INDEX BOUNDS FOR VALUE SETS OF POLYNOMIALS OVER FINITE FIELDS

Transcription

1 INDEX BOUNDS FOR VALUE SETS OF POLYNOMIALS OVER FINITE FIELDS GARY L. MULLEN, DAQING WAN, AND QIANG WANG We dedcate ths paper to the occaso of Harad Nederreter s 70-th brthday. Hs work o permutato poyomas over fte feds, ad more geeray, hs work so may areas of fte feds ad ther appcatos, has bee a huge ad astg sprato to a of us. Abstract. We provde a upper boud for the cardaty of the vaue set of a uvarate poyoma over a fte fed terms of the dex of the poyoma. Moreover, we study whe a poyoma vector map varabes s a permutato poyoma map, aga usg the dex tupe of the map. Ths aso provdes a upper boud for the vaue set of a poyoma map varabes. 1. troducto Let F q be the fte fed of q eemets wth characterstc p. The vaue set of a poyoma g over F q s the set V g of mages whe we vew g as a mappg from F q to tsef. Ceary g s a permutato poyoma (PP) of F q f ad oy f the cardaty V g of the vaue set V g of g s q. Asymptotc formuas such as V g = λ(g)q + O(q 1/2 ), where λ(g) s a costat depedg oy o certa Gaos groups assocated to g, ca be foud Brch ad Swerto-Dyer [7] ad Cohe [16]. Later, Wams [39] proved that amost a poyomas f are poyomas satsfyg λ(g) = ! 3! +( 1)d 1 1, where d s the degree of the poyoma g. There d! are aso severa resuts o expct bouds for the cardaty of vaue sets f g s ot a PP over F q ; see for exampe [33, 34]. Perhaps the most we-kow resut s due to Wa [34] who proved that f g s ot a PP the (1) V g q q 1 d. Usg resuts from group theory, Gurack ad Wa [20] further proved that f (d, q) = 1 the V g (47/63)q +O d ( q). Some progress o ower bouds of V g ca be foud [17, 36]. The cassfcato of mma vaue set poyomas (poyomas satsfyg V g = q/d ) ca be foud [11, 19, 27], ad [8] for a the mma 2000 Mathematcs Subject Cassfcato. 11T06. Key words ad phrases. poyomas, vaue sets, permutato poyomas, fte feds. Research of Daqg Wa ad Qag Wag was partay supported by NSF, NSERC of Caada, ad Natoa Natura Scece Foudato of Cha (Grat No ). 1

2 2 GARY L. MULLEN, DAQING WAN, AND QIANG WANG vaue set poyomas F q [x] whose set of vaues s a subfed of F q. More recety, agorthms ad compexty computg V g have bee studed [13]. A of these resuts reate V g to the degree d of g. I ths paper, we take a dfferet approach to study vaue sets. We ote that ay o-costat poyoma g F q [x] of degree q 1 ca be wrtte uquey as g(x) = a(x r f(x (q 1)/ )) + b wth dex defed beow. Namey, wrte g(x) = a(x + a 1 x a k x k ) + b, where a, a j 0, j = 1,..., k. The case that k = 0 s trva. Thus, we sha assume that k 1. Wrte k = r, the vashg order of x at 0 (.e., the owest degree of x g(x) b s r). The g(x) = a ( x r f(x (q 1)/ ) ) + b, where f(x) = x e 0 + a 1 x e a k 1 x e k 1 + ar, = q 1 gcd( r, r 1,..., r k 1, q 1) := q 1, s ad gcd(e 0, e 1,..., e k 1, ) = 1. The teger = q 1 s caed the dex of h(x). s The cocept of the dex of ay poyoma was frst troduced [2] ad s cosey reated to the cocept of the east dex of a cycotomc mappg poyoma [14, 32]. Ceary, the study of the vaue set of g(x) = a(x r f(x (q 1)/ ))+b over F q s equvaet to studyg the vaue set of g(x) = x r f(x (q 1)/ ) = x r f(x s ) over F q. If (r, (q 1)/) = 1, we say g s reduced form. Otherwse, f (r, (q 1)/) = t, the g(x) = g (x t ) where g (x) = x r/t f(x s/t ) s reduced form. I fact a permutato poyoma g must be reduced form. We ote that permutato poyomas of the form x r f(x s ) were studed by Wa ad Ld [35] 1991 ad more recety by may others [1, 2, 4, 5, 6, 15, 37, 43, 44, 45]. For more backgroud matera o permutato poyomas we refer to Chap. 7 of [24]. For a detaed survey see [22, 23, 28, 31] ad recet resuts see [3, 9, 10, 12, 18, 21, 38, 40, 41, 42]. We refer to Secto 8.1 of [29] for a detaed dscusso of PPs ad Secto 8.3 of [29] for a dscusso of vaue sets of poyomas over fte feds. I Secto 2, we study the vaue set probem terms of the dex of the poyoma g. I Theorem 2.1 we prove that f g s ot a PP the (2) V g q q 1. Our resut mproves Wa s resut whe the dex of a poyoma s strcty smaer tha the degree d. We ote that the dex of a poyoma s aways smaer tha the degree d as og as q 1. For exampe, the dex of ay permutato boma s aways ess tha or equa to the degree. I fact, the dex of poyomas s cosey reated to the cocept of the east dex of cycotomc permutatos. These permutatos terms of cycotomc cosets were studed by Nederreter ad Wterhof [32] ad Wag [37, 38]. Aso Secto 2 a geerc formua for V g terms of the umber of certa dstct cycotomc cosets s gve Proposto 2.3.

