View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector ARTICLE IN PRESS Journal of Number Theory 105 (2004) 192–202 http://www.elsevier.com/locate/jnt Polynomial functions and permutation polynomials over some finite commutative rings$ Qifan Zhang College of Mathematics, Sichuan University, Chengdu 610064, China Received 18 July 2003 Communicated by S.-W. Zhang Abstract We extend some classical results on polynomial functions mod pl: We prove all results in algebraic methods avoiding any combinatorial calculation. As applications of our methods, we obtain some interesting new results on permutation polynomials in several variables over some finite commutative rings. r 2003 Elsevier Inc. All rights reserved. Keywords: Polynomial function; Permutation polynomial; Witt polynomial; Teichmu¨ ller element 1. Introduction We are interested in polynomial functions and permutation polynomials over finite commutative rings. Without loss of generality, we need only to consider local rings, for example, the ring Z=plZ: The polynomial functions over Z=plZ are widely used in the literature on finite combinatorics (see e.g. [1,2,11]). It is well-known that each function over a finite field is a polynomial function. One should ask when a function f over Z=plZ is a polynomial function. A clear necessary condition, easily l derived from Taylor’s formula, is that there exist functions fi over Z=p Z; i ¼ 0; 1; y; l À 1 such that for any x; sAZ=plZ; the following equality holds: lÀ1 f ðx þ psÞ¼f0ðxÞþf1ðxÞps þ ? þ flÀ1ðxÞðpsÞ : $This work is supported by NSFC (10128103,19901023). E-mail address: [email protected]. 0022-314X/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jnt.2003.09.009 ARTICLE IN PRESS Q. Zhang / Journal of Number Theory 105 (2004) 192–202 193 Decades ago Carlitz [4], Kempner [6] and Rosenberg [12] respectively proved that the condition is also sufficient. This is an interesting result, which asserts that all ‘‘analytic functions’’ over Z=plZ are polynomial functions, but the known proofs contain complicated calculations. In the present paper we devote a quick proof avoiding any combinatorial calculation. As a by-product we obtain another classical result (maybe not well-known, see e.g. [4]): For lpp; a polynomial FAZ½X defines the zero function over Z=plZ if and only if F is in the ideal ðp; X À X pÞl of Z½X: In fact, we will work on more general ring R=plR; where R is the ring of integers of l l an unramified extension of Qp: When R ¼ Zp; R=p R becomes Z=p Z: For a field k of characteristic p; there exists a series of Witt rings WlðkÞ; l ¼ 1; 2; y : They are defined by Witt polynomials plÀ1 plÀ2 ? lÀ1 Wl ¼ X1 þ pX2 þ þ p Xl; l ¼ 1; 2; y l in a suitable way (see [13] for details). In particular, WlðFqÞ is isomorphic to R=p R: l l The reason is that one has a natural bijection between Fq and R=p R: This bijection is the key of our proofs. In Section 4, we get two new theorems on permutation polynomials (Theorems 4.4 and 4.7) in a similar method. 2. Preliminaries From now on, fix a prime number p and a p-adic integers’ ring R with R=pRDFq; where q is a power of p: We will consider the ring R=plR: Firstly, let us make a convention on notations: l Convention. Let pl be the natural homomorphism from R=p RtoR=pR: If aAR; then aðlÞ denotes the image of a in R=plR: If f is a function from R to R with the property f ðx þ plsÞf ðxÞðmod plÞ; then f ðlÞ denotes the function from R=plRtoR=plR induced by f : If F is a polynomial over R; F ðlÞ denotes the function over R=plR induced by F: We often use the following proposition which is easy to prove: Proposition 2.1. If l41; and a; bAR satisfy a b ðmod plÞ; then (1) ap bp ðmod plþ1Þ; (2) aq bq ðmod plþ1Þ: It is well-known that every að1ÞAR=pR has a Teichmu¨ ller lifting, i.e. the unique root of the equation xq ¼ x in the set a þ pR: So the equation has q roots in R which form a residue system of R modulo p: We call these roots Teichmu¨ ller elements in R: Denote by T the set of all Teichmu¨ ller elements in R: Of