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Interpolation Domains
Journal of Algebra 225, 794᎐803 Ž2000. doi:10.1006rjabr.1999.8151, available online at http:rrwww.idealibrary.com on
Interpolation Domains Paul-Jean Cahen Faculte´ des Sciences de Saint Jerome, ´ ˆ 13397 Marseille Cedex 20 France E-mail: [email protected]
Jean-Luc Chabert Faculte´ de Mathematiques et d’Informatique, 80039 Amiens Cedex 01 France ´ E-mail: [email protected]
and Sophie Frisch Institut fur ¨ Mathematik C, Technische Uni¨ ersitat ¨ Graz, Steyrergasse 30, A-8010 Graz, Austria E-mail: [email protected] Communicated by E¨ a Bayer-Flickiger Received November 20, 1998
Call a domain D with quotient field K an interpolation domain if, for each choice of distinct arguments a1 , . . . , a n and arbitrary values c1 , . . . , c n in D, there exists an integer-valued polynomial f Žthat is, f g K w X x with f Ž D . : Ž D .., such that f Ž a i . s c i for 1 F i F n. We characterize completely the interpolation domains if D is Noetherian or a Prufer ¨ domain. In the first case, we show that D is an interpolation domain if and only if it is one-dimensional, locally unibranched with finite residue fields, in the second one, if and only if the ring IntŽ D . s f g K w X x< f Ž D . : D4 of integer-valued polynomials is itself a Prufer ¨ domain. We also show that an interpolation domain must satisfy a double-boundedness condition, and thereby simplify a recent characterization of the domains D such that IntŽ D . is a Prufer ¨ domain. 䊚 2000 Academic Press Key Words: interpolation; integer-valued polynomials; Dedekind and almost Dedekind domains; unibranched domains; double-boundedness.
794 0021-8693r00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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INTRODUCTION Let K be a field. By the Lagrange interpolation formula, there is a polynomial, with coefficients in K, assigning given values to given distinct elements in K. The same does not hold for a domain D Žwhich is not a field., as polynomials in Dw X x preserve congruences modulo every ideal of D. However, for certain domains D interpolation may be possible using integer-¨ alued polynomials, that is, polynomials f with coefficients in the quotient field of D, such that f Ž D . : D. For instance, if D s ⺪ is the ring of integers, it is clear that the binomial polynomials Ž nX . s X Ž X y 1 . ⭈⭈⭈n! Ž X y n q 1 . are integer-valued, hence, by linear combination, there is an integer-valued polynomial of degree n assigning given values to the integers 0 to n Žand, changing X to X y a, to any finite set of consecutive integers.. It follows from a result of Carlitz w4, Theorem 7.1x, together with Lagrange interpolation, that this is also the case for D s ⺖q w t x Ža proof using an interpolation sequence that runs through ⺖q w t x bijectively has been given by Wagner w7x.. Using such interpolation sequences one could extend this property to a discrete rank-one valuation domain with finite residue field Žgiving also an estimate for the degree of interpolation polynomials, w5x.. More generally, it is known that interpolation by integer-valued polynomials is possible in every Dedekind domain with finite residue fields w5x. In this paper we completely classify the domains for which interpolation by integer-valued polynomials is possible, among Noetherian and Prufer ¨ domains. We adopt the usual notation of IntŽ D . for the ring of integer-valued polynomials on the domain D, that is, IntŽ D . s f g K w X x< f Ž D . : D4 , where K denotes the quotient field of D, and we set the following definition: DEFINITION 1. The domain D is an interpolation domain if, for each choice of distinct arguments a1 , . . . , a n and arbitrary values c1 , . . . , c n in D, there exists f g IntŽ D . such that f Ž a i . s c i , for 1 F i F n. In the first section, we give necessary conditions. We find that every interpolation domain is one dimensional with finite residue fields and satisfies the following double-boundedness condition: for each nonzero z g D there is an integer n such that, for each maximal ideal ᒊ containing z, z f ᒊ n, and < Drᒊ < F n. In the second section, after giving some properties of localization, we characterize the interpolation domains in the Noetherian case: a one-dimensional Noetherian domain with finite residue fields is an interpolation
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domain if and only if it is locally unibranched Žwe recall the definition within the said section.. In the last section we let D be a Prufer ¨ domain. We first note that an interpolation Prufer domain is necessarily an almost Dedekind domain ¨ with finite residue fields. Under this condition, we show that the following conditions are equivalent: Ži. D is an interpolation domain, Žii. D satisfies the double-boundedness condition, Žiii. IntŽ D . is a Prufer ¨ domain. The equivalence of Žii. and Žiii. simplifies Alan Loper’s characterization of the domains D such that IntŽ D . is a Prufer ¨ domain w6x. As seen at the beginning, the case of a field is trivial by the Lagrange interpolation formula. In what follows, we always assume D to be a domain that is not a field.
