List of open problems

Transcription

1 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 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

2 111 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

3 REFERENCES BOOKS: [A] van der WAERDEN,B.L. Algebm, 5th ed., Springer MR 38#1968. [B] BOURBAI\I,N. Alqebre commutative, VII, Diviseurs, Hermann, MR 41#5339. [CF] CASSELS,J.W.S., FROHLICH,A. (editors), Alqebraic Number Theory, Academic Press, MR 35#6500. [DG] LANG,S. Fundamentals of Diophantine Geometry, Springer \983. MR 85j: [EATAN] NARKIEWICZ,W. Elementary and Analytic Theory of Algebraic Numbers, Springer & PWN, MR 91h: [H] HASSE,H. Zahlentheorie, Akadernie Verlag \949. MR J1 p.580. [IRF] BEARDON,A.F. Iteration of rational functions, Springer \991. MR 92j: [KR] KRONECKER L. Vorlesunqen iiber Zahlentheorie I, Leipzig [MD] McDONALD,B.R. Finite rings with identiu), I\'l.Dekker MR 50#7245. [MIT] GILMER,R. Multiplicative Ideal Theory. M.Dekker, MR 55#323. [SH] SHAFAREVITCH,I.R. Fundamentals of Algebraic Geomet1'y, 2nd ed., Moscow MR 90g: (Russian) PAPERS: ACZEL,J. [60] Ueber die Gleichlieii del' Polunomfunctionen auf Ringen, Acta Sci. Math., 21, 1960, MR 24#AI291a. ANDERSON,D.D.,ANDERSON,D.F.,ZAFRULLAH,M. [91] Rings between D[X] and I<[X], Houston J.Math., 17, 1991, MR 92c:J3014. ANDERSON,D.F., BOUVIER,A., DOBBS,D.E., FONTANA,M., KABBAJ, S. [88] On laffard domains, Expos. Math., 5, 1988, MR 89c: ARTIN,E.,WHAPLES,G. [45] Axionratic characterization of fields by the product formula [or valuations, Bull. Arner. Math. Soc., 51, 1945, MR7p.111. AVANISSIAN,V.,GAY,R. [75] Sur une transformation des [onctionnelles analytiques et ses applications aur [onciions eniieres de plusieurs variables, Bull. Soc. Math. France, 103, 197fi, : MR fi:3#848.

4 BAKER,A. [67] A note on inteqral integer-valued functions of several variables, Proc. Cambridge Phil. Soc., 63, 1967, MR 35#4168. BAKER,I.N. [59a] Fizpoints and iterates of entire functions, Math. Zeitschr., 71,1959, MR 21# [59b] Some entire functions with fixpoints of every order, J. Austral. Math. Soc. 1, 1959/61, MR 22#4838a. - [60] The existence of fixpoints of entire functions, Math, Zeitschr., 73, 1960, MR 22#4838b.- - [64] Fixpoints of polynomials and rational functions, J. London Math. Soc., 39, 1964, MR 30#230. BARSKY,D. [72] Polsjnomes dont les derioees sont a oaleurs entieres et fonctions k-lipschitziennes sur un anneau local, These 3 cycle, Univ. Paris VII, [73] Fonctions k-lipschitziennes sur till anneau local et polynomes aualeurs entieres, Bull. Soc. Math. France, 101, 1973, MR 51#8080. BASS,H. [63] Big projective modules are free, Illinois.J. Mat.h., 7, 1963, MR 26#1341. BATRA,A.,MORTON,P. [94] Algebraic dimamics ofpolynomial maps on the algebmic closure of a finite field,!, Rocky Mount. J. Math., 24, 1994, BENZAGHOU,B. [70] Algebres de Hadamard, Bull. Soc. Math. France, 98, 1970, MR 44#1658. BERGWEILER,W. [91] Periodic points of entire functions: proof of a conjecture of Baker, Complex Variable Theory Appl., 17, 1991, [93] Iteration of meromorphic functions, Bull. Amer. Math. Soc., 29, 1993, BEZIVIN,J.-P. [84] Une genemlisation a plusieurs variables d'un resultat de Gel'[ond, Analysis, 4,1984, MR 86 : BODUCH,J. [90] Polynomial cycles in rings of alqebraic integers, MA thesis, Wroclaw University (Polish) BRAWLEY,J.V., MULLEN,G.L. [92] Functions and polynomials over Galois rings, J. Number Th., 41,1992, IVIR 93d: BREWER,J.,KLINGLER L. [91] The rinq of inteqer-uolued polynomials of a semi-local principal-ideal domain, Lin. Alg. Appl., 15, 1991, MR 92g: BRIZOLIS,D. [74] On the ratios of integer-valued polynomials over any algebraic number field, American Mat.h. Monthly, 81,1974, MR 50# [75] Hilbert rings of tnieqer-ualued polynomials, Comm. Algebra, 3, 1975, MR 52# [76] Ideals in rings of integer-valued polynomials,,1. reine angew. Math., 285, 1976, MR 54# [79] A theorem on ideals in Priijer rings of integer- valued polynomials, Comm. Algebra, 7, 1979, MR 80j:l:3013. BRIZOLIS,D., STRAUS,E.G. [76] A basis for the ring of double integervalued polynomials,.j. reine angew. Math.. 286/287, 1976, MR 54#5168.

