Polynomial functions on finite commutative rings

Every function on a nite residue class ring D=I of a Dedekind domain D is induced by an integer-valued polynomial on D that preserves congruences mod I if and only if I is a power of a prime ideal. If R is a nite commutative local ring with maximal ideal P of nilpotency N satisfying for all a; b 2 R, if ab 2 Pn then a 2 P k , b 2 P j with k + j min(n;N), we determine the number of functions (as well as the number of permutations) on R arising from polynomials in R[x]. For a nite commutative local ring whose maximal ideal is of nilpotency 2, we also determine the structure of the semigroup of functions and of the group of permutations induced on R by polynomials in R[x]. INTRODUCTION Let R be a nite commutative ring with identity. Every polynomial f 2 R[x] de nes a function on R by substitution of the variable. Not every function ':R ! R is induced by a polynomial in R[x], however, unless R is a nite eld. (Indeed, if the function with '(0) = 0 and '(r) = 1 for r 2 R n f0g is represented by f 2 R[x], then f(x) = a1x+ : : :+ anxn and for every non-zero r 2 R we have 1 = f(r) = (a1 + : : :+ anrn 1)r, which shows r to be invertible.) 1 This prompts the question how many functions on R are representable by polynomials in R[x]; and also, in the case that R = D=I is a residue class ring of a domain D with quotient eld K , whether every function on R might be induced by a polynomial in K[x]? We will address these questions in sections 2 and 1, respectively. Other related problems are to characterize the functions on R arising from polynomials in R[x] by intrinsic properties of these functions (such as preservation of certain relations), and to determine the structure of the semigroup of polynomial functions on R and that of the group of polynomial permutations of R. In section 4, we will answer the second question in the special case that R is a local ring whose maximal ideal is of nilpotency 2. Apart from that, the only result I am aware of is Nobauer's expression of the group of polynomial permutations on Zpn as a wreath product GoSp , with G a rather inscrutable subgroup (characterized by conditions on the coe cients of the representing polynomials) of the group of polynomial permutations on Zpn 1 [11]. (There is a wealth of literature on the functions induced by polynomials on nite elds, some of it concerning the structure of the subgroup of Sq generated by special polynomials, see e.g. [8] and its references. Methods from the theory of nite elds do not help much with nite rings, however, except when the rings are algebras over a nite eld, see [2].) A characterization of polynomial functions by preservation of relations has been given for R=Zn by Kempner [6]. For nite commutative rings in general there is the criterion of Spira [17] that a function is representable by a polynomial if and only if all the iterated divided di erences that can be formed by subsets of the arguments and the respective values are in R. In what follows, all rings are assumed to be commutative with identity, the natural numbers are written as N = f1; 2; 3; : : :g, and the non-negative integers as N0 = f0; 1; 2; : : :g. 1 FUNCTIONS INDUCED ON RESIDUE CLASS RINGS BY INTEGER−VALUED POLYNOMIALS In this section we give the answer, for Dedekind rings, to a question asked by Narkiewicz in his \Polynomial Mappings" book [9]. For 2 R= Z, the `if' direction has been shown (for several variables, cf. the corollary) by Skolem [16], the `only if' direction by R edei and Szele [12, 13]. If D is a domain with quotient