Polynomial Functions and Permutation Polynomials Over Some Finite Commutative Rings

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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].

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

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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. Let FðXÞ¼ i¼0 aiX be a polynomial over R: If doq; and FðxÞ l l 0 ðmod p Þ for all xAR; then F½X0 ðmod p Þ; i.e. each ai is congruent to 0.

Proof. Without loss of generality, assume FðXÞa0: Express F as F ¼ pnG with % % Gc0 ðmod pÞ: Let G be the reduction of G in Fq½X: Then G is a non-zero polynomial ð1Þ % ð1Þ over Fq whose degree is less than q: There exists a AR=pR such that Gða Þa0: Namely, GðaÞc0 ðmod pÞ: But pnGðaÞ¼FðaÞ0 ðmod plÞ: So nXl: This completes the proof of the lemma. & ARTICLE IN PRESS

Q. Zhang / Journal of Number Theory 105 (2004) 192–202 197

i i Theorem 3.3. Let Pi denote the ring of polynomial functions from R=p RtoR=p R; let J be the ideal of R½X generated by p and X À X q: Then for lpq lðlþ1Þ q (1) the cardinality of Pl is m 2 ; where m ¼ q is the number of the functions from Fq to Fq; (2) as rings, we have

l R½X=J DPl:

ðiÞ i Proof. (1) Let Ai denote the abelian group of functions from T to R=p R: For a ðiÞ system of functions hi AAi; i ¼ 1; y; l one can deﬁne a polynomial function from R=plR to R=plR according to (3.1). This gives an abelian group homomorphism from Al AlÀ1 ? A1 to Pl: Theorem 3.1 implies this map is surjective. Lemma 3.2 assures that the map is also injective. By Proposition 2.2, we can see the Q lðlþ1Þ l i i 2 cardinality of Ai is m : So the cardinality of Pl is i¼1 m ¼ m : (2) It is clear that each polynomial in Jl deﬁnes the zero function from R=plR to l l R=p R: So there exists a natural surjective homomorphism c from R½X=J to Pl: We will complete the proof by counting the set R½X=Jl: For iX1; JiÀ1=Ji is an R½X=J module in a natural way. Furthermore, this module is generated by i elements. So the cardinality of JiÀ1=Ji is jJiÀ1=Jijpmi: Using the basic exact sequence

0-JiÀ1=Ji-R½X=Ji-R½X=JiÀ1-0 we have

Yl Yl lðlþ1Þ l iÀ1 i i jR½X=J j¼ jJ =J jp m ¼ m 2 ¼jPlj: i¼1 i¼1

So c is an isomorphism. This ﬁnishes the proof of Theorem 3.3. &

4. Applications to permutation polynomials over R=plR

Note that the preceding results are easy to be extended to several variable case. We have considered only the functions in one variable for simplicity of writing. Now we l consider permutation polynomials over R=p R in nð41Þ variables. Let pl;n be the l n n ðlÞ ðlÞ ð1Þ ð1Þ natural map from ðR=p RÞ to ðR=pRÞ deﬁned by pl;nðx1 ; y; xn Þ¼ðx1 ; y; xn Þ: A function f from ðR=plRÞn to R=plR is called a p-function if for any x; sAðR=plRÞn we have f ðx þ psÞ¼f ðxÞ: In other words, f is a p-function if and only if there exists ARTICLE IN PRESS

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n l a map r from ðR=pRÞ to R=p R such that f ¼ r3pl;n: So we have the following proposition (the extension of Proposition 2.2):

Proposition 4.1. Ap-function f from ðR=plRÞn to R=plR corresponds to a unique map f from ðR=pRÞn to ðR=pRÞl according to the following commutative diagram:

Moreover, if f is deﬁned by l polynomials F1; y; FlAR½X1; y; Xn; then f is deﬁned by the polynomial WlðF1; y; FlÞ: Now let us recall some concepts:

Definition 4.1. Let A be a finite commutative ring. A system of polynomials F1; y; FmAA½X1; y; Xn are said to be orthogonal if they induce a uniform map from An to Am: In particular, the system is called a permutation polynomial vector when m ¼ n: A polynomial FAA½X1; y; Xn is called a permutation polynomial if it alone forms an orthogonal system. A map f between two finite sets S and B is said to be uniform if for every aAB; the cardinality of the set f À1ðaÞ is the same. A subsystem of a permutation polynomial vector is called a strong orthogonal system.