5 114 de BRUIJN,N.G. [55) Some classes of inteqer-valued functions, Indag. Math., 17, 1955, MR 17 p.128. BUCK,C.R. [48) Integral valued entire functions. Duke l\iath. J., 15, 1948, MR 10 p.693. BUNDSCHUH,P. [80) Arithmetische Eiqenscluiften qanzer Ftmktionen mehreren Variablen, J. reine angew. Math., 313, 1980, MR 8li:l0037. CAHEN,P.-J. [72) Polsjnomes a valeurs entieres, Canad. J. Math., 24,1972, MR 46# [75) Polimomes et derioees a ualeurs entiers, Ann. Sci. Univ. Clermont II, ser. math., 10, 1975, MR 53# [78) Fractions rationnelles a oaleurs entieres, Ann. Sci. Univ. Clermont II, ser. math., 16, 1978, MR 80e: [90] Dimension de 1'anneau des polunomes d oaleurs eniieres, Manuscr. Math., 67, 1990, MR 91e: [91] Polunomes a ualeurs entieres SUI' un (l1l1lea u non analytiquement irreductible, J. reine angew. Math., 418, 1991, MR 92i: [93] Parties pleines d'un anneau noetherien, J. Algebra, 157, 1993, CAHEN,P.-J., CHABERT, J.-L. [71) Coefficients et oaleurs d'un polynome, Bull. Sci. math., (2), 95, 1971, MR 45#5126. CAHEN,P.J.,HAOUAT,Y. [88) Polsjnomes a ualeurs eniieres sur un anneau de pseudoualuation, Manuscr. Math., 61, MR 89f: CARLITZ,L. [59] A note on integral-valued polynomials, Indag. Math., 21, 1959, , MR 21# [63] Permutations in a finite field, Acta Sci. Math., 24, 1963, MR 28#81. - [64] Functions and polynomials (mod pn), Acta Arith., 9, 1964, MR 29#4721. CARLSON,F. [21] Uber qanzuieriiqe Funktionen, Math. Zeitschr., 11, 1921, CHABERT,J.-L. [71) Anneaux de "polunomes avaleurs eniieres" et anneaux de Fatou, Bull. Soc. Math. France, 99, 1971, MR 46# [77] Les ideauz premiers de I'anneau des polunomes d valeurs entieres, J. reine angew. Math., 293/294, 1977, MR 56# [78] Polunomes a ualeurs eniieres et proprieu de Skolem, J. reine angew. Math., 303/304, 1978, , MR 80d: [79a] Anneaux de Skolem, Archiv Math., 32, 1979, MR 81k: [79b] Polunomes a ualeurs eniieres atnsi que leurs deriuees, Ann. Sci. Univ, Clermont II, ser. math., 18, 1979, MR 81d: [82] La propriete de Skolem forte, VI Congr. Int. Math. d'expression latine, Gauthier-Villars, 1982, MR 83i: [83] Le Nullstellensatz de Hilbert et II'S polsjnomes (; valeurs entieres. Monatsh. Math., 95, 1983, MR 86j: [87] u«anneau de Prufer, J. Algebra, 107, 1987, MR 88i:13022.