eld K , a polynomial f 2 K[x] is called integer-valued on D if f(d)2D for all d 2D. We write Int(D) for the set of all integer-valued polynomials on D. If I is an ideal of a domain D, we say that a polynomial f 2 Int(D) induces a function ':D=I ! D=I if '(d+ I) = f(d)+ I is well de ned, i.e., if c d mod I implies f(c) f(d) mod I . THEOREM 1. Let R be a Dedekind domain and I an ideal of R of nite index. Every function ':R=I!R=I is induced by a polynomial f 2 Int(R) if and only if I is a power of a prime ideal of R. Proof. The case of a nite eld or of I = R = P 0 is trivial, so we consider R in nite and I 6= R. Let P be a prime ideal with I P . Assume that the characteristic function of f0g on R=I is induced by a polynomial f 2 Int(R), then f(r) 1 mod I for r 2 I and f(r) 0 mod I for r 62 I . We show that I must be a power of P . Suppose otherwise, then Pn 6 I for all n 2 N . Let c 2 R and g 2 R[x] such that f(x) = g(x)=c, and n = vP (c). Since g is in R[x], the function r 7! g(r) on R preserves congruences mod every ideal of R, in particular mod Pn+1 . It follows that r s mod Pn+1 implies f(r) f(s) mod P . Now consider an element r 2 Pn+1 n I . On one hand, f(r) 62 P , since f(r) f(0) mod P and f(0) 1 mod I ; on the other hand, since r 62 I , we have f(r)2 I P , a contradiction. To show that every function on R=Pn (P a prime ideal of nite index) is induced by a polynomial in Int(R), it su ces to show this for the charcteristic function of f0g on the residue class ring. For this, we need only construct a polynomial f 2 Int(R) satisfying f(r)2P for r 62 Pn and f(r) 62 P for r 2 Pn ; an appropriate power ~ f(x) = f(x)m will then satisfy ~ f(r) 2 Pn for r 62 Pn and ~ f(r) 1 mod Pn for r 2 Pn . Let a1; : : : ; aqn 1 2 R be a system of representatives of the residue classes of Pn other than Pn itself, and let a0 2 Pn 1 n Pn . Put h(x) =Qqn 1 k=0 (x ak) and =Pnj=1 h qn qj i= qn 1 q 1 , then for all r 2Pn 3 we have vP (h(r)) = 1, while vP (h(r)) for all r 2 R n Pn . Now let Q= fQ2 Spec(R) jQ 6=P ; 9k ak 2Qg and for Q2Q de ne mQ = maxfm 2 N j 9k ak 2 Qmg. Pick c 2 R such that c 62 P and c2QmQ+1 for all Q2Q, and set bk= c 1ak and g(x)=Qqn 1 k=0 (x bk). We now set f(x) = g(x)=g(0) and claim that f 2 Int(R) and that for all r 2 R, f(r) 2 P if and only if r 62 Pn . To verify this, we check that for all Q 2 Spec(R) and all r 2 R, vQ(g(r)) vQ(g(0)) and that vP (g(r)) > vP (g(0)) for r 2 R n Pn , while vP (g(r)) = vP (g(0)) for r 2 Pn . First consider those Q2Spec(R) with vQ(c)>0. We have vQ(bk)<0 for all k and therefore vQ(g(r))=Pqn 1 k=0 vQ(bk)=vQ(g(0)) for all r2R. Now consider a Q 2 Spec(R) with vQ(c) = 0 and Q 6= P , then vQ(bk)= 0 for all k, and for all r 2R we have vQ(g(r)) 0= vQ(g(0)). Concerning P , we observe that vP (r bk) = vP (c 1(cr ak)) = vP (cr ak), such that vP (g(r)) = vP (h(cr)). Since vP (cr) = vP (r), this implies vP (g(r)) for r 2 R n Pn and vP (g(r)) = 1 for r 2 Pn . If K is the quotient eld of a domain D and I an ideal of D, we say that f 2 K[x1; : : : ; xm] induces a function ': (D=I )m ! D=I if '(d1 + I; : : : ; dm + I) = f(d1; : : : ; dm) + I makes sense, i.e., if f(d1; : : : ; dm) 2 D for all (d1; : : : ; dm) 2 Dm and f(d01; : : : ; d0m) f(d1; : : : ; dm) mod I whenever d0i di mod I for 1 i m. COROLLARY. If R is a Dedekind domain, P a maximal ideal of nite index and n 2 N then every function f : (R=Pn)m!R=Pn is induced by a polynomial f 2K[x1; : : : ; xm] (K being the quotient eld of R). Proof. It su ces to have a polynomial f 2K[x1; : : : ; xm] that induces the characteristic function of (0; 0; : : : ; 0) mod Pn . As R=P is a eld, there exists a g 2 R[x1; : : : ; xm] such that g(r1; : : : ; rm) 1 mod P if ri 2 P for 1 i m and g(r1; : : : ; rm) 0 mod P otherwise. By the Theorem, there exists h 2 Int(R) such that h(r) 2 P if r 2 Pn and h(r) 62 P otherwise. Now f(x1; : : : ; xm) = g(h(x1); : : : ; h(xm)) satis es f(r1; : : : ; rm) 62 P i ri 2 Pn for 1 i m, and a suitable power of g(x) = f(x)k nally satis es g(r1; : : : ; rm) 1 mod Pn if ri 2 Pn for 1 i m and g(r1; : : : ; rm) 0 mod Pn otherwise, as required. 