3 INDEX BOUNDS FOR VALUE SETS OF POLYNOMIALS OVER FINITE FIELDS 3 Let g : F q F q be a poyoma map varabes defed over F q, where s a postve teger. Deote by V g the umber of dstct vaues take by g(x 1,..., x ) as (x 1,..., x ) rus over F q. It s cear that V f q. If V f = q, the f s a permutato poyoma vector, see [24, Chapter 7]. Motvated by a ope probem rased by Lpto [25] hs computer scece bog, we exteded Wa s resut o upper bouds of vaue sets for uvarate poyomas to poyoma maps varabes [30]. More specfcay, we wrte g as a poyoma vector: (3) g(x 1,..., x ) = (g 1 (x 1,..., x ),..., g (x 1,..., x )), where each g (1 ) s a poyoma varabes over F q. The poyoma vector g duces a map from F q to F q. By reducg the poyoma vector g moduo the dea (x q 1 x 1,..., x q x ), we may assume that the degree of g each varabe s at most q 1 ad we may further assume that g s a o-costat map to avod the trva case. Let d deote the tota degree of f the varabes x 1,..., x ad et d = max{d 1,..., d }. The d satsfes 1 d (q 1). I partcuar, we proved { } Theorem 1.1. [30] If V g < q, the V g q m (q 1), q. d I Secto 3, we exted the cocept of dex of a uvarate poyoma to dex tupes for mutvarate poyomas g (x 1,..., x ) for 1 ad the poyoma vector map g(x 1,..., x ) respectvey. We remark that ay mutvarate poyoma g (x 1,..., x ) behaves as a mooma each subset of F q F q that s parttoed by the cycotomc cosets determed by the dex tupe ( () 1,..., () ). Smary, each coordate g (x 1,..., x ) of ay poyoma vector map g(x 1,..., x ) behaves as a mooma whe we vew the vector map as a cycotomc mappg. It turs out the dex tupe ( 1,..., ) of g(x 1,..., x ) ca be obtaed from dex tupes ( () 1,..., () ) of g (x 1,..., x ) s. Namey, = cm( (1),..., () ) for 1. The we study the extreme cases for the vaue set probem for poyoma maps of varabes. Namey, we descrbe whe a poyoma map g varabes s a permutato poyoma map. Essetay, each coordate poyoma of a permutato vector map behaves as a mooma terms of oy oe varabe each subset of F q F q that s parttoed by the cycotomc cosets determed by the dex tupe ( 1,..., ), aog wth other expct codtos as descrbed Theorem 3.5. I other words, each permutato vector map varabes cossts of uvarate cycotomc mooma permutatos together wth aother permutato o coordate varabes. As a coroary, we obta Theorem 1.2. Let g be a poyoma vector map from F q to F q wth the dex tupe ( 1,..., ) ad = max{ 1,..., } > 1. If V g < q the V g q q 1. Ths aso provdes aother aswer to Lpto s probem o the exstece of a Pcard jump for poyoma maps (roughy, f g msses oe vaue F q the g msses qute

4 4 GARY L. MULLEN, DAQING WAN, AND QIANG WANG a few). A exampe meetg ths upper boud s aso provded. We aso ote that our ew boud mproves the boud Theorem 1.1 whe d >. 2. Vaue sets of uvarate poyomas As expaed the troductory secto, the vaue set probem for a uvarate poyoma s equvaet to that for the poyoma g(x) = x r f(x s ). Here we wat to emphasze the parameter dex stead of the degree d. We ote from [37] that g(x) = x r f(x s ) s a PP f ad oy f gcd(r, s) = 1, ad f(ζ ) 0 for 0 1 where ζ s a prmtve -th root of uty, ad g(x) duces a permutato amog a the cycotomc cosets {C 0, C 1,, C 1 } = F q/(f q). I fact, we ca aways wrte g(x) terms of cycotomc mappgs g(x) = c x r f x C, where c = f(ζ ); more detas ca be foud [14, 32, 37, 38]. I partcuar, a poyoma g(x) F q [x] s caed a orthomorphsm f both g(x) ad g(x) x are permutato poyomas. Gve a fte, oempty set of postve tegers R, a poyoma g(x) s caed a R-orthomorphsm f g (r) (x) s a orthomorphsm of F q for a r R. (Here g (r) deotes