6 [88] Ldeaux de polsmomes et ideaux de ualeurs. l\lanuscr. Math., 60, 1988, MR 89j: [91] Anneaux de polsjnomes a ualeurs entieres et anneallx de Priijer, Comptes Rendus Acad. Sci. Paris, 312,1991, MR 92d: [93] Derioees et differences dioisees a oaleurs entiers, Acta Arith., 63, 1993, CHABERT, J.-L., GERBOUD,G. [93] Polunonies a ualeurs entieres et binomes de Fermat, Canad. J. Math., 45, 1993,6-26. CHASSE,G. [86] Combinatorial cycles of a polynomial map over a commutative field, Discrete Math., 61, 1986, MR 87k: CHATELET,F. [67] Les ideaux de I'anneau des polunomes d'une variable ci coefficients entiers, Alg. Zahlentheorie, 43-51, Bibl. Inst. Mannheim MR 36#2611. DICKSON,L.E. [10] Definite forms in a finite field, Trans. Amer. Math. Soc., 10, 1909, DUEBALL,F. [49] Bestimmung von Polynomen aus ihren Werten (mod pn), Math. Nachr., 3, 1949, MR 11 p.715. EVERTSE,J.H. [84] On equations in S units and the Thue Mahler equation, Inv. math., 75, 1984, MR 85f:ll048. FATOU,D. [19] Sur les equations [onctionnelles, Bull. Soc. Math. France, 47, 1919, ; 48, 1920,33-94,208-: [26] Sur I'iieration des [onctions transcendanies entieres, Acta Mathematica, 47, 1926, FOSTER,A.L. [67] Semiprimal algebras: Churacterization and normal decomposition, Math. Zeitschr., 99, 1967, MR 35# [70] Congruence relations and functional completeness ill universal algebras; structure theory of hemiprimols, I, Math. Zeitschr., 113, 1970, MR 41#5271. FUKASAWA,S. [26] Uber ganzwertige ganze Funktionen, T6hoku Math. J., 27,1926, [28] Uber ganzwertige qanze Funktionen, Tohoku Math. J., 29,1928, GELFOND,A.O. [29a] Sur un theoreme de M.P6Iya, At.t.i della Reale Acad. Naz. dei Lincei, Rendiconti cl.sci.fis.mat.nat., ((3), 10, 1929, [29b] Sur les proprietes arithmeiiques des [onciions entieres, Tohoku Math. J., 30, 1929, [33] On functions integral valued at points of a geometric progression, Mat. Sbornik, 40, 1933, (Russian) - [67] On functions attaining integral values, Mat. Zametki, 1, 1967, MR 35#6630. (Russian) GERBOUD,G. [86] Etude de certains polunonies a valeurs eniiers, These 3 cycle, Universite de Provence [88a] Exemples d 'anneaux A pour lesquels ((X)) est une base du n nen A module des polimomes entieres sur A, Cornptes Rendus Acad. Sci. Paris, 307, 1988,1-4. MR 89h:13025.