1. NECESSARY CONDITIONS We first note that the interpolation property amounts to the possibility of assigning arbitrary values to every pair of distinct elements. PROPOSITION 1.1. The following assertions are equi¨ alent for a domain D. Ži. D is an interpolation domain. Žii. For each pair of distinct elements a, b in D, there exists f g IntŽ D . such that f Ž a. s 0 and f Ž b . s 1. Žiii. For each pair of distinct elements a, b in D, and for each maximal ideal ᒊ of D, there exists f g IntŽ D . such that f Ž a. g ᒊ and f Ž b . f ᒊ. Proof. Ži. « Žiii. Obvious. Žiii. « Žii. Let ᑯ be the set of polynomials g g IntŽ D . such that g Ž a. s 0. Then ᑯ is an ideal of IntŽ D .. Let ᑯ Ž b . s g Ž b .< g g ᑯ 4 . Then ᑯ Ž b . is an ideal of D. Assuming Žiii., for each maximal ideal ᒊ, there is f g IntŽ D . such that f Ž a. g ᒊ and f Ž b . f ᒊ. Then g s f y f Ž a. belongs to ᑯ and is such that g Ž b . f ᒊ. Hence ᑯ Ž b . is not contained in any maximal ideal of D. Therefore ᑯ Ž b . s D, and in particular, there exists f g ᑯ such that f Ž b . s 1. Žii. « Ži. Use linear combinations of products of the polynomials existing by Žii.. Recall that, for each element a g D, and each maximal ideal ᒊ of D, the set ᑧ a s f g IntŽ D .< f Ž a. g ᒊ 4 is a maximal ideal of IntŽ D . Žwith residue field isomorphic to Drᒊ .. The third condition for the Proposition 1.1 is clearly equivalent to saying that these ideals are distinct, for distinct elements of D. This simple remark is the key for our characterization of
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Noetherian interpolation domains wSect. 2x. Hence, although it is just a rewording of the previous result, we give it as a corollary. COROLLARY 1.2. A domain D is an interpolation domain if and only if, for each pair of distinct elements a, b in D, and for each maximal ideal ᒊ of D, the ideals ᑧ a and ᑧ b of IntŽ D . are distinct. Remark 1.3. From Proposition 1.1, it is also easy to generalize interpolation to several indeterminates: if D is an interpolation domain, there is an integer-valued polynomial in n indeterminates assigning given values to distinct arguments in D n. Given a pair of distinct elements a, b in D n, it is enough to find a polynomial such that f Ž a. s 0 and f Ž b . s 1. Writing a s Ž ␣ 1 , . . . , ␣ n . and b s Ž  1 , . . . , n ., then ␣ i / i for some i, hence there is an integer-valued polynomial f Ž X i . in one indeterminate, such that f Ž ␣ i . s 0 and f Ž i . s 1. Such a polynomial can be considered as an integer-valued polynomial in n indeterminates. Next we address the issue of finding interpolating polynomials of prescribed degree. For this, we first establish a property of continuity. PROPOSITION 1.4. Let D be a domain, ᑾ be an ideal, and a g D. If f g IntŽ D . is of degree n, then z g ᑾ n implies Ž f Ž a q z . y f Ž a.. g ᑾ. Proof. It suffices to prove the assertion for z of the special form z s ty, with t g ᑾ and y g ᑾ ny 1. Indeed, each z g ᑾ n is a finite sum z s Ý ris1 z i of such special elements, hence Ž f Ž a q z . y f Ž a.. is a finite sum of elements of the form Ž f Ž bi q z i . y f Ž bi ... Replacing f Ž X . by f Ž X q a. y f Ž a., we assume that f Ž0. s 0, and show that f Ž ty . g ᑾ. The result is obvious if n s 0. For n G 1, we let g Ž X . s f Ž tX . y t n f Ž X .. Then g is a polynomial of degree at most n y 1, and such that g Ž0. s 0. By induction, we have g Ž y . g ᑾ. The result follows, since f Ž ty . s g Ž y . q t n f Ž y . and t n g ᑾ. In particular, for every ideal ᑾ of D, each f g IntŽ D . is a uniformly continuous function from D to D, in the ᑾ-adic topology: if degŽ f . F n, then z g ᑾ n h implies Ž f Ž a q z . y f Ž a.. g ᑾ h. We give two necessary conditions for the existence of an interpolating polynomial of given degree. LEMMA 1.5. Let D be a domain and z be a nonzero element of D for which there exists a polynomial f g IntŽ D . of degree n with f Ž0. s 0 and f Ž z . s 1. Then, for e¨ ery prime ideal ᒍ of D, we ha¨ e Ž1. Ž2.
z f ᒍ n, if z g ᒍ, then < Drᒍ < F n.
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Proof. Ž1. Since we have Ž f Ž z . y f Ž0.. f ᒍ, it follows from Proposition 1.4 that z f ᒍ n. Ž2. Assume that n - < Drᒍ <. Choosing n q 1 elements mutually incongruent modulo ᒍ in D, and writing f as the Lagrange interpolation polynomial assigning the same values as f to these elements, we see that f g Dᒍ w X x Žsee also w1, Corollary I.3.3x.. Therefore 1 s f Ž z . y f Ž0. g zDᒍ . We reach a contradiction if z belongs to ᒍ. If z is a nonzero element and ᒊ a maximal ideal of D, we denote by wᒊ Ž z . the order of z with respect to ᒊ Žthat is, the largest integer n such that z g ᒊ n .. For an interpolation domain, this order is well defined. PROPOSITION 1.6. Let D be an interpolation domain. Then D is one dimensional with finite residue fields. Moreo¨ er, for each nonzero element z g D, there is an integer n such that, for each maximal ideal ᒊ containing z, we ha¨ e the double boundedness condition < Drᒊ < F n and wᒊ Ž z . F n. Proof. Assume D is an interpolation domain. For each nonzero element z g D, there exists a polynomial f g IntŽ D ., of some degree n, such that f Ž0. s 0, and f Ž z . s 1. It follows from the previous lemma that, for each prime ideal ᒍ containing z, z f ᒍ n and < Drᒍ < F n. In particular, each nonzero prime ideal has a finite residue field Žsince it contains a nonzero element., and D is one dimensional. Remarks 1.7. Ž1. The proof of Lemma 1.5 suggests that, if D is an interpolation domain, then IntŽ D . is not contained in Dᒍ w X x for any prime ideal ᒍ. In fact, IntŽ D . is not contained in Bw X x for any overring B of D Žthat is, any ring B containing D and strictly contained in K .. Indeed, consider a nonzero prime ideal ᑪ of B. Then ᒍ s ᑪ l D is a nonzero prime ideal of D. If IntŽ D . : Bw X x, then for each f g IntŽ D ., if a and b are congruent modulo ᒍ in D, we have Ž f Ž b . y f Ž a.. g ᑪ l D s ᒍ. Ž2. If D is a one-dimensional local Noetherian domain with finite residue field, the conditions of Proposition 1.6 are satisfied. The characterization we give in the Noetherian case will show that these conditions are not sufficient wSect. 2x. Ž3. The double-boundedness condition of Proposition 1.6 is very similar to the condition given by Loper for the characterization of the domains D such that IntŽ D . is a Prufer ¨ domain w6x. We come back to this issue in the last section.
