(received April 9, 1967) Let p denote a prime number and let k P

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1 ON EXTREMAL OLYNOMIALS Kenneth S. Williams (eceived Apil 9, 1967) Let p denote a pime numbe and let k denote the finite field of p elements. Let f(x) E k [x] be of fixed degee d 2 2. We suppose that p is also fixed, lage compaed with d, say, p 2 po(d). By V(f) we denote the numbe of distinct values of 1. f(x), x E k. We call f maximal ~f V(f) = p and quasi-maximal 2 if but it is not known unde what conditions the convese holds. V(f) -- p t O(1). Clealy a maximal polynomial is quasi-maximal dv(f) 2 p, the minimum possible value of V(f) is - d When f(x) = x and p E 1 (mod d), V(f) = d > [y] As t 1. is in fact the actual minimum. If V(f) = t 1 we call f a d minimal polynomial and if V(f) L- t O(1) a quasi-minimal polyd nomial. Clealy a minimal polynomial is a quasi-minimal poly- nomial and Moclell has noted in an addendum to 171 that the convtse is tue fo pzp (d). It seems easonable to conjectue that a 0 quasi-maximal polynomial is maximal fo p 2 p (d). 0 It is the pupose of this pape to genealize the ideas of quasi-maximal and quasi-minimal. We set Dickson [h] calls such a polynomial a substitution polynomial. 2 We shall see late that these ae the exceptional polynomials of Davenpot and Lewis [5]. (See Coollay 1 and Theoem 2. ) Canad. Math. Bull. vol. 10, no. 4, 1967

2 and call f(x) an extemal polynomial of index if, in the (unique) clecomposition of f"(x, y) into ieducible factos in kp[x y], thee ae B linea factos and no non-linea absolutely ieducible 4 factos. Clealy 0 - of index 1 when p 5 3 (mod 4) since < 1 5 d-1. Fo example, f(x) -- x is extemal is ieducible but not absolutely ieducible. When p s 1 (mod 4) 2 the-e exists w E k such that w = -1 so that 4 hence f(x) = x is extemal of index 3 in this case. On the othe 3 hand, f(x) = x + x is not an extemal polynomial as 3 (x tx) - (y ty) = x txy t y tl x- Y is absolutely ieducible in li [x, y] fo any pime p > 3. THEOREM 1. I f(x) is extemal of index B then oof. As f(x) is extenlal of index we can wite whee each 2, (x, y) i~ linea so that 1 (possibly 0) is the index 1 of f and each h (x, y) is ieducible but not al~solutely ieducible J in k [, )-]. Clealy no two of gl. g ae associates and none is associated with (x-y). Let

3 and suppose that some a. = 0. Then 1 f(x) - f(y) = (x-y)(b.y+ci)g(x, 1 y) fo some g(x, y) E k [x, y]. Now b. # 0, othewis g. would 1 1 not be linea, so on taking y = - c. /b. we have 1 1 contadicting d 2. Hence no a. = 0 and sin~ilaly no b. = Q Set a = n a., di = bi/a. 1 and e, = c,/a, so Lhat i=i Q 111 f'?(x, y).- a n (xtd.~ 1 i =1 + e,) n h.(x, y). jzl J Now let N ( = 2, 3,..., d) denote the, numbe of solutjons of f(x ) = f(x2) =... = f(x) 1 with x. # x (i j, 1, j ). This systemhas the same numbe l j of solutions as the system = n (X + d.x + e.) n h.(xl,x) = 0 i =I j =I J with x. f x. (i # j, Zli, j ~.) Now it is known (see fo example J [i]) that if f(x, Y) E k [x, y] is ieducible but not absolutely ieducible then f(x, y) = 0 has O(1) solutions. Hence N

4 diffes fom the numbe N' of solutions, with X. # X. 1 J (i # j, z si,.j<), of by only O(1). Since fo any i and j with ifj, i<i, - jf x td.yte. = x td.y te. = J J has 0 o 1 solutions (gi, g. ae not associates) J whee N(i2, i3,..., i) denotes the numbe of solutions of (2 x td. x te =x +d x. te, = o with xi # X. (i fj, 21i, j1). Now J 1 i I x Sd x tei = x td. x -ke, =O l i m 1 1 n 1 ITI m n n with i =i gives x = x so n n m n Let N'(i..., i ) denote the numbe of solutions of (2) without 2' the conditions x, # x, (i f j, 2(i, jz). As l J x td. x te. = O (2< k< ) i l l < k I<

5 has one solution x fo each x k 1' Now, as the numbe of solutions of (whee nlf n, 25m, nl) is 0 o 1, giving Now let M ( fo which the equation f(x) = y has pecisely distinct oots in k. Then 1, 2,..., d ) denote the numbe of y E k and d

6 Thus so that d N as equied. COROLLARY 1. is quasi-maximal. If f(x) is extemal of index 0 then f f COROLLARY 2. If f(x) is extemal of index d- 1 then is quasi-minimal.