the fucto g composed wth tsef r tmes.) We ote that [32] Nederreter ad Wterhof proved severa exstece resuts for cycotomc orthomorphsms ad cycotomc R-orthomorphsms of fte feds. Here we oy cocetrate o the permutato behavor of g(x). Theorem 2.1. Let g(x) = ax r f(x s ) + b (a 0) be a poyoma reduced form (.e., gcd(r, s) = 1) over F q wth dex. The V g > q q 1 f ad oy f g s a PP of F q. Proof. Wthout oss of geeraty, we ca assume a = 1 ad b = 0. Hece g(0) = 0. The codtos gcd(r, s) = 1 ad f(ζ ) 0 guaratee that a the mages of eemets each C are dstct ozero eemets. Because C = s = q 1, we cocude that V g > q q 1 f ad oy f there are more tha 1 ozero dstct mage sets of cycotomc cosets. Sce there are exacty ozero dstct mage sets of cycotomc cosets, we deduce that V g > q q 1 f ad oy f g s a PP of F q. Ths resut mproves Wa s resut (.e., Equato (1)) for arbtrary poyomas wth dex q 1. Ideed, f the dex q 1, the s q + 1 ad thus the degree d s + 1 >. Our resut aso works at east as good as Wa s resut [34] f we wat to verfy a arbtrary boma over a prme fed s a permutato usg the cotrapostve ower boud. Ideed, et g(x) = x d + ax m wth d > m > 0, be a arbtrary permutato boma over a prme fed F p. It s proved by Masuda ad Zeve [26] that gcd(d m, p 1) p 3/4 1/2 (> p 1). Here gcd(d m, p 1) turs out to be equa to s. So the dex = p 1 p 3/4 + 1/2 s (< p + 1) ad thus s > p 1. The d = m + es 1 + s p 3/4 + 1/2.

5 INDEX BOUNDS FOR VALUE SETS OF POLYNOMIALS OVER FINITE FIELDS 5 Hece p p 1 p p 1. However, we ote that d s strcty greater tha for ay d m > 1 or e > 1 as above. Coroary 2.2. Let g(x) = ax r f(x q 1 ) + b (a 0) be ay poyoma over F q wth dex > 1 ad s = q 1. Assume V g < q. The (a) If gcd(r, s) = 1 the V g q q 1. (b) If gcd(r, s) = t > 1 the V g q t Therefore we aways have V g q q 1. Proof. Wthout oss of geeraty, we ca assume a = 1 ad b = 0. Hece g(0) = 0. The case of gcd(r, s) = 1 foows from Theorem 2.1. If gcd(r, s) = t > 1, the g(x) = g 1 (x t ) for some poyoma g 1 F q [x]. Thus, V g V x t = q We ote t that q = q (t 1)(q 1) = q (q 1 q 1 ). Because t > 1 ad > 1, we must t t t have q 1 q 1 q 1 q 1 = q 1 q 1. Thus we have V t 2 2 g q q 1 both cases. I fact, we ca obta the foowg formua for the cardaty of the vaue set. Proposto 2.3. Let g(x) = ax r f(x s ) + b (a 0) be ay poyoma over F q wth dex = q 1 ad et gcd(r, s) = t. Let ξ be a fxed prmtve eemet of F s q. The V g = c s t + 1, or V g = c s t, where c = {(ξ r f(ξ s )) t = 0,..., 1}. Proof. Wthout oss of geeraty, we ca assume a = 1 ad b = 0. Hece g(0) = 0. Let C 0 be the subgroup of F q cosstg of a the -th powers of F q ad D 0 be the subgroup of F q cosstg of a the t-th powers. Let C = ξ C 0 for = 0,..., 1 be cycotomc cosets of F q duced by C 0. Note that g(x) = c x r whe x C, where c = f(ξ s ) for = 0,..., 1. We aso ote that x r maps C 0 oto D 0 whch cotas s dstct eemets. So t xr maps each coset C = ξ C 0 oto ξ r D 0. Therefore g maps C oto ξ r f(ξ s )D 0, whch coud be ether the set {0} or oe of the ozero cycotomc cosets of dex t. We observe that c s the umber of dstct cycotomc cosets of the form ξ r f(ξ s )D 0. Hece we have V g = c s +1 or c s, the atter happes t t whe some of c s g(x) = c x r equa g(0) = 0. From here t s straghtforward to obta a geerc ower boud s for ay ozero (r,s) poyoma. However, ths ower boud ca be mproved depedg o how much formato we kow about the coeffcets of g order to say more about ξ r f(ξ s ). We aso refer to [17] for a matrx method whch ca be used to obta a ower boud for the cardaty V g of the vaue set of a uvarate poyoma g over F q.