2. NOETHERIAN INTERPOLATION DOMAINS We open this section with a property of localization. Recall that, for a multiplicative subset S of a domain D, we have the containment
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Sy1 IntŽ D . : IntŽ Sy1 D . w1, Proposition I.2.2x. In general, this containment is strict, but if D is Noetherian, we have an equality w1, Theorem I.2.3x. LEMMA 2.1. Let D be an interpolation domain. Then Sy1 D is an interpolation domain for each multiplicati¨ e subset S of D. Proof. Let ars, brs be two distinct elements of Sy1 D, where a, b g D and s g S. By hypothesis, there exists a polynomial f g IntŽ D . such that f Ž a. s 0 and f Ž b . s 1. Set g Ž X . s f Ž sX ., then g Ž ars . s 0, g Ž brs . s 1, and g g IntŽ D . : IntŽ Sy1 D .. The conclusion follows from Proposition 1.1. We turn now to the Noetherian case. PROPOSITION 2.2. Let D be Noetherian. Then D is an interpolation domain if and only if Dᒊ is an interpolation domain for e¨ ery maximal ideal ᒊ of D. Proof. It remains to show that, if each Dᒊ is an interpolation domain, then so is D. Let a / b in D, and ᒊ be a maximal ideal of D. By hypothesis, there is a polynomial f g IntŽ Dᒊ . such that f Ž a. s 0 and f Ž b . s 1. Since D is Noetherian, we have IntŽ D . ᒊ s IntŽ Dᒊ .: there is s g D, s f ᒊ such that sf g IntŽ D .. Clearly sf Ž a. g ᒊ and sf Ž b . f ᒊ. The conclusion follows from Proposition 1.1. Remark 2.3. For non-Noetherian D, it may be that each Dᒊ is an interpolation domain while D is not. For instance, if D is an almost Dedekind domain with finite residue fields, then each Dᒊ is a discrete valuation domain with finite residue field, hence an interpolation domain Žas noted in the introduction, see also Theorem 2.4 below., but D does not necessarily satisfy the double-boundedness condition of Proposition 1.6 Žsee Sect. 3 below.. By Proposition 1.6 we may assume that D is a one-dimensional domain with finite residue fields and by Proposition 2.2 that D is local with maximal ideal ᒊ. Under these hypotheses, it is known that the ideals ᑧ a s f g IntŽ D .< f Ž a. g ᒊ 4 are distinct if and only if D is unibranched, that is, the integral closure DX of D is a local ring Žor equivalently a rank-one discrete valuation domain.; and that, if D is not unibranched, there are only finitely many distinct maximal ideals of the form ᑧ a w1, Theorem V.3.1 and Proposition V.3.10x. From Corollary 1.2 it then follows that D is an interpolation domain if and only if it is unibranched. In the global case, we derive immediately the following characterization. THEOREM 2.4. Let D be a Noetherian domain. Then D is an interpolation domain if and only if it is one dimensional with finite residue fields and locally unibranched.
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Remarks 2.5. Ž1. Assume that D is a local one-dimensional Noetherian ˆ the domain, with maximal ideal ᒊ and finite residue field. Denote by D completion of D in the ᒊ-adic topology. Integer-valued polynomials are ˆ Given a / b, uniformly continuous functions and can be extended to D. there is clearly a continuous function such that Ž a. s 0 and Ž b . s 1. ˆ to Dˆ were arbitrarily uniformly If the continuous functions from D approximated by integer-valued polynomials, analogously to the classical Stone᎐Weierstrass theorem, it would be easy to find f g IntŽ D . such that f Ž a. s 0 and f Ž b . s 1. This approximation property holds if and only if D is analytically irreducible w1, Theorem III.5.3x Žanalytically irreducible means ˆ is a domain and implies that D is unibranched.. that D Ž2. If D is a one-dimensional Noetherian local ring with finite residue field, either D is an interpolation domain, and the ideals ᑧ a are all distinct, or D is not unibranched, and there are only finitely many ideals of the form ᑧ a w1, Proposition V.3.10x. In general, for a quasi-local domain D, it may be that there are infinitely many ideals of the form ᑧ a which are not all distinct. This is obviously the case if < Drᒊ < is infinite, in which case IntŽ D . s Dw X x is not an interpolation domain, while ᑧ a s ᑧ b if and only if a ' b Žmod ᒊ .. But here is a less trivial example. Let V be a valuation domain, with maximal ideal ᒊ, such that Vrᒊ is finite and ᒊ is a principal ideal. If the dimension of V is strictly greater than one, then V is not an interpolation domain. However, letting ᒍ s F⬁1 ᒊ n, then ᒍ / Ž0. and ᑧ a s ᑧ b in IntŽ V . if and only if a ' b Žmod ᒍ . w2, Theoreme ´ ` 2.2x. COROLLARY 2.6. If D is a Noetherian interpolation domain, then each o¨ erring of D is an interpolation domain. Proof. Let B be an overring of the Noetherian interpolation domain D. It follows from Theorem 2.4 and from the Krull᎐Akizuki theorem that B is a one-dimensional Noetherian domain with finite residue fields. Moreover, for each maximal ideal ᒋ of B, Bn contains Dᒊ where ᒊ s ᒋ l D. Hence the integral closure BnX of Bn is an overring of the X X integral closure Dᒊ of Dᒊ . Since Dᒊ is a valuation domain, then so is BnX .
¨ 3. PRUFER DOMAINS We finally turn to the case where D is a Prufer domain: For each ¨ maximal ideal ᒊ, Dᒊ is a valuation domain. If D satisfies the doubleboundedness condition of Proposition 1.6, Dᒊ is a rank-one discrete valuation domain with finite residue field. In other words, D is an almost Dedekind domain with finite residue fields. We derive the following characterization
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THEOREM 3.1. Let D be a Prufer ¨ domain. Then D is an interpolation domain if and only if, for each nonzero element z g D, there is an integer n such that, for each maximal ideal ᒊ containing z, we ha¨ e the doubleboundedness condition < Drᒊ < F n and wᒊ Ž z . F n. Proof. It follows from Proposition 1.6 that the double-boundedness condition is necessary. Conversely, as we said above, it implies that D is an almost Dedekind domain with finite residue fields. To each maximal ideal ᒊ of D corresponds an essential valuation ¨ ᒊ of D, and for z g D, the order wᒊ Ž z . is simply ¨ ᒊ Ž z .. Consider then a / b in D and denote by V the set of essential valuations of D such that ¨ Ž b y a. ) 0. We shall construct a polynomial f g IntŽ D . such that f Ž a. s 0 and ¨ Ž f Ž b .. s 0 for each ¨ g V . From Proposition 1.1 this will complete the proof since, on the other hand, g s X y a is such that g Ž a. s 0 and ¨ Ž g Ž b .. s 0 for each essential valuation ¨ f V . Since there is an integer n such that ¨ mŽ b y a. F n for each ᒊ containing Ž b y a., there is an integer e such that e¨ Ž y . is a multiple of ¨ Ž b y a. for each y g D and each ¨ g V Žfor instance, e s lcm ¨ Ž b y a.< ¨ g V 4.. Since there is an integer n such that < Drᒊ < F n for each ᒊ containing Ž b y a., there is also an integer q such that ¨ Ž y qy 1 y 1. ) 0 for each ¨ g V and each y g D such that ¨ Ž y . s 0 Žfor instance, q y 1 s lcm q¨ y 1 < ¨ g V 4 , where q¨ s < Drᒊ ¨ <.. We define a sequence of polynomials by e
f 0 Ž X . s Ž X y a . , and
fn Ž X . s
f n-1 Ž X . Ž f ny1 Ž X .
qy1
y 1.
bya
e
.
