Divisibility properties of generalized Laguerre polynomials

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andnF(x):= 'Lajdjx"-Jj=Ofor integers aD, at, ... , all' In the sequel we will denote by 111,112, ... effectivelycomputable absolute positive real constants. We have the following result.Theorem 1. Let e = 1/5 if a < 0 and e = 1/112 if a > O. Let aD, al, ... , an be anyintegers with laoI= IallI= 1. Then there exists a constant 11 I such thatfor all k withn111 < k
the polynomial F (x ) does not have a factor ofdegree k.The proofs of Theorems I and 3 split in two parts. First by using p-adicarguments, especially the p-adic Newton polygon, we reduce the problem to findinga prime p having certain properties. By considering several cases depending on kthe proofwill be finished.In the next section we introduce the Newton polygon with respect to a prime pand give some auxiliary results on this polygon, as well as on primes in certainintervals. Afterwards, we will give the proofs of Theorem I and 3, respectively.2. NEWTON POLYGONS AND PRELIMINARIES ON PRIMESFor a prime p let vp be a p-adic valuation, i.e. for a positive integer n we have thatvl'(n) is the largest integer such that pvp(n) In (we will also use the notation pVp(lI) lin,for short) and vp(O) = 00. We shall also write v for "» when it will be clear fromthe context which p we are taking. Let g(x) = LJ-=o b[x":j E Z[x] with bobll :j:. O.The p-adic Newton polygon (or just Newton polygon) for g(x) with respect to theprime p is now defined as the polygonal path formed by the lower edges of theconvex hull ofthe pointsThe left-most endpoint is (0, v(bo)) and the right-most endpoint is (n, v(bll)).Moreover, the endpoints of each edge belong to the above set and the slopes ofthe edges strictly increase from left to right.Then Filaseta [3, Lemma 2] proved the following result.Lemma 1. Let k and i be integers with k > e;:: 0 and k ~ n12. Suppose thatIlg(x) = Lbjx"- j E Z[x]j-=Oand p is a prime such that p t bo, p Ibj for all j E Ii + I, ... , n} and the right-mostedge ofthe Newton polygon for g(x) with respect to p has slope < 11k. Thenforany integers ao, aI, ... , all with laoI= IallI=I, the polynomial11G(x) = Lajbjx ll - jj-=Ocannot have a factor with degree in the interval [i + 1,k].We apply this lemma in the case L~a)(x), in fact we will use g(x) = lex) andG(x) = F(x).The next result is an estimate on the difference between consecutive primes andit will be used in the proof of Theorem I.221
Page 2 and 3: for ex E lIt n EN. An important ins
Page 6 and 7: Lemma 2 (Lou and Yao [12]), Denote
Page 8 and 9: so far together with the hypothesis
Page 10 and 11: Now we turn to the second case and
Page 12 and 13: with v q (b q ) maximal. We mention
Page 14 and 15: Hence, by the same lemma we get tha

Divisibility properties of generalized Laguerre polynomials

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