7 We now pove the conveses of coollaies 1 and 2. THEOREM 2. Lf f(x) is quasi-maximal then f(x) is extemal of index 0. oof. As f(x) is quasi-maximal V(f) = p + O(1). Set M=M M sothatfom(3) we have 2 d Eliminating M1 we have M = O(1) so that each M.(i 3 2) is 1 O(1). Hence N = O(1). Now if?(x, y) has t absolutely 2 ieducible factcs (linea o non-linea) in k [x, y] then by a esult of Lang and Weil (see fo example Lemma 8 in [ill), f"(x, y) = 0 has tp + O(p 'I2) solutions. Hence t = 0 as equied. THEOREM 3. Lf f(x) is quasi-minimal then f(x) is extemal of index d - 1. oof. This was poved by Modell in [7]. Finally we calculate the numbe Vn(f) of esidues of an extemal polynomial in the sequence 1, 2,..., h, whee h 5 p. (Hee we ae identifying the elements of k with the esidues 1, 2,..., p (mod p). ) We equie a lemma. LEMMA. If f(x) is an extemal polynomial of index 1 then, fo = 2,..., d, unifomly in t f 0, the implied constant depending only on d. (e(u) denotes exp(2~iu/p)).

8 oof. Fom the poof of the estimation of N in Theoem 1 we see that - 1 e(tf(x )) = > X..., X =o 1<i..., i <B x td x te. 1' - 2' 1 i xi#x. (i#j) i fi J m n -... f(x )=...=f(x ) m # n =x td x te 1 l i i 25m, nl = 0 I?,lI I x f + o(11 by a deep esult of Calitz and Uchiyama [3]. THEOKEM 4. If f(x) is an extenal polynomial of index the numbe V (f) of esidues of f(x) (mod p) in the h is given by set (1, 2,..., h) oof. solutions of Let N (h) ( = 2, 3,..., d ) denote the nunbc of with yt {1,2,..., h) and xi#x. (ifj). Then J h N (h) = C x ' y=l x,..., x 1 1, whee the dash ( ' ) denotes summation ove xi,..., x satisfying x, #x. (i#j) and f(x ) =... =f(x)= y. Thus 1 J 1

9 by the lemma and the familia esult p-l h C 1 C e(-tz)((p logp. t=l z=l Hence appealing totheoem 1 we obtain N(h) = 1 (f-l)... (1 - ( - ~ ) ) h + ~ log ( p) ~ l ~ ~ Now if M (h) ( = 1, 2,..., d) denotes the numbe of y E {I, 2,..., h) fo which the equation f(x) = y has pecisely distinct oots in k we have and The fist of these is obvious and the second is due to Modell [8]. Coesponding to (4) we have N (h) = C s(s- l)... (s - (- l))ms(h) s =

10 and the est of the poof is the same as in Theoem 1 with Vh(f), M (h). N (h), h eplacing V(f), Mj N, p espectively. This poves a conjectue of the autho [9] in the case of extemal polynomials. When the index 1 is 2 1 it shows that the least 112 positive non-esidue of f(x) (mod p) is O(p log p). This has been poved fo moe geneal polynomials, without obtaining an asymptotic fomula fo Vh(f), by Bombiei and Davenpot [2], using the ecent wok of Bombiei on the L-functions coesponding to multiple exponential sums. REFERENCES B. J. Bich and D. J. Lewis, $-adic foms. Jou. Indian Math. Soc., 23 (1959), E. Bombiei and H. Davenpot, On two poblems of Modell. Ame. J. Math., 88 (1966), L. Calitz and S. Uchiyama, Bounds fo exponential sums. Duke Math. Jou., 24 (1957), J. H. H. Chalk and K. S. Williams, The distibution of solutions of conguences. Mathematika, 12 (1965), H. Davenpot and D. J. Lewis, Notes on conguences I. Quat. J. Math. Oxfod (2), 14 (1963), L. E. Dickson, Linea goups. Dove ublications, Inc., N.Y. (1958), L. J. Modell, A conguence poblem of E. G. Staus. Jou. Lond. Math. Soc., 38 (1963), L. J. Modell, On the least esidue and non-esidue of a polynomial. Jou. Lond. Math. Soc., 38 (1963), K. S. Williams, The distibution of the esidues of a quatic polynomial. To appea in the Glasgow Math. Jou. Caleton Univesity, Ottawa

H.W.GOULD West Virginia University, Morgan town, West Virginia 26506

H.W.GOULD West Virginia University, Morgan town, West Virginia 26506 A F I B O N A C C I F O R M U L A OF LUCAS A N D ITS SUBSEQUENT M A N I F E S T A T I O N S A N D R E D I S C O V E R I E S H.W.GOULD West Viginia Univesity, Mogan town, West Viginia 26506 Almost eveyone