6 6 GARY L. MULLEN, DAQING WAN, AND QIANG WANG 3. Permutato poyoma vectors Let us frst cosder a mutvarate poyoma g(x 1,..., x ) over F q. As the uvarate case, we ca wrte g(x 1,..., x ) = x r 1 1 x r f(x s 1 1,..., x s ) + b where g(0,..., 0) = b, ad r 1,..., r are vashg orders of x 1,..., x g(x 1,..., x ) b at 0 respectvey (.e., the owest degree of x g(x 1,..., x ) b s r ), ad each s s the greatest commo dvsor of a the expoets of x from a mooma terms after factorg x r, together wth q 1 for 1 (.e., s s the greatest commo dvsor of a the expoets of x f(x s 1 1,..., x s ) together wth q 1). We ote that r 0 ths case stead of r 1 for uvarate poyomas. Let = q 1 s wth 1. The ( 1,..., ) s caed the dex tupe of the mutvarate poyoma g(x 1,..., x ). Exampe 3.1. Let g(x 1, x 2 ) = x 4 1x 5 2 x 2 1x x 5 2 over F 7. So (r 1, r 2 ) = (0, 5) s the par of vashg orders of x 1 ad x 2 at 0 respectvey. We ca wrte g(x 1, x 2 ) = x 5 2(x 4 1+x 2 1+3) = x 0 1x 5 2f(x 2 1, x 6 2) wth f(x 1, x 2 ) = x 2 1+x 1 +3 because s 1 = gcd(4, 2, 0, 6) = 2 ad s 2 = gcd(0, 0, 0, 6) = 6. Namey, ( 1, 2 ) = (3, 1) s the dex tupe of g. Smary, h(x 1, x 2 ) = 3x 1 x 3 2 2x 1 over F 7 ca be wrtte as h(x 1, x 2 ) = x 1 (3x 3 2 2) = x 1 1x 0 2f(x 6 1, x 3 2) where f(x 1, x 2 ) = 3x 2 2. Namey, r 1 = 1, r 2 = 0, s 1 = 6, s 2 = 3, 1 = 1, ad 2 = 2. Fay, t(x 1, x 2 ) = 3x 2 1x 3 2 2x 3 1x over F 7 ca be wrtte as t(x 1, x 2 ) = x 2 1x 2 (3x 2 2 2x 1 ) + 5 = x 1 x 2 f(x 1, x 2 2) + 5 where f(x 1, x 2 ) = 3x 2 2x 1. Namey, r 1 = 1, r 2 = 1, s 1 = 1, s 2 = 2, 1 = 6, ad 2 = 3. Defto 3.2. The mutvarate poyoma g(x 1,..., x ) = x r 1 1 x r f(x (q 1)/ 1 1,..., x (q 1)/ ) + b s sad to be dex form f r 1,..., r are vashg orders of x 1,..., x at 0 respectvey, g(0,..., 0) = b, ad ( 1,..., ) s the dex tupe of g. Wthout oss of geeraty, we assume g(0,..., 0) = 0 ad s = q 1 for 1. Hece g(x 1,, x ) = x r 1 1 x r f(x s 1 1,..., x s ). For each 1, et C,0 be the mutpcatve subgroup of F q cotag a the - th powers ad et C,j be the j -th coset of C,0 F q where 1 j 1. Let ξ be a fxed prmtve eemet F q ad ζ = ξ s be a prmtve -th root of uty where 1. Hece C,j = ξ j C,0. Moreover, f x C,j the x s = ζ j. If r > 0, the g(x 1,..., x 1, 0, x +1,..., x ) = 0. Otherwse, g(x 1,..., x 1, 0, x +1,..., x ) may ot be zero. Hece, for a gve dex tupe ( 1,..., ), we ca partto F q F q as a uo of A 1 A where A s ether the set {0} or oe of the cosets C,j determed by the dex. We defe costats a 1,..., a over these

7 INDEX BOUNDS FOR VALUE SETS OF POLYNOMIALS OVER FINITE FIELDS 7 sets A 1,..., A as foows: a = { ζ j f A = C,j, 0 f A = {0}. The g(x 1,..., x ) = x r 1 1 x r f(x s 1 1,..., x s ) ca be wrtte as a cycotomc mappg as foows: { 0 f (x1,..., x (4) g(x 1,..., x ) = ) = (0,..., 0), f(a 1,..., a )x r 1 1 xr f (x 1,..., x ) A 1 A. Because a 1,..., a are costats over A 1 A, we remark that ay mutvarate poyoma g behaves as a mooma f(a 1,..., a )x r 1 1 xr the subset A 1 A, determed by the partto of F q F q accordg to the dex tupe of g. We ote that the smar cocept for uvarate poyomas ca be foud [32, 37, 38]. Exampe 3.3. Cosder g(x 1, x 2 ) = x 2 1 x 2(4x 3 1 x2 2 2x4 2 ) over F 7. We ca wrte g(x 1, x 2 ) = x 2 1 x 2f(x 3 1, x2 2 ) wth the dex tupe (2, 3), where f(x 1, x 2 ) = 4x 1 x 2 2x 2 2. Let ξ = 3 be the fxed prmtve eemet F 7. So ζ 1 = ξ 6/2 = 6 ad ζ 2 = ξ 6/3 = 2. We ca partto F 7 to ether C 1,0 = {1, 2, 4} ad C 1,1 = {3, 5, 6} correspodg to 1 = 2, or C 2,0 = {1, 6}, C 2,1 = {3, 4}, ad C 2,2 = {2, 5} correspodg to 2 = 3. The F 7 F 7 ca be parttoed to the uo of a these A 1 A 2 s, where A 1 deotes ay oe of the sets {0}, C 1,0, ad C 1,1, ad A 2 deotes ay oe of the sets {0}, C 2,0, C 2,1 ad C 2,2. Hece g(x 1, x 2 ) ca be represeted by (5) g(x 1, x 2 ) = 0 f (x 1, x 2 ) = (0, 0), 0 f (x 1, x 2 ) {0} C 2,0, 0 f (x 1, x 2 ) {0} C 2,1, 0 f (x 1, x 2 ) {0} C 2,2, 0 f (x 1, x 2 ) C 1,0 {0}, 2x 2 1 x 2 f (x 1, x 2 ) C 1,0 C 2,0, 0 f (x 1, x 2 ) C 1,0 C 2,1, 5x 2 1 x 2 f (x 1, x 2 ) C 1,0 C 2,2, 0 f (x 1, x 2 ) C 1,1 {0}, x 2 1 x 2 f (x 1, x 2 ) C 1,1 C 2,0, 5x 2 1 x 2 f (x 1, x 2 ) C 1,1 C 2,1, x 2 1 x 2 f (x 1, x 2 ) C 1,1 C 2,2, where the coeffcets of x 2 1 x 2 a these braches are computed by usg f(a 1, a 2 ) = 4a 1 a 2 2a 2 2 where a 1 = 0, 1, 1 ad a 2 = 0, 1, 2, 4 respectvey. By a abuse of otato, et us ow cosder g as a poyoma vector map varabes from F q to F q : (6) g(x 1,..., x ) = (g 1 (x 1,..., x ),..., g (x 1,..., x )), where each g (1 ) s a poyoma varabes over F q. Usg the prevous defto of the dex tupe ad cycotomc mappgs, for each 1, we ca wrte

8 8 GARY L. MULLEN, DAQING WAN, AND QIANG WANG each g the dex form. Namey, g (x 1,..., x ) = x r() 1 1 x r() f (x s() 1 1,..., x s() ) + b, wth the dex tupe ( () 1,..., () ) ad b F q. Wthout oss of geeraty, we assume further that b = 0 for a 1. Hece = g(x 1,..., x ) ( x r(1) 1 1 x r(1) For each 1, we et f 1 (x s(1) 1 1,..., x s(1) ),..., x r() 1 1 x r() f (x s() 1 1,..., x s() s = gcd(s (1),..., s () ) ad = q 1. s The we ca ( 1,..., ) the dex tupe of the poyoma vector map g varabes. Let ζ = ξ s be a prmtve -th root of uty ad C,j be the j -th coset of C,0 F q where 1 j. We ote aga that F q F q ca be parttoed as a uo of A 1 A where A s ether the set {0} or oe of the cosets C,j determed by the dex tupe ( 1,..., ). Aga, as defed before, we et { ζ j a = f A = C,j, 0 f A = {0}. Hece, f (x 1,..., x ) A 1 A the ) ). = g(x 1,..., x ) ( x r(1) 1 1 x r(1) f 1 (a s(1) 1 /s 1 1,..., a s(1) /s ),..., x r() 1 1 x r() f (a s() 1 /s 1 1,..., a s() /s ) ). Let c = f (a s() 1 /s 1 1,..., a s() /s ). The g(x 1,..., x ) maps (A 1,..., A ) to ( ) c 1 A r(1) 1 1 A r(1),..., c A r() 1 1 A r(), where we use the coveto 0 0 = 1 ad A r = {x r x A}. Exampe 3.4. Let g(x 1, x 2 ) = (x 2 (x x ), x 1(3x )) be a map from F 7 F 7 to tsef. Usg the prevous deftos, we obta r (1) 1 = 0, r (1) 2 = 1, r (2) 1 = 1, r (2) 2 = 0, s (1) 1 = 2, s (1) 2 = 6, s (2) 1 = 6, ad s (2) 2 = 3. Moreover, f 1 (x 2 1, x6 2 ) = x x , f 2 (x 6 1, x3 2 ) = 3x Hece s 1 = gcd(2, 6) = 2, s 2 = gcd(6, 3) = 3, 1 = 3 ad 2 = 2. Therefore we use cycotomc cosets of orders 3 ad 2 respectvey the partto of F 7. Namey, C 1,0 = {1, 6}, C 1,1 = {3, 4} ad C 1,2 = {2, 5} are the cycotomc cosets of order 3, C 2,0 = {1, 2, 4}, ad C 2,1 = {3, 5, 6} are cycotomc cosets of order 2. The F 7 F 7 s parttoed to {0} {0}, {0} C 2,0, {0} C 2,1, C 1,0 {0}, C 1,1 {0}, C 1,2 {0}, C 1,0 C 2,0, C 1,0 C 2,1, C 1,1 C 2,0, C 1,1 C 2,1, C 1,2 C 2,0, C 1,2 C 2,1. Note that a 1 {0, 1, 2, 4} ad a 2 {0, 1, 6}. We ca check that f 1 (a 1, a 2 2 ) 0 ad f 2(a 3 1, a 2) 0. For exampe, g(x 1, x 2 ) = (x 2 (x x ), x 1(3x )) maps C 1,1 C 2,1 to C 2,1 C 1,0 because