We see by induction that ¨ Ž f nŽ y .. is a non-negative multiple of ¨ Ž b y a. for each y g D and each ¨ g V , and hence, that f n g IntŽ D .. On the other hand, we clearly have f nŽ a. s 0 for all n. Finally, if ¨ Ž f ny1Ž b .. ) 0, qy 1 Ž . x e . s 0, hence ¨ Ž f nŽ b .. s ¨ Ž f ny1Ž b .. y ¨ Ž b y a.. we have ¨ Žw f ny 1 b y1 For n F e, we then have ¨ Ž f n Ž b . . s ¨ Ž f 0 Ž b . . y n¨ Ž b y a . s e¨ Ž b y a . y n¨ Ž b y a . .
We conclude that f e is the polynomial we are looking for. If IntŽ D . is a Prufer ¨ domain, then D is an almost Dedekind domain with finite residue fields w1, Proposition VI.1.5x. On the other hand, the double-boundedness condition is very similar to the condition given by Loper to characterize the almost Dedekind domains D such that IntŽ D . is a Prufer ¨ domain w6x. Let us state Loper’s condition Žas he did himself. in the case where the characteristic of D is 0: IntŽ D . is a Prufer ¨ domain if and only if, for each prime element p of ⺪, there is an integer n, such that, for each essential valuation ¨ of D such that ¨ Ž p . ) 0, ¨ Ž p . F n and
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< Drᒊ < F n. ŽIn the case where the characteristic of D is a nonzero prime number q, he then gives a similar condition replacing ⺪ by the ring of polynomials ⺖q w t x.. Our condition is a priori stronger than Loper’s, since we do not restrict ourselves to the prime elements of a subring of D. In fact, it follows from the next result that both conditions are equivalent. COROLLARY 3.2. Let D be a Prufer ¨ domain. Then D is an interpolation domain if and only if IntŽ D . is a Prufer ¨ domain. Proof. If IntŽ D . is a Prufer ¨ domain, then D is an almost Dedekind domain with finite residue fields w1, Proposition VI.1.5x. Let ᒊ be a maximal ideal of D, b be an element of D, and consider the maximal ideal ᑧ b s h g IntŽ D .< hŽ b . g ᒊ 4 of IntŽ D .. Clearly, the localization IntŽ D . ᑧ b is contained in the valuation domain W s g K Ž X .< Ž b . g Dᒊ 4 . Since IntŽ D . is a Prufer ¨ domain, IntŽ D . ᑧ b is itself a valuation domain, and we have the equality IntŽ D . ᑧ b s W w3, remark 2.5x. Since Dᒊ is a rank-one discrete valuation domain with finite residue field, it is an interpolation domain. Hence, for a / b, there exists a polynomial f g IntŽ Dᒊ . such that f Ž a. s 0 and f Ž b . s 1. Obviously, IntŽ Dᒊ . : W, thus f g IntŽ D . ᑧ b, and there is g g IntŽ D ., g f ᑧ b such that gf g IntŽ D .. Clearly gf Ž a. g ᒊ and gf Ž b . f ᒊ. It follows from Proposition 1.1 that D is an interpolation domain. Conversely, if D is an interpolation domain, it satisfies the doubleboundedness condition, and it is known that this condition, under Alan Loper’s weaker form, implies that IntŽ D . is a Prufer ¨ domain w1, Proposition VI.4.4 and Remark VI.4.5x. Remarks 3.3. Ž1. This proof via interpolation domains avoids Loper’s difficult argument that the double-boundedness condition is necessary for IntŽ D . to be a Prufer ¨ domain w6x. Moreover, it shows that this condition holds in its stronger form. Ž2. For a Dedekind domain D with finite residue fields, it is implicit in Loper’s characterization that IntŽ D . is a Prufer ¨ domain if and only if it is not contained in Dᒊ w X x for any maximal ideal ᒊ of D w6x. This can be related to the fact that, if D is an interpolation domain, then IntŽ D . is not contained in Bw X x for any overring B of D wRemark 1.7 Ž1.x. Similarly to the Noetherian case wCorollary 2.6x, we finally have the following. COROLLARY 3.4. If the Prufer ¨ domain D is an interpolation domain, then each o¨ erring of D is an interpolation domain. Proof. If B is an overring of an almost Dedekind domain D, each maximal ideal ᒊ of B is above a maximal ideal ᒋ s ᒊ l D of D, and
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Bᒊ s Dᒋ . If D is an interpolation domain, the double-boundedness condition satisfied by D is then also clearly satisfied by B. We end this paper with a question raised by Corollaries 2.6 and 3.4. QUESTION 3.5. Assume that D is an interpolation domain. Is each o¨ erring of D an interpolation domain? In particular, is the integral closure of D an interpolation domain?
REFERENCES 1. P.-J. Cahen and J.-L. Chabert, Integer-valued polynomials, Amer. Math. Soc. Sur¨ eys and Monographs 48, Providence, 1997. 2. P.-J. Cahen and Y. Haouat, Polynomes ˆ `a valeurs entieres ` sur un anneau de pseudo-valuation, Manuscripta Math. 61 Ž1988., 23᎐31. 3. J.-L. Chabert, Integer-valued polynomials, Prufer ¨ domains and localization, Proc. Amer. Math. Soc. 118 Ž1993., 1061᎐1073. 4. L. Carlitz, Finite sums and interpolation formulas over GFw p n , x x, Duke Math. J. 15 Ž1948., 1001᎐1012. 5. S. Frisch, Interpolation by integer-valued polynomials, J. Algebra 211 Ž1999., 562᎐577. 6. A. Loper, A classification of all domains D such that IntŽ D . is a Prufer ¨ domain, Proc. Amer. Math. Soc. 126 Ž1998., 657᎐660. 7. C. G. Wagner, Interpolation series for continuous functions on -adic completions of GFŽ q, x ., Acta Arith. 17 Ž1971., 389᎐406.
