LOCAL MINIMAL POLYNOMIALS OVER FINITE FIELDS

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1 Maria T* Acosta-de-Orozeo Departent of Matheatics, Southwest Texas State University, San Marcos, TX Javier Goez-Calderon Departent of Matheatics, Penn State University, New Kensington Capus, New Kensington, PA (Subitted June 1994) 1. INTRODUCTION Let F q denote the finite field of order q = p e, where q is an odd prie. If f(x) is a polynoial of degree d>\ over F then it is clear that d + l<v(f) = \{f(x):xef}\<q, where [w] denotes the greatest integer less than or equal to w. We say that fix) polynoial ifv(f) = q, and f(x) is a inial value set polynoial if V(f) = q-l d + 1. perutation A polynoial fix, y) with coefficients in F q is a local perutation (inial value set) polynoial over F if f(a, x) and f(x, b) are perutation (inial value set) polynoials in x for all a and b in F q. Local perutation polynoials have been studied by Mullen in [5] and [6]. In this note we will consider local inial polynoials of sall degree (< <Jq) on both x and y. We will show that there are only five classes of local inial polynoials. Naely, (a) f(x,y) = ax Y n + bx + cy n +d,,n\(g-1), (b) f(x,y) = (ax + by + c) +d,\(q-l), (c) f(x,y) = ax 2 Y"+bX 2 +cx + dy"+e,n\(q-l), (d) f(x,y) = ax Y 2 +by 2 +cy + dx +e,\(q-l) 9 d (e) f{x,y) = ax 2 Y 2 +bx 2 +cy 2 +dx + ey + gxy + h. where X = (x- x Q ) and Y = (y - y 0 ) with x 0, y 0 in F q. 2. THEOREM ANB PROOF Miinial value set polynoials have been studied by several authors. L. Carlitz, D. J. Lewis, W. H. Mills, and E. Strauss [2] showed that, when q is a prie and d- deg(/) <q, all inial value set polynoials with V(f) > 3 have the for f{x) - a(x + b) d +c with d dividing q-l. Later, W. H. Mills [4] gave a coplete characterization of inial value set polynoials over arbitrary finite fields with d <Jq. A weakened for of Mills's results can be stated as follows: 1996] 139

2 Lea 1 (Mills): If F is a finite field with q eleents and f(x) is a onic polynoial over F of degree d prie to q, then iply d<^ and V(f) = q-\ + 1 d\(q-\) and f(x) = (x + b) d +c. For other related results, see [1] and [3]. We are now ready for our result. Theore 2: Let F q denote a finite field of order q- p e!, where/? is an odd prie. Let n /=0 y'=0 denote a polynoial with coefficients in F q. Assue that, n, n-l, and -\ are relatively prie to q and \<,n<^q. Assue a n (x)b (y) & 0 for all x, y in F. Then f(x,y) is a local inial polynoial if and only if f(x, y) has one of the following fors: (a) f(x,y) = ax Y n +bx +cy"+d,rn,n\(q-l\ (h) f(x,y) = (ax + by + cr+d,\(q-l), (c) f(x, y) = ax 2 Y n +bx 2 +cx + dy n +e,n\(q -I), (d) f(x,y) = ax Y 2 + by 2 +cy + dx +e,\(q-1), and (e) f(x,y) = ax 2 Y 2 +bx 2 +cy 2 +dx + ey + gxy+h. where X = (x-x 0 ) and Y- (y-y Q ) with x 0,y 0 inf q. Proof: If f(x, y) is one of the fors (a)-(e), then it is easy to see that f(x, y) is a local inial value set polynoial. Now, let n denote a local inial value set polynoial over F q satisfying: (i) \<,n<jq, (ii) {ni - \)(n -1), q) = 1, (HI) ^ ( x ^ O O ^ O f o r a l l x ^ i n ^. Also, and without loss of generality, assue that <n and n>3 [n = 2 gives for (e)]. Then, by Lea 1, f(x 9 y) = a n (xi. «a (x)) n"a" n \x) (i) K x+m4t' + *bo-- ^ "»*«(y) «r*r 1 0')' (2) 140 [MAY