4 Note that the theorem and its proof still hold if we replace Dedekind ring by Krull ring, prime ideal by height 1 prime ideal, and restrict I to ideals with div(I) 6= R. 2 THE NUMBER FORMULAS For a commutative nite ring R, let us denote by F(R) the set (or semigroup with respect to composition) of functions on R induced by polynomials in R[x], and by P(R) the subset (or group) of those polynomial functions on R that are permutations. When considering the functions induced on a nite commutative ring R by polynomials in R[x], we can restrict ourselves to local rings, since every nite commutative ring is a direct sum of local rings, and addition and multiplication (and therefore evaluation of polynomials in R[x]) are performed in each component independently. For residue class rings of the integers, we know jF(Zpn)j = pPnk=1 p(k) and jP(Zpn)j = p!pp(p 1)ppPnk=3 p(k); where p is a prime and p(k) is the minimal m 2 N such that pk m! (in other words, the minimal m 2 N such that p(m) k, with p(m) =Pj 1 hm pj i). The most lucid proof, in my opinion, of these two formulas is that by Keller and Olson [5], to whom the second one is due. Kempner's earlier proof [6] of the formula for jF(Zpn)j is rather more involved. Singmaster [15] and Wiesenbauer [18] gave proofs for R=Zm which do not use reduction to the local ring case. Brawley and Mullen [3] generalized the formulas to Galois rings (rings of the form Z[x]=(pn; f), where p is prime and f 2 Z[x] is irreducible over Zp, see [7]) and Ne caev [10] to nite commutative local principal ideal rings. We will give a proof along the lines of Keller and Olson of a generalization of the formulas to a class of local rings (the suitable rings de ned below) that properly contains the rings considered by Brawley, Mullen and Ne caev. DEFINITION. Let R be a nite commutative local ring R with maximal ideal P and N 2 N minimal with PN = (0). We call R 5 \suitable", if for all a; b 2 R and all n 2 N , ab 2 Pn =) a 2 P k and b 2 P j with k + j min(N;n): Note that every nite local ring R with maximal ideal P such that P 2 = (0) is suitable, as well as every nite local ring whose maximal ideal is principal. We may think of this property as inducing a valuation-like mapping v:R ! HN , by v(r) = k if r 2 P k n P k+1 and v(0) = 1, where (HN ;+) results from the non-negative integers by identifying all numbers greater or equal N ; it is the semigroup with elements f0; 1; : : : ; N 1; N=1g and i+j=min(i+j;N), where the operations on the right are just the usual ones on non-negative integers. DEFINITION. If R is a nite local ring and P its maximal ideal, for n 0, let (n) = (R;P )(n) =Xj 1 n [R : P j ] and let (n)= (R;P )(n) be the minimal m2N such that (R;P )(m) n. (If R and P are understood, we suppress the subscript (R;P ) of and .) REMARK. Note that (R;P )(n) is nite if and only if n< jRj; we will never use (R;P ) outside that range. Also note that, since [R=P k : P j=P k] = [R : P j] for j k, we have (R;P )(n) = (R=Pk;P=Pk)(n) in the range where both values are nite, that is for n < [R : P k]. THEOREM 