Interpolation Domains Paul-Jean Cahen Faculte´ des Sciences de Saint Jerome, ´ ˆ 13397 Marseille Cedex 20 France E-mail: [email protected]
Jean-Luc Chabert Faculte´ de Mathematiques et d’Informatique, 80039 Amiens Cedex 01 France ´ E-mail: [email protected]
and Sophie Frisch Institut fur ¨ Mathematik C, Technische Uni¨ ersitat ¨ Graz, Steyrergasse 30, A-8010 Graz, Austria E-mail: [email protected] Communicated by E¨ a Bayer-Flickiger Received November 20, 1998
Call a domain D with quotient field K an interpolation domain if, for each choice of distinct arguments a1 , . . . , a n and arbitrary values c1 , . . . , c n in D, there exists an integer-valued polynomial f Žthat is, f g K w X x with f Ž D . : Ž D .., such that f Ž a i . s c i for 1 F i F n. We characterize completely the interpolation domains if D is Noetherian or a Prufer ¨ domain. In the first case, we show that D is an interpolation domain if and only if it is one-dimensional, locally unibranched with finite residue fields, in the second one, if and only if the ring IntŽ D . s f g K w X x< f Ž D . : D4 of integer-valued polynomials is itself a Prufer ¨ domain. We also show that an interpolation domain must satisfy a double-boundedness condition, and thereby simplify a recent characterization of the domains D such that IntŽ D . is a Prufer ¨ domain. 䊚 2000 Academic Press Key Words: interpolation; integer-valued polynomials; Dedekind and almost Dedekind domains; unibranched domains; double-boundedness.
794 0021-8693r00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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INTRODUCTION Let K be a field. By the Lagrange interpolation formula, there is a polynomial, with coefficients in K, assigning given values to given distinct elements in K. The same does not hold for a domain D Žwhich is not a field., as polynomials in Dw X x preserve congruences modulo every ideal of D. However, for certain domains D interpolation may be possible using integer-¨ alued polynomials, that is, polynomials f with coefficients in the quotient field of D, such that f Ž D . : D. For instance, if D s ⺪ is the ring of integers, it is clear that the binomial polynomials Ž nX . s X Ž X y 1 . ⭈⭈⭈n! Ž X y n q 1 . are integer-valued, hence, by linear combination, there is an integer-valued polynomial of degree n assigning given values to the integers 0 to n Žand, changing X to X y a, to any finite set of consecutive integers.. It follows from a result of Carlitz w4, Theorem 7.1x, together with Lagrange interpolation, that this is also the case for D s ⺖q w t x Ža proof using an interpolation sequence that runs through ⺖q w t x bijectively has been given by Wagner w7x.. Using such interpolation sequences one could extend this property to a discrete rank-one valuation domain with finite residue field Žgiving also an estimate for the degree of interpolation polynomials, w5x.. More generally, it is known that interpolation by integer-valued polynomials is possible in every Dedekind domain with finite residue fields w5x. In this paper we completely classify the domains for which interpolation by integer-valued polynomials is possible, among Noetherian and Prufer ¨ domains. We adopt the usual notation of IntŽ D . for the ring of integer-valued polynomials on the domain D, that is, IntŽ D . s f g K w X x< f Ž D . : D4 , where K denotes the quotient field of D, and we set the following definition: DEFINITION 1. The domain D is an interpolation domain if, for each choice of distinct arguments a1 , . . . , a n and arbitrary values c1 , . . . , c n in D, there exists f g IntŽ D . such that f Ž a i . s c i , for 1 F i F n. In the first section, we give necessary conditions. We find that every interpolation domain is one dimensional with finite residue fields and satisfies the following double-boundedness condition: for each nonzero z g D there is an integer n such that, for each maximal ideal ᒊ containing z, z f ᒊ n, and < Drᒊ < F n. In the second section, after giving some properties of localization, we characterize the interpolation domains in the Noetherian case: a one-dimensional Noetherian domain with finite residue fields is an interpolation
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domain if and only if it is locally unibranched Žwe recall the definition within the said section.. In the last section we let D be a Prufer ¨ domain. We first note that an interpolation Prufer domain is necessarily an almost Dedekind domain ¨ with finite residue fields. Under this condition, we show that the following conditions are equivalent: Ži. D is an interpolation domain, Žii. D satisfies the double-boundedness condition, Žiii. IntŽ D . is a Prufer ¨ domain. The equivalence of Žii. and Žiii. simplifies Alan Loper’s characterization of the domains D such that IntŽ D . is a Prufer ¨ domain w6x. As seen at the beginning, the case of a field is trivial by the Lagrange interpolation formula. In what follows, we always assume D to be a domain that is not a field.