9 INDEX BOUNDS FOR VALUE SETS OF POLYNOMIALS OVER FINITE FIELDS 9 g(x 1, x 2 ) behaves as the map (2x 2, 5x 1 ) over C 1,1 C 2,1. Ideed, f 1 (2, 6) = = 2 ad f 2 (2, 6) = = 5. Theorem 3.5. Let g be a poyoma vector map from F q to F q defed by g(x 1,..., x ) = (g 1 (x 1,..., x ) + b 1,..., g (x 1,..., x ) + b ), where b 1,..., b F q ad g has dex tupe ( 1,..., ) such that for each 1, g (x 1,, x ) = x r() 1 1 x r() f (x s() 1 1,..., x s() s a poyoma varabes over F q the dex form wth dex tupe ( () 1,..., () satsfyg g (0,..., 0) = 0 ad s () j = q 1 s = q 1 () j ) for 1 j. Let s = gcd(s (1),..., s () ) ) ad for 1. The g s a permutato of F q f ad oy f the foowg hods: (1) For a 1, we must have f (a s() 1 /s 1 1,..., a s() /s ) 0 as og as ot a a s are zero, where a = 0 or a = ξ s j wth 0 j 1 ad ξ s a fxed prmtve eemet of F q. (2) The matrx R :=. r (1) 1 r (1) 2 r (1) r (2) 1 r (2) 2 r (2).. r () 1 r () 2 r () each row ad each coum; Moreover, for each ozero r (k) cotas exacty oe ozero etry for we must have gcd(r (k), s (k) ) = 1. (3) g duces a bjecto betwee the set of a the parts A 1 A of the partto of F q F q correspodg to the dex tupe ( 1,..., ), ad the set of a the parts A 1 A of the partto of F q F q correspodg to the dex tupe ( 1,..., ), where ( 1,..., ) T = P (1,..., ) T ad the permutato matrx P s assocated wth R defed by p j = 1 f r () j 0. Proof. Wthout oss of geeraty, we ca assume that b 1 = b 2 = = b = 0. Assume that g s a permutato poyoma vector from F q to F q. It s easy to see that codto (1) hods. Otherwse, at east two eemets F q are mapped to the tupe cosstg of a 0 s. We ow prove codto (2). Frst, each row of R must cota at east oe ozero etry. Otherwse, suppose the -th row s the zero row, a the tupes (x 1,..., x ) satsfyg that the -th etry x A must be mapped to tupes wth the same -th etry, cotradctg that g s a permutato. Moreover, each row cotas exacty oe ozero etry. Ideed, wthout oss of geeraty, suppose the frst row cotas two ozero etres r (1) 1 ad r (1) 2. The tupes of the form {0} A 2 A ad A 1 {0} A are both mapped to {0} A 2 A, whch s a cotradcto. Smary, each coum of R shoud aso cota exacty oe ozero etry. Ideed, f oe coum s a zero coum, for exampe the frst coum, the g(x 1, x 2,..., x ) = g(x 1, x 2,..., x ) for ay x 1, x 1 C 1,j 1, cotradctg that g s a permutato map of F q. If oe coum cotas at east two ozero etres, for exampe r (1), the tupes of 1, r(2) 1