2. Let R be a suitable nite local ring with maximal ideal P , q = [R : P ], and N 2 N minimal, such that PN = (0). Then jF(R)j= (N) 1 Y j=0 [R : PN (j)]; where (n)=Pj 1 h n [R:P j ]i and (n) is the minimal m2N such that (m) n. Also, for N > 1, jP(R)j = q! (q 1)q q2q jF(R)j : 6 If [P k 1 : P k] = q for 1 k N , the formulas simplify to jF(R)j= qPNk=1 q(k) and jP(R)j = q!qq(q 1)q qPNk=3 q(k); where q(m) =Pj 1 hm qj i and q(k) is the minimal m 2 N such that q(m) k. We will prove the expression for jF(R)j at the end of the next section, and that for jP(R)j at the end of section 4. 3 A CANONICAL FORM FOR THE POLYNOMIAL REPRESENTING A FUNCTION DEFINITION. Let R be a commutative nite local ring with maximal ideal P of nilpotency N . We call a sequence (ak)1k=0 of elements in R a P -sequence, if for 0 n N ak aj 2 Pn () [R : Pn] k j; and if (ak) is a P -sequence, we call the polynomials hxi0 = 1 and hxin = (x a0) : : : (x an 1) for n > 0 the \falling factorials" constructed from the sequence (ak). A P -sequence (ak) for R is easy to construct inductively: Let a0; : : : ; a[R:P ] 1 be a complete set of residues mod P with a0 = 0. Once ak has been de ned for k < [R : Pn 1] (while n N ), de ne ak for [R : Pn 1] k < [R : Pn] as follows: let b0 = 0, b1 , : : : , b[Pn 1:Pn] 1 be a complete set of residues of Pn 1 mod Pn ; then, for k = j[R : Pn 1] + r with 0 r < [R : Pn 1] and 1 j < [Pn 1 : Pn], let ak = bj + ar . After a0; : : : ; ajRj 1 have been de ned (necessarily a complete enumeration of the elements of R), continue the sequence jRj-periodically. In the following Lemma, we use the convention that P1 = (0). 7 LEMMA. Let R be a suitable nite local ring with maximal ideal P of nilpotency N , and hxin the falling factorial of degree n constructed from a P -sequence (ak). Then for all n 2 N0 , 8r 2 R hrin 2 P (n) and if (n) < N then hanin 62 P (n)+1: Proof. If n jRj (equivalent to (n) =1) then, since a0; : : : ; ajRj 1 enumerate all elements of R, hrin = 0 for all r. If n < jRj then (n) =PNk=1 h n [R:Pk]i, while hrin 2 P e , where e = N 1 Xk=1 k fj j 0 j < n; r aj 2 P k n P k+1g + +N fj j 0 j < n; r aj 2 PNg = N Xk=1 fj j 0 j < n; r aj 2 P kg and (by de nition of suitable) hrin is in no higher power of P if e j , we have f(aj) cjhajij mod Pn . Also, hajij is in no higher power of P than P (j) . Therefore f(aj) 2 Pn implies cj 2 Pn (j) . COROLLARY 1. In the situation of the Proposition, for 0 j < (n), let Cj be a complete set of residues mod Pn (j) . Then every function on R=Pn arising from a polynomial in R[x] arises from a unique polynomial of the form f(x) = (n) 1 Xj=0 cjhxij with cj 2 Cj : For R = Zpn , other canonical forms for the functions representable by polynomials have been given by Dueball [4], Aizenberg, Semion and Tsitkin [1] and Rosenberg [14] (the latter for polynomials in several variables). COROLLARY 2. In the situation of the Proposition, if n> 0 then for every function induced on the residue classes of Pn 1 by a polynomial in R[x], there are exactly (n) 1 Y j=0 [Pn (j) 1 : Pn (j)] di erent polynomial functions on the residue classes of Pn that reduce to the given function mod Pn 1 . If [P k 1 : P k] = q for 1 k N then the expression simpli es to q q(n) , where q(n) is the minimal m 2 N such that q(m) =Pj 1 h n qj i n. Proof of the formula for jF(R)j in Theorem 2: jF(R)j= (N) 1 Y j=0 [R : PN (j)] follows immediately from Corollary 1 with n=N . In the special case that [P k 1:P k]=q for 1 k N , writing sk for the number of di erent functions on R=P k arising from polynomials in R[x], we see from Corollary 2 that