1. NECESSARY CONDITIONS We first note that the interpolation property amounts to the possibility of assigning arbitrary values to every pair of distinct elements. PROPOSITION 1.1. The following assertions are equi¨ alent for a domain D. Ži. D is an interpolation domain. Žii. For each pair of distinct elements a, b in D, there exists f g IntŽ D . such that f Ž a. s 0 and f Ž b . s 1. Žiii. For each pair of distinct elements a, b in D, and for each maximal ideal ᒊ of D, there exists f g IntŽ D . such that f Ž a. g ᒊ and f Ž b . f ᒊ. Proof. Ži. « Žiii. Obvious. Žiii. « Žii. Let ᑯ be the set of polynomials g g IntŽ D . such that g Ž a. s 0. Then ᑯ is an ideal of IntŽ D .. Let ᑯ Ž b . s g Ž b .< g g ᑯ 4 . Then ᑯ Ž b . is an ideal of D. Assuming Žiii., for each maximal ideal ᒊ, there is f g IntŽ D . such that f Ž a. g ᒊ and f Ž b . f ᒊ. Then g s f y f Ž a. belongs to ᑯ and is such that g Ž b . f ᒊ. Hence ᑯ Ž b . is not contained in any maximal ideal of D. Therefore ᑯ Ž b . s D, and in particular, there exists f g ᑯ such that f Ž b . s 1. Žii. « Ži. Use linear combinations of products of the polynomials existing by Žii.. Recall that, for each element a g D, and each maximal ideal ᒊ of D, the set ᑧ a s f g IntŽ D .< f Ž a. g ᒊ 4 is a maximal ideal of IntŽ D . Žwith residue field isomorphic to Drᒊ .. The third condition for the Proposition 1.1 is clearly equivalent to saying that these ideals are distinct, for distinct elements of D. This simple remark is the key for our characterization of
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Noetherian interpolation domains wSect. 2x. Hence, although it is just a rewording of the previous result, we give it as a corollary. COROLLARY 1.2. A domain D is an interpolation domain if and only if, for each pair of distinct elements a, b in D, and for each maximal ideal ᒊ of D, the ideals ᑧ a and ᑧ b of IntŽ D . are distinct. Remark 1.3. From Proposition 1.1, it is also easy to generalize interpolation to several indeterminates: if D is an interpolation domain, there is an integer-valued polynomial in n indeterminates assigning given values to distinct arguments in D n. Given a pair of distinct elements a, b in D n, it is enough to find a polynomial such that f Ž a. s 0 and f Ž b . s 1. Writing a s Ž ␣ 1 , . . . , ␣ n . and b s Ž  1 , . . . , n ., then ␣ i / i for some i, hence there is an integer-valued polynomial f Ž X i . in one indeterminate, such that f Ž ␣ i . s 0 and f Ž i . s 1. Such a polynomial can be considered as an integer-valued polynomial in n indeterminates. Next we address the issue of finding interpolating polynomials of prescribed degree. For this, we first establish a property of continuity. PROPOSITION 1.4. Let D be a domain, ᑾ be an ideal, and a g D. If f g IntŽ D . is of degree n, then z g ᑾ n implies Ž f Ž a q z . y f Ž a.. g ᑾ. Proof. It suffices to prove the assertion for z of the special form z s ty, with t g ᑾ and y g ᑾ ny 1. Indeed, each z g ᑾ n is a finite sum z s Ý ris1 z i of such special elements, hence Ž f Ž a q z . y f Ž a.. is a finite sum of elements of the form Ž f Ž bi q z i . y f Ž bi ... Replacing f Ž X . by f Ž X q a. y f Ž a., we assume that f Ž0. s 0, and show that f Ž ty . g ᑾ. The result is obvious if n s 0. For n G 1, we let g Ž X . s f Ž tX . y t n f Ž X .. Then g is a polynomial of degree at most n y 1, and such that g Ž0. s 0. By induction, we have g Ž y . g ᑾ. The result follows, since f Ž ty . s g Ž y . q t n f Ž y . and t n g ᑾ. In particular, for every ideal ᑾ of D, each f g IntŽ D . is a uniformly continuous function from D to D, in the ᑾ-adic topology: if degŽ f . F n, then z g ᑾ n h implies Ž f Ž a q z . y f Ž a.. g ᑾ h. We give two necessary conditions for the existence of an interpolating polynomial of given degree. LEMMA 1.5. Let D be a domain and z be a nonzero element of D for which there exists a polynomial f g IntŽ D . of degree n with f Ž0. s 0 and f Ž z . s 1. Then, for e¨ ery prime ideal ᒍ of D, we ha¨ e Ž1. Ž2.
z f ᒍ n, if z g ᒍ, then < Drᒍ < F n.
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Proof. Ž1. Since we have Ž f Ž z . y f Ž0.. f ᒍ, it follows from Proposition 1.4 that z f ᒍ n. Ž2. Assume that n - < Drᒍ <. Choosing n q 1 elements mutually incongruent modulo ᒍ in D, and writing f as the Lagrange interpolation polynomial assigning the same values as f to these elements, we see that f g Dᒍ w X x Žsee also w1, Corollary I.3.3x.. Therefore 1 s f Ž z . y f Ž0. g zDᒍ . We reach a contradiction if z belongs to ᒍ. If z is a nonzero element and ᒊ a maximal ideal of D, we denote by wᒊ Ž z . the order of z with respect to ᒊ Žthat is, the largest integer n such that z g ᒊ n .. For an interpolation domain, this order is well defined. PROPOSITION 1.6. Let D be an interpolation domain. Then D is one dimensional with finite residue fields. Moreo¨ er, for each nonzero element z g D, there is an integer n such that, for each maximal ideal ᒊ containing z, we ha¨ e the double boundedness condition < Drᒊ < F n and wᒊ Ž z . F n. Proof. Assume D is an interpolation domain. For each nonzero element z g D, there exists a polynomial f g IntŽ D ., of some degree n, such that f Ž0. s 0, and f Ž z . s 1. It follows from the previous lemma that, for each prime ideal ᒍ containing z, z f ᒍ n and < Drᒍ < F n. In particular, each nonzero prime ideal has a finite residue field Žsince it contains a nonzero element., and D is one dimensional. Remarks 1.7. Ž1. The proof of Lemma 1.5 suggests that, if D is an interpolation domain, then IntŽ D . is not contained in Dᒍ w X x for any prime ideal ᒍ. In fact, IntŽ D . is not contained in Bw X x for any overring B of D Žthat is, any ring B containing D and strictly contained in K .. Indeed, consider a nonzero prime ideal ᑪ of B. Then ᒍ s ᑪ l D is a nonzero prime ideal of D. If IntŽ D . : Bw X x, then for each f g IntŽ D ., if a and b are congruent modulo ᒍ in D, we have Ž f Ž b . y f Ž a.. g ᑪ l D s ᒍ. Ž2. If D is a one-dimensional local Noetherian domain with finite residue field, the conditions of Proposition 1.6 are satisfied. The characterization we give in the Noetherian case will show that these conditions are not sufficient wSect. 2x. Ž3. The double-boundedness condition of Proposition 1.6 is very similar to the condition given by Loper for the characterization of the domains D such that IntŽ D . is a Prufer ¨ domain w6x. We come back to this issue in the last section.
