List of open problems

List of open problems We give here a list of open problems which appear in the main text, where attributions and comments may be found. I. Characterize commutative rings which have a representation ring...1.5 II. Characterize those pairs of subsets S\,.(h of a domain R for which Int(S!> R) = Int(S2, R) holds,, Il.5 III. Describe domains R with Int(R) = R[X] I1.7 IV. Describe domains having a regular basis.. V.3 V. Let k 2. Describe the ring SdZ) of all polynomials P with rational coefficients, such that P, pi,,p(k) map Z in Z VI.3 VI. The same question for rings other than Z VII. For which domains R is the ring Int(R) Noetherian? VIII. Describe domains R which have an associated Skolem ring IX. Determine sets having the Skolem property in the general case VI.3 VII.l VIlA VIL5 X. Characterize non-noetherian domains R, for which Int(R) is a Priifer domain VII.9 XI. Describe the strong 2-generators in Int(Z) ".. VILlO XII. Do there exist Dedekind domains with class-group having at least 3 elements which are not D-rings? VIII.2 XIII. Do there exist Dedekind domains which are not localizations of Dedekind domains which are not D-rings? VII 1.2 XIV. Does every field having the property (P) have also the property (SP)? IX.3 XV. Is the property (P) preserved under finit.e extensions?,. IX.3 XVI. Give a constructive description of fields with the property (P) or (SP) IX.3 XVII. Does the field generat.ed by the quadratic root.s of all primes have the property (P)? Does it have the propert.y (SP)? IX.3

111 XVIII. The analogue of problems XV, XVI and XVII for the property (R) IX.8 XIX. Is the property (P) equivalent to (P) or (SP)? X.7 XX. Is the property (K) equivalent to (P), (SP)or (P)? XI.4 XXI. Describe the possible cycle-lengths for polynomials in algebraic number fields of fixed degree 2: 3 XII. 9

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Absolute height 81 absolute value 77 absolute values, equivalent 77 AczeI,J.25 admissile map 68 almost Dedekind domain 23 Anderson,D.D.23 Anderson,D.F.23,55 annihilator 24 archimedean absolute value 77 arithmetically dense set 20 Artin,E. 79,80 Avanissian,V. 15 INDEX Baker,A. 15 Baker,I.N. 99,102 Baron,G.105,107,108 Barsky,D. 45 Bass,H.33 Benzaghou,B. 60 Bergweiler, W. 102 Bessis,D. 109 Besivin.Jc-P. 15 Bezout ring 48 Bhattacharyya,P. 102 Boduch,J. 107 Bouvier,A. 55 Brawley.J.V. 9 Brewer,J.59 Brizolis,D. 21,41,45-47,49,51,52,55-57,59,61-63,65 Bruijn,N.G.de 45 Buck,C.R. 15 Bundschuh,P. 15 Cahen,P.-J. 15,20,21,23-25.29,31,45,46,.55,56,64 Carlitz,L. 7,11,44 Carlson,F. 15

125 Chabert,J.-L. 15,19,21,23-25,45,4,49,51,52,55-60,65,66 Chasse.G. 98 Chatelet.F. 53 cohomology groups 37 compatible function 46 completely invariant set 67 cycle 97 cycle length 97 Cycl(F) 68 cyclic element 68 cyclic elements 97 cyclic P6lya fields 39 CyclK(J) 97 Cycln(F) 68 cyclotomic P6lya fields 36 Dickson,L.E. 96 dimension 55 Dobbs,D.E.55 D-ring 51,62 Dueball,F. 9 Equivalent absolute values 77 Evertse,J.H. 106 Fatou ring 60 Fatou,D. 102 Fatou,D.67 Feher,J.46 finite F-cycle 68 fixed divisor 26 fixpoint of order n 97 Fontana,M. 55 Foster,A.L. 46 F(R) 1 Fukasawa,S. 15 full subset 20 fully invariant set 67 function, compatible 46 Galois ring 9 Gauss's Lemma 82 Gay,R. 15 Gelfond,A.O. 15 Gerboud,G. 17-19,23,37 Geronimo,S. 109 Gihner,R. 20,23,48,59,65 Goldman,O.51

