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
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 3 2 2 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. g2... - ae associates and none is associated with (x-y). Let
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. = 0. 1 1 Q Set a = n a., di = bi/a. 1 and e, = c,/a, so Lhat 1 1 1 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 1 1 1 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
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. = 0 1 1 1 1 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 1 1 2 1 2 2 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 < 1 - - k I<
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
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.
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)).
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 2 1 2 2 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
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 =
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), 11-32. E. Bombiei and H. Davenpot, On two poblems of Modell. Ame. J. Math., 88 (1966), 61-70. L. Calitz and S. Uchiyama, Bounds fo exponential sums. Duke Math. Jou., 24 (1957), 37-41. J. H. H. Chalk and K. S. Williams, The distibution of solutions of conguences. Mathematika, 12 (1965), 176-192. H. Davenpot and D. J. Lewis, Notes on conguences I. Quat. J. Math. Oxfod (2), 14 (1963), 51-60. L. E. Dickson, Linea goups. Dove ublications, Inc., N.Y. (1958), 54-64. L. J. Modell, A conguence poblem of E. G. Staus. Jou. Lond. Math. Soc., 38 (1963), 108-110. L. J. Modell, On the least esidue and non-esidue of a polynomial. Jou. Lond. Math. Soc., 38 (1963), 451-453. K. S. Williams, The distibution of the esidues of a quatic polynomial. To appea in the Glasgow Math. Jou. Caleton Univesity, Ottawa