i.e. is rational or a quadratic algebraic number. But it is known that if £ is a k-th primitive root of unity, then its degree is <£>(&), where <p(k) is the Euler-function. ip(k) < 2 if and only if k G {1,2, 3,4,6}. From the proof of (a) it is clear that k > 2. If k = 3 then t = —1, if k = 4 then t = — | and if fc = 6 then t = — As is primitive k -th root of unity if n + 1 = j — k, thus F

n(x) has a rational root if and only if3|n+lor4|n + l, i.e. gcd(n + 1,12) > 3. In the next step some properties of the sériés {/

n(x, 2/)}^°

=_

0

0 are presented. Lemma 3. The series {f

n{x,y)}

(^L_

0

0 has the property 6on • fn(x, y) = y

n~

l mod 2 • fn — í (x, y) + X- fn—2 y)tin G Z, where , f 0, if n / 0 ifn = 0