2. NOETHERIAN INTERPOLATION DOMAINS We open this section with a property of localization. Recall that, for a multiplicative subset S of a domain D, we have the containment
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Sy1 IntŽ D . : IntŽ Sy1 D . w1, Proposition I.2.2x. In general, this containment is strict, but if D is Noetherian, we have an equality w1, Theorem I.2.3x. LEMMA 2.1. Let D be an interpolation domain. Then Sy1 D is an interpolation domain for each multiplicati¨ e subset S of D. Proof. Let ars, brs be two distinct elements of Sy1 D, where a, b g D and s g S. By hypothesis, there exists a polynomial f g IntŽ D . such that f Ž a. s 0 and f Ž b . s 1. Set g Ž X . s f Ž sX ., then g Ž ars . s 0, g Ž brs . s 1, and g g IntŽ D . : IntŽ Sy1 D .. The conclusion follows from Proposition 1.1. We turn now to the Noetherian case. PROPOSITION 2.2. Let D be Noetherian. Then D is an interpolation domain if and only if Dᒊ is an interpolation domain for e¨ ery maximal ideal ᒊ of D. Proof. It remains to show that, if each Dᒊ is an interpolation domain, then so is D. Let a / b in D, and ᒊ be a maximal ideal of D. By hypothesis, there is a polynomial f g IntŽ Dᒊ . such that f Ž a. s 0 and f Ž b . s 1. Since D is Noetherian, we have IntŽ D . ᒊ s IntŽ Dᒊ .: there is s g D, s f ᒊ such that sf g IntŽ D .. Clearly sf Ž a. g ᒊ and sf Ž b . f ᒊ. The conclusion follows from Proposition 1.1. Remark 2.3. For non-Noetherian D, it may be that each Dᒊ is an interpolation domain while D is not. For instance, if D is an almost Dedekind domain with finite residue fields, then each Dᒊ is a discrete valuation domain with finite residue field, hence an interpolation domain Žas noted in the introduction, see also Theorem 2.4 below., but D does not necessarily satisfy the double-boundedness condition of Proposition 1.6 Žsee Sect. 3 below.. By Proposition 1.6 we may assume that D is a one-dimensional domain with finite residue fields and by Proposition 2.2 that D is local with maximal ideal ᒊ. Under these hypotheses, it is known that the ideals ᑧ a s f g IntŽ D .< f Ž a. g ᒊ 4 are distinct if and only if D is unibranched, that is, the integral closure DX of D is a local ring Žor equivalently a rank-one discrete valuation domain.; and that, if D is not unibranched, there are only finitely many distinct maximal ideals of the form ᑧ a w1, Theorem V.3.1 and Proposition V.3.10x. From Corollary 1.2 it then follows that D is an interpolation domain if and only if it is unibranched. In the global case, we derive immediately the following characterization. THEOREM 2.4. Let D be a Noetherian domain. Then D is an interpolation domain if and only if it is one dimensional with finite residue fields and locally unibranched.
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Remarks 2.5. Ž1. Assume that D is a local one-dimensional Noetherian ˆ the domain, with maximal ideal ᒊ and finite residue field. Denote by D completion of D in the ᒊ-adic topology. Integer-valued polynomials are ˆ Given a / b, uniformly continuous functions and can be extended to D. there is clearly a continuous function such that Ž a. s 0 and Ž b . s 1. ˆ to Dˆ were arbitrarily uniformly If the continuous functions from D approximated by integer-valued polynomials, analogously to the classical Stone᎐Weierstrass theorem, it would be easy to find f g IntŽ D . such that f Ž a. s 0 and f Ž b . s 1. This approximation property holds if and only if D is analytically irreducible w1, Theorem III.5.3x Žanalytically irreducible means ˆ is a domain and implies that D is unibranched.. that D Ž2. If D is a one-dimensional Noetherian local ring with finite residue field, either D is an interpolation domain, and the ideals ᑧ a are all distinct, or D is not unibranched, and there are only finitely many ideals of the form ᑧ a w1, Proposition V.3.10x. In general, for a quasi-local domain D, it may be that there are infinitely many ideals of the form ᑧ a which are not all distinct. This is obviously the case if < Drᒊ < is infinite, in which case IntŽ D . s Dw X x is not an interpolation domain, while ᑧ a s ᑧ b if and only if a ' b Žmod ᒊ .. But here is a less trivial example. Let V be a valuation domain, with maximal ideal ᒊ, such that Vrᒊ is finite and ᒊ is a principal ideal. If the dimension of V is strictly greater than one, then V is not an interpolation domain. However, letting ᒍ s F⬁1 ᒊ n, then ᒍ / Ž0. and ᑧ a s ᑧ b in IntŽ V . if and only if a ' b Žmod ᒍ . w2, Theoreme ´ ` 2.2x. COROLLARY 2.6. If D is a Noetherian interpolation domain, then each o¨ erring of D is an interpolation domain. Proof. Let B be an overring of the Noetherian interpolation domain D. It follows from Theorem 2.4 and from the Krull᎐Akizuki theorem that B is a one-dimensional Noetherian domain with finite residue fields. Moreover, for each maximal ideal ᒋ of B, Bn contains Dᒊ where ᒊ s ᒋ l D. Hence the integral closure BnX of Bn is an overring of the X X integral closure Dᒊ of Dᒊ . Since Dᒊ is a valuation domain, then so is BnX .
¨ 3. PRUFER DOMAINS We finally turn to the case where D is a Prufer domain: For each ¨ maximal ideal ᒊ, Dᒊ is a valuation domain. If D satisfies the doubleboundedness condition of Proposition 1.6, Dᒊ is a rank-one discrete valuation domain with finite residue field. In other words, D is an almost Dedekind domain with finite residue fields. We derive the following characterization
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THEOREM 3.1. Let D be a Prufer ¨ domain. Then D is an interpolation domain if and only if, for each nonzero element z g D, there is an integer n such that, for each maximal ideal ᒊ containing z, we ha¨ e the doubleboundedness condition < Drᒊ < F n and wᒊ Ž z . F n. Proof. It follows from Proposition 1.6 that the double-boundedness condition is necessary. Conversely, as we said above, it implies that D is an almost Dedekind domain with finite residue fields. To each maximal ideal ᒊ of D corresponds an essential valuation ¨ ᒊ of D, and for z g D, the order wᒊ Ž z . is simply ¨ ᒊ Ž z .. Consider then a / b in D and denote by V the set of essential valuations of D such that ¨ Ž b y a. ) 0. We shall construct a polynomial f g IntŽ D . such that f Ž a. s 0 and ¨ Ž f Ž b .. s 0 for each ¨ g V . From Proposition 1.1 this will complete the proof since, on the other hand, g s X y a is such that g Ž a. s 0 and ¨ Ž g Ž b .. s 0 for each essential valuation ¨ f V . Since there is an integer n such that ¨ mŽ b y a. F n for each ᒊ containing Ž b y a., there is an integer e such that e¨ Ž y . is a multiple of ¨ Ž b y a. for each y g D and each ¨ g V Žfor instance, e s lcm ¨ Ž b y a.< ¨ g V 4.. Since there is an integer n such that < Drᒊ < F n for each ᒊ containing Ž b y a., there is also an integer q such that ¨ Ž y qy 1 y 1. ) 0 for each ¨ g V and each y g D such that ¨ Ž y . s 0 Žfor instance, q y 1 s lcm q¨ y 1 < ¨ g V 4 , where q¨ s < Drᒊ ¨ <.. We define a sequence of polynomials by e
f 0 Ž X . s Ž X y a . , and
fn Ž X . s
f n-1 Ž X . Ž f ny1 Ž X .
qy1
y 1.
bya
e
.