7 116 - [88b] Polunom.es a ualeurs eniieres sur I 'anneau des eniiers de GOliSS, Comptes Rendus Acad. Sci, Paris, 307, 1988, :3/.')-:378, MR 89m:l [89] Construction sur WI anneall de Dedekind d'une base requliere de polunomes a ualeurs entieres, Manuscr, Math., ("i5, 1989, l\-1r 90h: [93] Substituobilit«d 'un a717leau de Dedekuul, Comptes Hendus Acad. Sci. Paris, 317,1993, GILMER,R. [89] Sets that determine inteqer-ualued polynomials, J. Number Th., 33, 1989, MR 90g: [90] Priijer domains and rings of inteqer-ualued polynomials,.t. Algebra, 129,1990, MR 91b: GILMER,R., HEINZER,W., LANTZ,D. [92] The Noetherian property ill rings of integer-valued polynomials, Trans. Amer. Math. Soc., 338, 1993, MR 93j: GILMER,R., HEINZER,W., LANTZ,D., SMITH,W. [90] The ring of integer-valued polynomials of a Dedekind domain, Proc. Amer. Math. Soc., 108, 1990, MR 90h: GILMER,R., SMITH,W.W. [83] Finitely qenerated ideals of the ring of integer-valued polynomials,.t. Algebra, 81,1983, MR 85b: [85] Integer-valued polunomials and the strong iuio-qenerator propertu, Houston J.Math. 11, 1985,6.')-74. MR 86b:1301O. GOLDMAN,O. [51] Hilbert rings and the Hilbert Nullstellensatz, Math. Zeitschr., 54, 1951, MR 13 p.427. GRAMAIN,F. [78a] Fonciions entieres arithmetiques, Serninaire Pierre Lelong - Henri Skoda (Analyse), Aimee 1976/77, Lecture Notes in Math. 694, 1978, MR 80f: [78b] Fonctions entieres arithnietiques, Sern. Delange-Pisot-Poitou, /78, exp.8. MR 80b: [80] Sur le theoreme de Fukasauia - Gel 'fond - G1'IlIl1an - Masser, Sem, Delange-Pisot-Poitou, 1980/81,67-86, Birkhauser MR 85d: [81] Sur le theoreme de Fukasauia - Gel'[ond, Inv. math., 63, 1981, MR 83g: [90] Fonctions entieres d 'une ou plusieurs variables complexes prerwnl des oaleurs entieres sur une progression qeometrique, Cinquante ans de polynomes, Fifty years of polynomials, Lecture Notes in Math. 1415,123-1:37, Springer MR 91e:ll086. GRAMAIN,F., MIGNOTTE,M. [83] Fonctions entieres arithmeiiques, Progress Math., 31, 1983, MR 85d: GRAMAIN,F., SCHNITZER,F.J. [89] Gan:e ganzwedige Funklionen: historische Bemerkungen, Complex methods in part.ial differential equations, , Akademie Verlag MR 91j: GRUMAN,L. [80] Proprietes arithmeiiques des [onciions entieres, Bull. Soc. Math. France, 108, 1980, MR 82g: GUNJI,H., McQUILLAN D.L. [69] On polynomials wilh integer coefficients, J. Number Th., 1, 1969, , MR 40#2648.

8 117 - [70] On a class of ideals in an alqebraic mnnberfieldi ), Number Th., 2, 1970, , MR 41# [75] On rings with a certain dil'isibility property, Michigan J. Math" , MR.53# [78] Polynomials with inieqral values, Proc. Roy. Irish Acad., A78, 1978, 1-7. MR 57# HALL,R.R. [71] On pseudo-polynomials, Mathematika, 18, 1971, MR 45 #3369. HALTER-KOCH,F., NARI(IEWICZ,W. [92a] Finiteness properties ofpolynomial mappings, Math. Nachr., 159, 1992, [92b) Polynomial maps with non-trivial commoll factor, Sem. Th. Nombres, Bordeaux, 4, 1992, [92c] Commutative rings and binomial coefficients, Mouatsh. Mat.h., 114, 1992, [94] Polynomial cycles in finitely generated domains, Monatsh. Math., to appear. HAOUAT,Y. [86) Anneaux de polsjnom.es ci oaleurs eniieres Sill' un anneau de valuation ou de Seidenberg, Ann. Sci. Univ. Clermont-Ferrand II, ser. math., 23, 1986, MR 88h: HAOUAT,Y.,GRAZZINI,F. [77) PolY7115me.s et dijjerence.s dioisees. Comptes Rendus Acad. Sci. Paris, 284,1977, AI171-AI173. MR 5.5# [78] DijJerence.s finies dioisees sur un mweau 8(2), Coruptes Hendus Acad. Sci. Paris, 286, 1978, A72:3-A725. l'vlr 81e: [79] Polunomes de Borsks), Ann. Scient. Univ. Clermont II, ser.rnath.. 18, 1979, MR 81b: HARDY,G.H. [16) On a theorem of All'. C.P(Slya, Proc. Cambridge Phil. Soc., 19, , HEITMANN,R.C. [76] Generating ideals in Priifer domains, Pacific J. Math., 62,1976, MR 53# HENSEL,IC [96] Ueber den qrossien qenieinsameu Theiler aile I' Zahlen, welche durch eine qanze Function von n Veriilllierlichen darsiellbar sind,.1. reine angew. Math., 116, 1896, IVANYI,A. [72) On multiplicative junctions with congruence property, Ann. Univ. Sci. Budapest, 15, 1972, Mil 48# ACOB,G. [76] Polunomes representant la [onction Ill/lie SUI' UII anueau commutatij uniiaire, Comptes Rendus Acad. Sci. Paris, 283, 1976, MR 55# [80] Anneau de [onct.ions polunomes d 'un anneau conunuiati] uniiaire, Comm. Algebra, 1980, MR 82j: ANKOWSKI,L., MARLEWSI(I,A. [90) On the rational polynomials having the same image o] the rational number set, Funct. et Approx., 19, 1990, MR 92b:ll012. KELLER,G.,OLSON,F.R. [68) Counting polynomial functions (mod p"), Duke Math..1.,35,1968, MR 38#1079.