10 10 GARY L. MULLEN, DAQING WAN, AND QIANG WANG the form {0} A 2 A are mapped to tupes of the form {0} {0} A, cotradctg that g s a permutato map. Moreover, f r (k) > 0 the we cosder two dstct tupes whch dffer oy the coordate x but both vaues coordate x are the same coset C,j. These tupes must be mapped to dfferet mages, ths forces that gcd(r (k), s (k) ) = 1. Usg codto (2), we wrte ) g(x 1,..., x ) = (x r(1) 1 1 f 1 (x s(1) 1 1,..., x s(1) ),..., x r() f (x s() 1 1,..., x s() ) so that 1,..., s a permutato of 1,..., duced by the permutato matrx P. Let A 1 A be a part the partto of F q F q determed by the dex tupe ( 1,..., ). Reca that c = f (a s() 1 /s 1 (A 1,..., A ) to (c 1 A r(1) 1 1 ),..., c A r() 1,..., a s(). Because gcd(r () /s ). The g(x 1,..., x ) maps, s () ) = 1 for each, the mage becomes (A 1,..., A ), whch gves oe part A 1 A of the partto of F q correspodg to the dex tupe ( 1,..., ). Hece codto (3) hods. The coverse aso hods usg the same argumets as above. We ow cosder the foowg exampes to ustrate Theorem 3.5. Exampe 3.6. Let g(x 1, x 2 ) = (x 2 (x x ), x 1(3x )) be a map from F 7 F 7 to tsef as show Exampe 3.4. Obvousy f 1 (a 1, a 2 2 ) = a a ad[ f 2 (a 3 1, ] a 2) = 0 1 3a where a 1 {0, 1, 2, 4} ad a 2 {0, 1, 6}. Here, P = R = s a 1 0 permutato matrx. Moreover, gcd(r (1) 2, s(1) 2 ) = 1 ad gcd(r(2) 1, s(2) 1 ) = 1. Furthermore, g maps a part A 1 A 2 of the partto F 7 F 7 correspodg to the dex tupe (3, 2) to a part A 2 A 1 of the same sze correspodg to the dex tupe (2, 3). Ideed, g maps {0} {0} {0} {0} (4x {0} C 2,0) 2,0 C 2,0 {0} (4x {0} C 2,0) 2,1 C 2,1 {0} (0,x C 1,0 {0} 1 ) {0} C 1,0 (0,x C 1,1 {0} 1 ) {0} C 1,1 (0,x C 1,2 {0} 1 ) {0} C 1,2 (2x C 1,0 C 2,4x 1 ) 2,0 C 2,0 C 1,1 C 1,0 C 2,1 (2x 2,5x 1 ) C 2,1 C 1,2 C 1,1 C 2,0 (2x 2,4x 1 ) C 2,0 C 1,2 C 1,1 C 2,1 (2x 2,5x 1 ) C 2,1 C 1,0 C 1,2 C 2,0 (x 2,4x 1 ) C 2,0 C 1,0 C 1,2 C 2,1 (x 2,5x 1 ) C 2,1 C 1,1 By Theorem 3.5, g s a permutato of F 2 7. Exampe 3.7. Let g(x 1, x 2 ) = (x 2, x 1 (2+x 3 2 (x4 1 2x3 2 ))) be a map from F 7 F 7 to tsef. So s (1) 1 = 6, s (1) 2 = 6, s (2) 1 = 2, ad s (2) 2 = 3. Hece s 1 = gcd(6, 2) = 2 ad s 2 = gcd(6, 3) = 3.

11 INDEX BOUNDS FOR VALUE SETS OF POLYNOMIALS OVER FINITE FIELDS 11 Thus 1 = 3 ad 2 = 2. So C 1,0 = {1, 6}, C 1,1 = {3, 4}, ad C 1,2 = {2, 5} are the cycotomc cosets of order 3, C 2,0 = {1, 2, 4}, ad C 2,1 = {3, 5, 6} are cycotomc cosets of order 2. The F 7 F 7 s parttoed to {0} {0}, {0} C 2,0, {0} C 2,1, C 1,0 {0}, C 1,1 {0}, C 1,2 {0}, C 1,0 C 2,0, C 1,0 C 2,1, C 1,1 C 2,0, C 1,1 C 2,1, C 1,2 C 2,0, C 1,2 C 2,1. Moreover, f 1 (x 6 1, x6 2 ) = 1 ad f 2(x 2 1, x3 2 ) = 2 + x3 2 (x4 1 2x3 2 ). Note that a 1 {0, 1, 2, 4} ad a 2 {0, 1, 2}. We ca easy check that f 1 (a[ 3 1, a2 2 ) ] 0 ad f 2(a 1, a 2 ) 0 as og as oe 0 1 of a 1 ad a 2 s ozero. Furthermore, R =. I fact, g maps 1 0 {0} {0} {0} {0} (x {0} C 2,0) 2,0 C 2,0 {0} (x {0} C 2,0) 2,1 C 2,1 {0} (0,2x C 1,0 {0} 1 ) {0} C 1,2 (0,2x C 1,1 {0} 1 ) {0} C 1,0 (0,2x C 1,2 {0} 1 ) {0} C 1,1 C 1,0 C 2,0 (x 2,x 1 ) C 2,0 C 1,0 C 1,0 C 2,1 (x 2,6x 1 ) C 2,1 C 1,0 C 1,1 C 2,0 (x 2,4x 1 ) C 2,0 C 1,2 C 1,1 C 2,1 (x 2,3x 1 ) C 2,1 C 1,2 C 1,2 C 2,0 (x 2,2x 1 ) C 2,0 C 1,1 C 1,2 C 2,1 (x 2,5x 1 ) C 2,1 C 1,1 By Theorem 3.5, g s a permutato of F 2 7. We remark that from the proof of Theorem 3.5, each coordate poyoma of the permutato vector map s a mutvarate cycotomc mappg, whch tur behaves as a mooma ( oe varabe) o every dvdua coset. I other words, each permutato vector map varabes cossts of uvarate cycotomc permutatos together wth aother permutato o coordate varabes. Ths fact may hep