# L. Rédei's Lacunary Polynomials Over Finite Fields PDF

By L. Rédei

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No misunderstandings will arise. ,/? —1) of two equal polynomials, is based on the component representation; in most cases we immediately go over (without further reference) to the reduced components. Component comparison is often better than coefficient comparison; its superiority originates in the finiteness of the number of the components, so that component comparison may be effective even in cases where coefficient comparison fails. In particular, it can be a useful tool for the solution of polynomial equations, leading to a system of polynomial equations in which the (reduced) components of the unknowns occur as new unknowns.

14. The reciprocal a, /^-polynomial equation and the a, 6-power series equation In order to facilitate the solution of the a, 6-polynomial equation xa{x)p + b{x)p = χ"α(χ)ρ-2 + α(χ)ρ-3ο(χ) (1) let us use on it the transformation x -- — by which we shall understand that, x instead of a(x) and b(x), the reciprocal polynomials â(x) = x ... £ 2 a\^\, ,. E(x) = x* are introduced as new unknowns. From (2) the similar formulae (2) b\-\ a O O - x V ä j l j , *(*) = jc'r«15[l) (3) can be inferred. e. _ι_ Ιζλ = x~r~iP~2)^ P 2 ä(x)p+x _ Σζλ P 2 B(x)p = â(x)p-2 + x~(P~2)~^ â(x)p-*B(x), â(x)p + xh(x)p = α(χ)ρ-2 + χ"α(χ)ρ-3Β(χ) r = ^; â(x)9E(x)tF[x]; 5(0) = 1 ; 5<>, βο ^ (4) r _^L where the conditions, indicated in brackets, follow from those in (1) and from (2).

Let xk— 1 (k\q — 1) be called the special Euler binomial of degree k. This binomial lies in Fp[x]. 28 I PRELIMINARIES AND FORMULATION OF PROBLEMS I, II, III NOTE 1. , q—2 }, then the question concerns polynomials of the form f(x) = α0 + α 1 χ+···+α ί Ζ _ 2 χ ί *- 2 («o *q-2 €F). a0 Ui then it follows from the generalized theorem of König-Rados (see Rédei [9]) that f(x) has exactly q — 1 — r zeros in F which are different from one another and from 0. Concerning a similar theorem see Rédei-Turân [10].