q q(k)sk 1= sk . Therefore qPNk=1 (k)= sN = jF(R)j in that case. 9 4 THE GROUPP(R/P2) We want to determine the structure of the group P(R=P 2) with respect to composition of functions, R being a suitable nite local ring as above. To simplify notation, we consider the group P(R), where R is a nite local ring with maximal ideal P of nilpotency N = 2. Some notational conventions: We write the group of invertible elements of a monoid M as M . If M is a monoid and H a monoid acting on a set S then the wreath product M oH is the monoid de ned on the set H MS by the operation (h; (ms)s2S)(g; (ls)s2S) = (hg; (mg(s)ls)s2S): If M acts on a set T then the standard action of M oH on S T is (h; (ms)s2S)(x; y) = (h(x);mx(y)): Note that an element (h; (ms)s2S) is in (M oH) if and only if h2H and ms 2M for all s 2 S , and that therefore (M oH) 'M oH . If D is a commutative ring and M a D-module, we write A D (M) for the semigroup with respect to compostion of transformations of M of the form x 7! ax + b with a 2 D and b 2 M . We have jA D (M)j= jD=Ann(M) M j. PROPOSITION 2. Let R be a nite local ring with maximal ideal P of nilpotency 2 and q = [R : P ]. Denote by QQ the semigroup of functions from a set of q elements to itself. Then F(R)' A R=P (P )oQQ and P(R) ' A R=P (P )oSq; and in particular, jF(R)j= qq jRjq and jP(R)j = q! (q 1)q jP jq: Proof. Fix a system of representatives Q of R mod P . We identify R with Q P by r 7! (s; t) with s 2 Q, t 2 P , such that r = s+ t. Let f 2 R[x]. We have f(r) = f(s+ t) = f(s) + f 0(s)t; 10 since this holds mod P 2 by Taylor's Theorem and P 2 = (0) in R. Now let '(s) be the representative in Q of f(s) + P , then f(s+ t) = '(s) + (f(s) '(s)) + f 0(s)t; with '(s)2Q and f(s) '(s)2P . We regard f 0(s) as being in R=P . (As it gets multiplied by t2P , only its residue class mod P matters). If we associate to f 2 R[x] the functions 'f :Q! Q and f :Q! A R=P (P ), where 'f (s) is the representative in Q of f(s) + P f (s) is the transformation x 7! af (s)x+ bf (s) on P , where af (s) 2 R=P is f 0(s) mod P , bf (s) = f(s) '(s) 2 P then 'f and f completely determine the function induced by f on R. Moreover, the function de ned on Q P by ' 2 QQ , a 2 (R=P )Q and b 2 PQ via (s; t) 7! '(s) + a(s)t + b(s) determines ', a and b uniquely, such that for f; g2R[x] inducing the same function on R we have 'g = 'f and g = f . Therefore f 7! ('f ; f ) depends only on the function induced by f 2 R[x] on R and de nes a homomorphism from F(R) to A R=P (P )oQQ , which takes the action of F(R) on R (identi ed with Q P ) to the standard action of AoQQ arising from the obvious actions of A on P and of QQ on Q. We have already seen that this homomorphism is injective. To check surjectivity, we show that every triple of functions ':Q! Q, b:Q! P and a:Q! R=P actually occurs as 'f , af and bf for some f 2 R[x]. Every pair of functions on R=P arises as f mod P and f 0 mod P for some polynomial f 2 R[x], because R=P is a nite eld. This takes care of 'f and af . Since the characteristic function of every residue class of P is induced by a polynomial in R[x] (just take a su ciently high power of a polynomial representing it mod P ), we can adjust f to take prescribed values on the s 2 Q, by adding a P -linear combination of these characteristic functions. This produces a prescribed bf without disturbing the values of f and f 0 mod P , since we only add a polynomial in P [x]. 