We see by induction that ¨ Ž f nŽ y .. is a non-negative multiple of ¨ Ž b y a. for each y g D and each ¨ g V , and hence, that f n g IntŽ D .. On the other hand, we clearly have f nŽ a. s 0 for all n. Finally, if ¨ Ž f ny1Ž b .. ) 0, qy 1 Ž . x e . s 0, hence ¨ Ž f nŽ b .. s ¨ Ž f ny1Ž b .. y ¨ Ž b y a.. we have ¨ Žw f ny 1 b y1 For n F e, we then have ¨ Ž f n Ž b . . s ¨ Ž f 0 Ž b . . y n¨ Ž b y a . s e¨ Ž b y a . y n¨ Ž b y a . .
We conclude that f e is the polynomial we are looking for. If IntŽ D . is a Prufer ¨ domain, then D is an almost Dedekind domain with finite residue fields w1, Proposition VI.1.5x. On the other hand, the double-boundedness condition is very similar to the condition given by Loper to characterize the almost Dedekind domains D such that IntŽ D . is a Prufer ¨ domain w6x. Let us state Loper’s condition Žas he did himself. in the case where the characteristic of D is 0: IntŽ D . is a Prufer ¨ domain if and only if, for each prime element p of ⺪, there is an integer n, such that, for each essential valuation ¨ of D such that ¨ Ž p . ) 0, ¨ Ž p . F n and
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< Drᒊ < F n. ŽIn the case where the characteristic of D is a nonzero prime number q, he then gives a similar condition replacing ⺪ by the ring of polynomials ⺖q w t x.. Our condition is a priori stronger than Loper’s, since we do not restrict ourselves to the prime elements of a subring of D. In fact, it follows from the next result that both conditions are equivalent. COROLLARY 3.2. Let D be a Prufer ¨ domain. Then D is an interpolation domain if and only if IntŽ D . is a Prufer ¨ domain. Proof. If IntŽ D . is a Prufer ¨ domain, then D is an almost Dedekind domain with finite residue fields w1, Proposition VI.1.5x. Let ᒊ be a maximal ideal of D, b be an element of D, and consider the maximal ideal ᑧ b s h g IntŽ D .< hŽ b . g ᒊ 4 of IntŽ D .. Clearly, the localization IntŽ D . ᑧ b is contained in the valuation domain W s g K Ž X .< Ž b . g Dᒊ 4 . Since IntŽ D . is a Prufer ¨ domain, IntŽ D . ᑧ b is itself a valuation domain, and we have the equality IntŽ D . ᑧ b s W w3, remark 2.5x. Since Dᒊ is a rank-one discrete valuation domain with finite residue field, it is an interpolation domain. Hence, for a / b, there exists a polynomial f g IntŽ Dᒊ . such that f Ž a. s 0 and f Ž b . s 1. Obviously, IntŽ Dᒊ . : W, thus f g IntŽ D . ᑧ b, and there is g g IntŽ D ., g f ᑧ b such that gf g IntŽ D .. Clearly gf Ž a. g ᒊ and gf Ž b . f ᒊ. It follows from Proposition 1.1 that D is an interpolation domain. Conversely, if D is an interpolation domain, it satisfies the doubleboundedness condition, and it is known that this condition, under Alan Loper’s weaker form, implies that IntŽ D . is a Prufer ¨ domain w1, Proposition VI.4.4 and Remark VI.4.5x. Remarks 3.3. Ž1. This proof via interpolation domains avoids Loper’s difficult argument that the double-boundedness condition is necessary for IntŽ D . to be a Prufer ¨ domain w6x. Moreover, it shows that this condition holds in its stronger form. Ž2. For a Dedekind domain D with finite residue fields, it is implicit in Loper’s characterization that IntŽ D . is a Prufer ¨ domain if and only if it is not contained in Dᒊ w X x for any maximal ideal ᒊ of D w6x. This can be related to the fact that, if D is an interpolation domain, then IntŽ D . is not contained in Bw X x for any overring B of D wRemark 1.7 Ž1.x. Similarly to the Noetherian case wCorollary 2.6x, we finally have the following. COROLLARY 3.4. If the Prufer ¨ domain D is an interpolation domain, then each o¨ erring of D is an interpolation domain. Proof. If B is an overring of an almost Dedekind domain D, each maximal ideal ᒊ of B is above a maximal ideal ᒋ s ᒊ l D of D, and
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Bᒊ s Dᒋ . If D is an interpolation domain, the double-boundedness condition satisfied by D is then also clearly satisfied by B. We end this paper with a question raised by Corollaries 2.6 and 3.4. QUESTION 3.5. Assume that D is an interpolation domain. Is each o¨ erring of D an interpolation domain? In particular, is the integral closure of D an interpolation domain?
REFERENCES 1. P.-J. Cahen and J.-L. Chabert, Integer-valued polynomials, Amer. Math. Soc. Sur¨ eys and Monographs 48, Providence, 1997. 2. P.-J. Cahen and Y. Haouat, Polynomes ˆ `a valeurs entieres ` sur un anneau de pseudo-valuation, Manuscripta Math. 61 Ž1988., 23᎐31. 3. J.-L. Chabert, Integer-valued polynomials, Prufer ¨ domains and localization, Proc. Amer. Math. Soc. 118 Ž1993., 1061᎐1073. 4. L. Carlitz, Finite sums and interpolation formulas over GFw p n , x x, Duke Math. J. 15 Ž1948., 1001᎐1012. 5. S. Frisch, Interpolation by integer-valued polynomials, J. Algebra 211 Ž1999., 562᎐577. 6. A. Loper, A classification of all domains D such that IntŽ D . is a Prufer ¨ domain, Proc. Amer. Math. Soc. 126 Ž1998., 657᎐660. 7. C. G. Wagner, Interpolation series for continuous functions on -adic completions of GFŽ q, x ., Acta Arith. 17 Ž1971., 389᎐406.