For basic knowledge on permutation polynomials, see [9]. Carlitz [3] showed that all orthogonal systems over finite fields is strong. But if A is not a field, then there do exist non-strong permutation polynomials (see [5]). Note that we often say l polynomials F1; y; FnAR½X1; y; Xn form an orthogonal system over R=p R if l their images in R=p R½X1; y; Xn do.

plÀ1 plÀ2 Proposition 4.2. Let F1; y; Fl be polynomials in R½X1; y; Xn: If F ¼ F1 þ pF2 þ lÀ1 l ? þ p Fl ¼ WlðF1; y; FlÞ; then F is a permutation polynomial over R=p R if and only if F1; y; Fl form an orthogonal system over R=pR:

Proof. Let f be the map from ðR=plRÞn to R=plR deﬁned by F and f the map from n l ðR=pRÞ to ðR=pRÞ deﬁned by F1; y; Fl: Thus, Diagram 4.1 is commutative. Since pl;n is uniform, f is uniform if and only if f is uniform. This completes the proof of Proposition 4.2. &

This easy proposition tells us when a p-function is uniform. Furthermore, it has the following consequences: ARTICLE IN PRESS

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Corollary 4.3. There exist uniform p-functions from ðR=plRÞn to R=plR if and only if lpn: Moreover if lpn; then the number of uniform p-functions is

qn! ðqnÀl!Þql

Theorem 4.4. There exists a polynomial FAR½X1; y; Xn such that F is a permutation polynomial over R=pnR; but F is not a permutation polynomial over R=pnþ1R:

Proof. Take

pn pnÀ1 ? nÀ1 p p p F ¼ X1 þ pX2 þ þ p Xn ¼ WnðX1 ; y; Xn Þ¼Wnþ1ðX1; y; Xn; 0Þ:

p p Clearly, X1 ; y; Xn form an orthogonal system over R=pR: So, by Proposition 4.2, F n is a permutation polynomial over R=p R: On the other hand, X1; y; Xn; 0 cannot form an orthogonal system over R=pR: So, by Proposition 4.2, F is not a permutation polynomial over R=pnþ1R:

A fundamental question is how to check whether a polynomial FAR½X1; y; Xn is a permutation polynomial over R=plR: One has a natural necessary condition that F is a permutation polynomial over R=plÀ1R based on the following commutative diagram:

where the vertical maps are the natural maps, which are uniform. Let rnðRÞ be the least positive integer such that each polynomial FAR½X1; y; Xn which is a permutation polynomial over R=prnðRÞR must be a permutation polynomial over every R=plR: One should ask:

Question. Is it true that rnðRÞoN?

Theorem 4.4 implies that rnðRÞXn þ 1: This is a generalization of a result in [15] which asserts that r2ðZpÞX3: By No¨ bauer’s theorem below we have r1ðZpÞ¼2: Let GðR=plR; nÞ denote the group of all permutation maps of ðR=plRÞn deﬁned by polynomials. This group is easy to understand in the following sense:

Proposition 4.5. Let F1; y; Fn be polynomials in R½X1; y; Xn; let l41: Then l F1; y; Fn form a permutation polynomial vector over R=p R if and only if they form a ARTICLE IN PRESS

200 Q. Zhang / Journal of Number Theory 105 (2004) 192–202 permutation polynomial vector over R=pR and the Jacobi matrix ð@Fi ðxÞÞ is of full rank @Xj at every point xAðR=pRÞn:

We omit the proof (an easy exercise of Hensel’s lemma) which is similar to the special case R ¼ Zp proved in [8]. The special case R ¼ Zp and n ¼ 1 is proved by No¨ bauer [10].No¨ bauer’s theorem is fundamental though it is easy to prove. A lot of work is motivated by generalizing No¨ bauer’s theorem. Many people try to understand the permutation polynomials over R=plR in n variables. But Theorem 4.4 implies that the case n41 is totally different from the case n ¼ 1: The group GðR=plR; nÞ; in a natural way, acts on the set of all maps from ðR=plÞn to R=plR deﬁned by permutation polynomials. Then we get some orbits, one of which consists of the maps deﬁned by strong permutation polynomials. This orbit can be characterized (see [7,16] for the case R ¼ Zp). In [14] we get the following general theorem:

Theorem 4.6. Let A be a ﬁnite local ring with maximum ideal m; and F1; y; Fk be polynomials over A in nðXkÞ variables. If A is not a ﬁeld, then F1; y; Fk form a strong orthogonal system over A if and only if they form an orthogonal system over A=m and the Jacobi matrix ð@Fi ðxÞÞ is of rank k at every xAR=pR: @Xj

In a recent paper [17] we characterize all orbits in the case n ¼ l ¼ 2 and R ¼ Zp: But in the general case it is difﬁcult to understand all orbits. Here we characterize another special orbit.

n Theorem 4.7. If lpn; then all uniform p-functions from ðR=plRÞ to R=plR form an orbit under the action of the group GðR=plR; nÞ: In other words, all permutation polynomials with the form WlðF1; y; FnÞ form an orbit.

Proof. Let f be a uniform p-function from ðR=plRÞn to R=plR and f the corresponding map from ðR=pRÞn to ðR=pRÞl: For a permutation map s of ðR=plRÞn deﬁned by n polynomials over R; denote by r the permutation map of ðR=pRÞn deﬁned by the same polynomials. We have the following commutative diagram:

So f 3s is the p-function corresponding to f3r: If g is another uniform p-function from ðR=plRÞn to R=plR corresponding to a map c from ðR=pRÞn to ðR=pRÞl; by Proposition 4.2, we can see that both f and c are uniform. So we can ﬁnd a permutation map r of ðR=pRÞn such that c ¼ f3r: Suppose r is deﬁned by ARTICLE IN PRESS

Q. Zhang / Journal of Number Theory 105 (2004) 192–202 201 polynomials F1; y; FnAR½X1; y; Xn: By Lemma 4.8 below, we can choose these @Fi polynomials F1; y; Fn such that ; i; j ¼ 1; y; n deﬁne any functions from @Xj n ðR=pRÞ to R=pR we need. Thus, we can choose these polynomials F1; y; Fn such that the Jacobi matrix ð@Fi ðxÞÞ is of full rank at every point xAðR=pRÞn: Let s @Xj denote the map from ðR=plRÞn to ðR=plRÞn deﬁned by polynomials F1; y; FnAR½X1; y; Xn: By Proposition 4.5, this s is bijective. Meanwhile our s and r make Diagram 4.3 be commutative. So f 3s ¼ g as both f 3s and g correspond to c ¼ f3r: This completes the proof of Theorem 4.7. &

Lemma 4.8. Let f0ðx1; y; xnÞ; f1ðx1; y; xnÞ; y; fnðx1; y; xnÞ be functions in n variables over the ﬁnite ﬁeld Fq; then there exists a polynomial FAFq½X1; y; Xn; n such that for any xAFq; the following equalities hold:

FðxÞ¼f0ðxÞ;

@F ðxÞ¼fiðxÞ; i ¼ 1; y; n: @Xi

Proof. Choose polynomials FiðXÞ in Fq½X1; y; Xn; i ¼ 0; 1; y; n such that for any n xAFq;

fiðxÞ¼FiðxÞ; i ¼ 1; y; n:

Construct a new polynomial

q q ? q FðXÞ¼F0 ðXÞþðX1 À X1 ÞF1ðXÞþ þðXn À Xn ÞFnðXÞ:

An easy calculation yields

@F @F @F q 1 ? q n ¼ Fi þðX1 À X1 Þ þ þðXn À Xn Þ : @Xi @Xi @Xi

So F is what we need. This completes the proof of Lemma 4.8. &

Remark. One can easily generalize the main results by substituting a general p-adic P l i 0 l iÀ1 p À integer ring R for R because a generalized Witt polynomial i¼1 p Xi also deﬁnes a bijection between ðR0=pR0Þl and R0=plR0; where p is a uniformizer of R0: ARTICLE IN PRESS

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Acknowledgments

The author greatly thanks the referee for many helpful comments and suggestions. He also acknowledges the support from the Morningside Center of Mathematics of Chinese Academy of Sciences.

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