course, T ðlÞ :¼ftðlÞjtATg is ARTICLE IN PRESS 194 Q. Zhang / Journal of Number Theory 105 (2004) 192–202 the set of the roots of the same equation in R=plR: We call the elements of T ðlÞ Teichmu¨ ller elements in R=plR: In fact, we can understand T by understanding T ðlÞ: If t is the Teichmu¨ ller lifting of að1Þ; then by Part 2 of Proposition 2.1, we have l 1 l 1 tq À aq À ðmod plÞ; i.e. lÀ1 t aq ðmod plÞ: ð2:1Þ n This implies that t ¼ lim aq : Now we define two maps: l ol : R=pR-R=p R; ð1Þ ðlÞ olðt Þ¼t for any tAT l l tl : ðR=pRÞ -R=p R; l 1 l 2 ð1Þ ð1Þ p À p À ? lÀ1 ðlÞ A tlðt1 ; y; tl Þ¼ðt1 þ t2 p þ þ tlp Þ ; for any ti T: Of course, ol is injective. It is a one-sided inverse of pl: Every aAR can be written in one and only one way as XN i aip ; i¼0 where aiAT; meanwhile for any i every tAT can be written in one and only one way i as a p th power of an element in T: So tl is bijective. By (2.1), the maps also can be defined as ð1Þ qlÀ1 ðlÞ olðx Þ¼ðx Þ and ð1Þ ð1Þ plÀ1 plÀ2 ? lÀ1 ðlÞ tlðx1 ; y; xl Þ¼ðx1 þ x2 p þ þ xlp Þ : In some sense both ol and tl are polynomial maps. They are, respectively, defined qlÀ1 by two polynomials over Z : X and Wl: Definition 2.1. A function f ðlÞ from R=plR to R=plR is called a p-function if for any x; sAR; f ðx þ psÞf ðxÞðmod plÞ: Obviously, a map f from R=pR to ðR=pRÞl and a function h from T ðlÞ to R=plR determine each other according to the following commutative diagram: ARTICLE IN PRESS Q. Zhang / Journal of Number Theory 105 (2004) 192–202 195 Meanwhile, there exists a unique p-function f from R=plR to R=plR such that ðlÞ ðlÞ f jT ðlÞ ¼ h: More precisely, f should be defined by f ðt þ psÞ¼hðt Þ or by the following commutative diagram: Checking the diagrams, we have the following proposition: Proposition 2.2. There are natural bijections between the following three sets: 1) the set of p-functions f from R=plRtoR=plR; 2) the set of functions h from T ðlÞ to R=plR; 3) the set of maps f from R=pR to ðR=pRÞl: Moreover, if f is defined by l polynomials F1; y; Fl in R½X; then the two functions f and h corresponding to f are defined by the polynomial WlðF1; y; FlÞ: Remark. Proposition 2.2 implies that all p-functions are polynomial functions. 3. Polynomial functions over R=plR Theorem 3.1. Let f ðlÞ be a function from R=plRtoR=plR; then the following conditions are equivalent: (1) f ðlÞ is a polynomial function, ðiÞ i i (2) there exist functions fi from R=p RtoR=p R; i ¼ 1; y; l such that for any x; sAR; lÀ1 l f ðx þ psÞflðxÞþflÀ1ðxÞps þ ? þ f1ðxÞðpsÞ ðmod p Þ; ðiÞ ðiÞ i (3) there exist functions hi from T to R=p R such that for any tAT and any sAR; lÀ1 l f ðt þ psÞhlðtÞþhlÀ1ðtÞps þ ? þ h1ðtÞðpsÞ ðmod p Þ: ðiÞ ðiÞ Proof. Taylor’s formula yields (1) ) (2). Taking hi ¼ fi jT ðiÞ ; we have proved (2) ) ðiÞ (3). At last, we prove that (3) ) (1). By Proposition 2.2, all hi are polynomial functions. So there exist polynomials H1; y; Hl in R½X such that i hiðtÞHiðtÞHiðt þ psÞðmod p Þ; i ¼ 1; y; l: ð3:1Þ ARTICLE IN PRESS 196 Q. Zhang / Journal of Number Theory 105 (2004) 192–202 Hence ðlÀiÞ ðlÀiÞ l hiðtÞðpsÞ Hiðt þ psÞðpsÞ ðmod p Þ; i ¼ 1; y; l: ð3:2Þ For any x ¼ t þ psAR; where tAT; by (2.1) we have lÀ1 t xq ðmod plÞ so lÀ1 ps x À xq ðmod plÞ: ð3:3Þ By (3.1) and (3.2), we have lÀ1 l f ðt þ psÞHlðt þ psÞþHlÀ1ðt þ psÞps þ ? þ H1ðt þ psÞðpsÞ ðmod p Þ: This together with (3.3) leads to qlÀ1 qlÀ1 lÀ1 l f ðxÞHlðxÞþHlÀ1ðxÞðx À x Þþ? þ H1ðxÞðx À x Þ ðmod p Þ: This completes the proof of Theorem 3.1. & By the same method, one can prove Theorem 3.10 below, which is similar to the old result. But Theorem 3.1 is useful to count the number of polynomial functions (see Theorem 3.3 below). Theorem 3.10. A function f ðlÞ from R=plRtoR=plR is a polynomial function if and only ðlÞ ðlÞ y ðlÞ l l A if there exist functions f0 ; f1 ; ; flÀ1 from R=p RtoR=p R such that for any x; s R; the following congruence holds: lÀ1 l f ðx þ psÞf0ðxÞþf1ðxÞpt þ ? þ flÀ1ðxÞðpsÞ ðmod p Þ: In order to prove the last theorem in this section, we need a lemma: P d i Lemma 3.2.