us to costruct permutato maps varabes from these smper coordate poyomas. Aso from the proof of Theorem 3.5, t s easy to see that f oe eemet a part of the partto of F q beogs to the vaue set the the whoe part of the partto beogs to the vaue set. Hece we aso obta Coroary 3.8. Let g be a poyoma vector map from F q to F q wth the dex tupe ( 1,..., ) ad = max{ 1,..., } > 1. If V g < q the V g q q 1. Proof. Uder the assumptos (1) ad (2) Theorem 3.5, the cardaty of the vaue set of g satsfes V g > q q 1 f ad oy f g duces a permutato of F q. Aso from the proof of Theorem 3.5, f assumpto (1) fas, the at east two cosets correspodg to oe coordate, say, are coapsed to the same mage set, therefore V g q q 1 q q 1. Whe assumpto (2) fas, the matrx R cotas a zero row or a zero coum, or at east two etres oe row or coum. Hece the same dscusso shows that at east two cosets are coapsed to oe mage set. Ths mpes aga that V g q q 1 j for some j ad thus V g q q 1. Fay, f the matrx R

12 12 GARY L. MULLEN, DAQING WAN, AND QIANG WANG cotas exacty oe etry each row ad coum, but (r (k), s (k) ) = t > 1 for some, we mss at east (t 1)(q 1) t = (q 1) q 1 t q 1 2 q 1 vaues the vaue set. Hece V g q q 1. As the ext smpe exampe shows, the upper boud for o permutatos ca be acheved. Exampe 3.9. Let g(x 1, x 2 ) = (x 1, x 2 ( x 3 2 (x )2 + 4) 2 ) be a map from F 7 F 7 to tsef. So s (1) 1 = 6, s (1) 2 = 6, s (2) 1 = 3, ad s (2) 2 = 3. Hece s 1 = gcd(6, 3) = 3 ad s 2 = gcd(6, 3) = 3. Thus 1 = 2 ad 2 = 2. Here we use cycotomc cosets of order 2: C 1,0 = C 2,0 = {1, 2, 4}, ad C 1,1 = C 2,1 = {3, 5, 6} to partto F 7 F 7 to {0} {0}, {0} C 2,0, {0} C 2,1, C 1,0 {0}, C 1,1 {0}, C 1,0 C 2,0, C 1,0 C 2,1, C 1,1 C 2,0, C 1,1 C 2,1. Moreover, f 1 (x 6 1, x6 2 ) = 1 ad f 2(x 3 1, x3 2 ) = ( x3 2 (x )2 + 4) 2. Note that f 2 (a 1, a 2 ) = ( a 2 (a 1 + 2) 2 + 4) 2 {0, 2, 4} where a 1 {0, 1, 6} ad a 2 {0, 1, 6}. I fact, g maps {0} {0} {0} {0} (0,0) {0} C 2,0 {0} {0} (0,x {0} C 2 ) 2,1 {0} C 2,1 (x C 1,0 {0} 1,2x 2 ) C 1,0 {0} (x C 1,1 {0} 1,2x 2 ) C 1,1 {0} (x C 1,0 C 1,4x 2 ) 2,0 C 1,0 C 2,0 C 1,0 C 2,1 (x 1,x 2 ) C 1,0 C 2,1 C 1,1 C 2,0 (x 1,2x 2 ) C 1,1 C 2,0 C 1,1 C 2,1 (x 1,4x 2 ) C 1,1 C 2,1 Here g maps the {0} C 2,1 to {0} {0}, ad a other parts of the partto correspodg to the dex tupe (2, 2) to dstct parts of the partto correspodg to the dex tupe (2, 2). Therefore V g = 46 = q 2 q 1 2. ackowedgemets We thak the referee ad Are Wterhof for ther hepfu suggestos. Refereces [1] A. Akbary, S. Aarc, ad Q. Wag, O some casses of permutato poyomas, It. J. Number Theory 4 (2008), o. 1, [2] A. Akbary, D. Ghoca, ad Q. Wag, O permutato poyomas of prescrbed shape, Fte Feds App. 15 (2009), [3] A. Akbary, D. Ghoca, ad Q. Wag, O costructg permutatos of fte feds, Fte Feds App. 17 (2011), o. 1, [4] A. Akbary ad Q. Wag, O some permutato poyomas, It. J. Math. Math. Sc. 16 (2005), [5] A. Akbary ad Q. Wag, A geerazed Lucas sequece ad permutato bomas, Proc. Amer. Math. Soc. 134 (2006), o 1, [6] A. Akbary ad Q. Wag, O poyomas of the form x r f(x (q 1)/ ), It. J. Math. Math. Sc., Voume 2007 (2007), Artce ID 23408, 7 pages.

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