Gramain,F. 15,24 Grazzini,F. 45 group of units 61 Gruman,L. 15 Gunji,H. 12,29,33,51,61-64 126 HO(G,A) 37 Hl(G,A) 37 Hall,R.R. 45 Halter-Koch,F. 17,18,70,73,75,88,96,108 Haouat,Y. 45,56 Hardy,G.H. 14 height 80 height of a polynomial 30 Heinzer,W.48,59 Heitmann,R.C. 58 Hensel,K. 29,30 Hilbert property 65 Hilbert ring 51 Hilbert's theorem 90 38 Infinite F-cycle 68 Int(A, R) 12 integer-valued polynomial 12 Int(R) 12 Int(R) 12 invariant set 67 IR 1 Ivanyi.A. 46 Jacob,G.34 Jacobson radical 66 Jacobson ring 51 Jaffard domain 55 Jankowski,L. 93 Kabbaj.S. 55 Keller,G.9 Kempner,A.J.9,12 Klingler,L. 59 Kochen,S. 64 Kronecker,L. 52 Krull dimension 55 Krull,W.51 Kubota,K.K. 85,93,95,96 Landau,E. 14 Lantz,D.48

127 Lantz,D.59 Laohakosol,V. 45 leading form 68 length of a cycle 97 Lenstra constant 104 Lenstra,H.W. 104 Leutbecher,A. 104 Lewis,D.J.30,64,73,95 Liardet,P. 73,75,83-85,95 Lind,D.A. 63,64 Linden,F.J.van der 40 linearly conjugated polynomials 102 localization 56 Loper,A.64 Marlewski,A. 93 Masley,J.M. 36 Masser,D. 15 Matlis,E. 58 McQuillan,D.L. 12,20,29,33,49,51,56-58,61-65 Mignotte,M. 15 modular function 45 module without polynomial torsion 24 Montgomery,H.L. 36 Morton,P.64,104,107,108 Moussa,P. 109 Mullen,G.9,10 Nagell,T. 12,13 Nakano.S. 23 Narkiewicz,W. 17,18,64,70,73,75,76,85-88,93,96,105 Necaev,A.A. 9 Niklasch,G. 104 Niven 10 non-archimedean absolute value 77 norm 77 normalized fraction 59 Northcott height 92 Northcott,D.G. 83,92 Nobauer,W.46 Olson,F.R. 9 order of a cyclic element 97 order of a fixpoint 97 Ostrowski,A. 31,35,36 Permutation polynomial 10 Pezda,T. 102,107

128 Phong,B.M. 46 Picard group 59 Pisot,C. 15 P6Iya,G. 12,14,15,26,31,36 Polya field 31,35,36 Polya field, cyclotomic 36 Polya field, quadratic 36,40 Polya field, quadratic 40 Polya field, cyclic 39 P(R) 1 Prestel,A. 64 product formula 77 property (K)??? property (P) 69 property (P)?? property (R)?? property 5 51,62 property (SP) 68 Priifer domain 56 pseudo-polynomials 45 Quadratic Polya field 36,40 Rausch,U.46 Rauzy,G.20 Redei,L.2-6,12 regular basis 31 representation pair 4 representation ring 4 R-fractional set 56 ring of polynomial functions 1 rings with property 5 51,52 Ripoli,C.C. 64 Robinson,R.M. 15 Rogers,K. 46 Roquette,P. 64 Rosenberg,I.G. 9 Rush,D.E. 58 Russo,P. 107 Ruzsa,I. 46 R[X] 1 5 00 41 Sally,J. 58 Sato,D. 14 Schnitzer,F.J. 15 Schulting,H.W. 59

129 Selberg,A. 15 Shapiro, H.N. 93 Shibata,F. 21,23 Silverman,J.H. 104,108 Singmaster,D. 33 S-integral106 s, 41 Skolem property 49,52 Skolem ring 48 Skolem,T.4,49,51,52,65,66 Smith,W.W.58,59 Somayajulu,A. 46 Spira,R. 9 Stackel,P. 14 Steinitz,E. 86 Stevens,H. 9,10 Straus,E.G. 14,41,46,47 strong 2-generator 59 strong Hilbert property 65 strong Skolem property ('i5 Subbarao,M.Y.46 Sugatani,T. 21 Sugatani,T. 23 S-unit 106 Szekeres,G. 53 Szele,T.2-6,12 Trotter,P.G. 53 Turk,J.30 Ubolsri,P. 45 Uchida,K. 36 universal function 45 Valuation domain 56 Vasconcelos,W. 58 Wagner,C.G.45 Walde,R. 107 Waldschmidt,M. 15 Walliser,R. 15 Warren,Leroy J. 10 Whaples,G.79 Whaples,G.80 Wiesenbauer J. 2,9 WPT24 Yoshida,K.21,23

Zafrullah,M. 23 Zantema,H. 36,37,39,40 130

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