11 If we restrict to polynomials representing permutations or, equiv-alently, to polynomials for which 'f is a permutation of Q andaf (s) 6= 0 + P for all s 2 Q, we get an isomorphism of P(R) andA R=P (P)oSq , which takes the action of P(R) on R (identi ed withQ P ) to the standard action of the wreath product on Q P arisingfrom the obvious actions of A R=P on P and of the symmetric groupSq on Q.REMARK. We may simplify the expression for P(R) by noting thatA R=P (P ) is isomorphic to the semi-direct product ((R=P ) ; )n(P;+)with (R=P ) acting on (P;+) through the scalar mulutiplication ofthe R=P -vectorspace structure on P .Proof of the formula for jP(R)j in Theorem 2: For n N , let sndenote the number of functions on the residue classes of Pn induced bypolynomials in R[x] and tn the number of them that are permutations.If n 2, a polynomial induces a permutation mod Pn if and only if itinduces a permutation mod P and its derivative is nowhere zero modP , cf. [7]. In particular, if n> 2, a polynomial induces a permutationmod Pn if and only if it induces one mod Pn 1 . Together with thefact that every class of polynomial functions mod Pn reducing tothe same function mod Pn 1 contains the same number of elements(Corollary 2 of Proposition 1), this implies that tntn 1 = snsn 1 for alln > 2, and therefore tn = t2s2 sn for all n 2.From Proposition 2 applied to R=P 2 we get t2 = q!(q 1)q[P :P 2]qand s2 = qq[R : P 2]q and the formula for jP(R)j follows. REFERENCES[1] N. Aizenberg, I. Semion, and A. Tsitkin, Polynomial represen-tations of logical functions, Automatic Control and ComputerSciences (transl. of Automatika i Vychislitel'naya Tekhnika,Acad. Nauk Latv. SSR (Riga)), 5 (1971), pp. 5{11 (orig. 6{13).[2] D. A. Ashlock, Permutation polynomials of Abelian group ringsover nite elds, J. Pure Appl. Algebra, 86 (1993), pp. 1{5.12 [3] J. V. Brawley and G. L. Mullen, Functions and polynomials overGalois rings, J. Number Theory, 41 (1992), pp. 156{166.[4] F. Dueball, Bestimmung von Polynomen aus ihren Wertenmod pn , Math. Nachr., 3 (1949/50), pp. 71{76.[5] G. Keller and F. Olson, Counting polynomial functions(mod pn), Duke Math. J., 35 (1968), pp. 835{838.[6] A. J. Kempner, Polynomials and their residue systems, Trans.Amer. Math. Soc., 22 (1921), pp. 240{266, 267{288.[7] B. R. McDonald, Finite Rings with Identity, Dekker, 1974.[8] G. L. Mullen and H. Niederreiter, The structure of a groupof permutation polynomials, J. Austral. Math. Soc. Ser. A, 38(1985), pp. 164{170.[9] W. Narkiewicz, Polynomial Mappings, vol. 1600 of Lecture Notesin Mathematics, Springer, 1995.[10] A. Nechaev, Polynomial transformations of nite commutativelocal rings of principal ideals, Math. Notes, 27 (1980), pp. 425{432. transl. from Mat. Zametki 27 (1980) 885-897, 989.[11] W. Nobauer, Gruppen von Restpolynomidealrestklassen nachPrimzahlpotenzen, Monatsh. Math., 59 (1955), pp. 194{202.[12] L. Redei and T. Szele, Algebraisch-zahlentheoretische Betracht-ungen uber Ringe I, Acta Math. (Uppsala), 79 (1947), pp. 291{320.[13], Algebraisch-zahlentheoretische Betrachtungen uber RingeII, Acta Math. (Uppsala), 82 (1950), pp. 209{241.[14] I. G. Rosenberg, Polynomial functions over nite rings, Glas.Mat., 10 (1975), pp. 25{33.[15] D. Singmaster, On polynomial functions (mod m), J. NumberTheory, 6 (1974), pp. 345{352.[16] Th. Skolem, Einige Satze uber Polynome, Avh. Norske Vid.Akad. Oslo, I. Mat.-Naturv. Kl., 4 (1940), pp. 1{16.13 [17] R. Spira, Polynomial interpolation over commutative rings,Amer. Math. Monthly, 75 (1968), pp. 638{640.[18] J. Wiesenbauer, On polynomial functions over residue class ringsof Z, in Contributions to general algebra 2 (Proc. of Conf.in Klagenfurt 1982), Holder-Pichler-Tempsky, Teubner, 1983,pp. 395{398.