3 for all x, y in F and, n \ (q ~ 1). Hence, a n{x)y + ^=M^ + f l b ( x K-i ( x ) _ftim =«rw w+^i+wr'w w J for all x, 3; in F q. Further, since \<<n<^jq, polynoials. Therefore, C i ( y ) W (3) equation (3) also establishes the equality of the K~ 2 (y)\ a:-\x)y" + a:-\x)a n _ x {x)y n ^ ^ - y + a 0 {x)at\x) = a" n -\x)\ Hence, n\a7 3 (yfe(y)^- 3, x O 0 a^ (x) divides ~ y + - >,«-2 and, consequently, ^~ 2 (x) divides a"lj(x). Now, if g(x) is an irreducible factor of a n (x) so that g c (x) a (x) but g" c+1 (x) a w (x), then g e (x) divides a n _ x {x) for soe integer e such that 1 < c(n - 2) < (w - l)e. Therefore, since deg(g(x)) > 2, e < c -1 iplies c(?? - 2) < (w - l)(c -1) or 7? - l < c < y < y, a contradiction. Thus, a M (x) divides <Vi(x). Case 1. a w _ 1 (x) = 0. Then, by (1), f(x,y) = a n (x)y n +a 0 (x) = V/=o y /=o /=o = (any n+a x! a n-\y 0)\ + a 0 - l + a W + % ) " \anf+a, y- X Hence, f(x, y) has the for (c) or > 3 and \-i a y + % or a n-\y + a 0 - l \ =(«nffl /+«oj -' _1 Ky+%) (4) for all 3; in F and / = 1,2,...,. So, if a w = 0, then a n _ x - 0 and we obtain f(x,y) = a { 0 x 1 M Q-i ^ O w - l +<w +%>v w %» -*0w» 1996] 141

4 where a Q a n0 & 0. On the other hand, if a n ^ 0, then, again by (4), a 0 a 0-l Therefore, either f(x, y) has the for (c) or I n \ / n \ x + a iy+a 0 ) +anqyn+aoq_{a y +a 0^y = (a n y n +a Q )\x + {a n y + +a<**f o )) + %) H - J=± a nj (a (a n y n +a 0 ) n y" +<*o) and f(x, y) has the for (a). Case 2. a^x^a^ix) * 0. Then, by (1), dog(a n (x))hri-l)dj^^)<. Hence, either d e g ( ^ ^ } ) = 0 or d e g ( ^ ^ ) = 1 and deg(a (x)) = 0. First, we assue that deg(^g^) = 1 and deg(a (x)) = 0. Thus" n -1 < < n and ; f(x, y\= A x (y + a x x.+ c x ) n + g(x), where A x a x ^ 0 and g(x) denotes a polynoial of degree less than or equal to n. Now, -n-\ gives b (y) = b n _ l (y) = na"~ l (y + c l ) + c 2, a contradiction to (iii). Thus, b (y)=b n (y) is a constant polynoial, deg( ^"ff) = 1 and where A 2 a 2 ^ 0 and h(y) denotes a polynoial of degree less than or equal to n = w. Therefore, there exist constants A 3, a 3, and c 3 such that «n /=0 7=0 W where #(x) = Z 7 =0 ^ obtain anc * ^00-2f=o ^ y Now we copare the coefficients of x"~ 7 y in (5) to A 3 "p = A 2\j)4 for i = 1,..., n - 1. Since in -1,9) = 1, it follows that A^- A 3 and a 2 =a 3. Thus, coparing the coefficients of x" -2 }>, c 2 = c 3. Therefore, g(x) = A(y) = t/ for soe constant d, and which has the for (b). Now we assue that deg(^ *\/) = 0. Then f(x,y) = A(x + ay + c) n +d 142 [MAY

5 for soe a <=F q. Therefore, f(x, y-a) =a n (x)y" + g(x), which is a polynoial already considered in Case 1. This copletes Case 2 and the proof for < n. lfn<, then a siilar arguent will provide for (d). The next exaple illustrates the necessity of the condition (n -1, q) - 1. Exaple: Then For a in F n, let f(x, y) denote the polynoial f(x, y) = 2x 4 + x 3 y + xy 3 + j 4 + lax 3 +ay 3 + 2a 3 x + a 3 y. f(x, y) = (x + y + a) 4 + x 4 +ax: 3 +a 3 x + 2a 4 = 2(x + 2j/ + a ) / + a 4. Therefore, since 4 80, f(x, y) is a local inial polynoial that is not in the list (a)-(e). REFERENCES 1. M. Acosta-de-Orozco & J. Goez-Calderon. "Polynoials with Minial Value Set over Galois Rings." International Journal of Matheatics and Matheatical Science 14 (1991): L. Carlitz, D. J. Lewis, W. H. Mills, & E. G. Strauss. "Polynoials Over Finite Fields with Minial Value Set." Matheatika 8 (1961): J. Goez-Calderon. "A Note on Polynoials with Minial Value Set Over Finite Fields." Matheatika 35 (1988): W. H. Mills. "Polynoials with Minial Value Sets." Pacific Journal of Matheatics 14 (1964): G. L. Mullen. "Local Perutation Polynoials Over Z p." The Fibonacci Quarterly 18.2 (1980): G. L. Mullen. "Local Perutation Polynoials in Three Variables Over Z p." The Fibonacci Quarterly 18.3 (1980): AMS Classification Nubers: 11T06, 12E ] 143