9 118 KEMPNER,A.J. [21] Polunomials and theirresidue systems, Trans. Amer. Math. Soc., 22, 1921, KOCHEN,S. [69] Integer-valued raiionol functions over the p-udic numbers: A p-adic analogue of the theory of real fields, Proc. Syrup. Pure Math. 12, Number Theory, AMS 1969, MR 41#1685. KRULL,W. [51] Jacobsonsche Ringe, Hilbertscher Nullstellensatz, Dimensionstheorie, Math. Zeitschr., 54, 1951, MR. 13 p.903. KUBOTA,K.K. [72a] Note on a conjecture of WNnrkiewicz,.J. Number Th., 4,1972, MR 46# [72b] Factors of polynomials under composition, J. Number Th., 4, 1972, MR 58# [73] Image sets of polynominls, Acta Arith., 23, 1973, MR 48#8459. KUBOTA,K.IC, LIARDET,P. [76] Refutation d'une conjecture de WNarkiewicz, Comptes Rendus Acad. Sci. Paris, 282, 1976, AI261-AI264. MR 54# LANDAU,E. [20] Note on Mr. Hardy's extension of a theorem of Mr. Polua, Proc. Cambridge Philos. Soc., 20,1920,14-15 =Collected Works, VII, , Thales-Verlag. LAOHAKASOL,V., UBOLSRI,P. [80] A short 1I0te all inteqral-valued polynomials, Southeast Asian Bull. Math., 4, 1980,43-47, l\lr 83k:l0036. LENSTRA,H.W.Jr. [77] Euclidean nuniberjields oflmyje deqree. lnv. math., 38, 1977, MR.55#2836. LEUTBECHER,A., NIKLASCH,G. [89] Oil cliques of exceptional units and Lenstra's construction of Euclidean fields, Number Theory, Lecture Notes in Math. 1380, , Springer MR 90i: LEWIS,D.J. [56] Ideals and polynomial [unctions, American.J. Math, 78, 1956, MR 19 p [72] Invariant sets of morphtsms in projective and affine number spaces, J. Algebra, 20,1972, MR 46#1746. LEWIS,D.J., MOR.TON,P. [81] Quotients of polynomials and a theorem of Pisot and Cantor, J. Fac. Sci. Tokyo, la, Mat.h., 28, 1981, MR 83f: LIARDET,P. [70] Transformations rationnelles et ensembles alqebriques, These 3 cycle, Marseille [71] Surles transformations polsjnomiales et rot.ionnellcs, Sern. Th. Nombres Bordeaux, 1971/72, expo 29. MR 52# [72] Sur une conjecture de Narkieuncz, Cornptes Hendus Acad. Sci. Paris, 274, 1972, MR 47# [75] Sur la stabilite rationnelle ou nlgebriqllf d 'ensembles de nombres olqebriques, These, Aix-Marseille II, LIND,D.A. [71] Which polunomials over an alqebraic number field map the algebraic inieqers into theniselves'l, American Math. Monthly, 78, 1971,

10 119 LINDEN,F.J. van del' [88] Inieqer valued polunouiials over [unction fields, Indag. Math., 50, 1988, MR 90a: LOPER,A. [88] 011 rinqs without a certain divi8ibility property, J. Number Th., 28, 1988, MR 89a: MASLEY,J.M.,MONTGOMERY,H.L. [76] Cyclotomic fields with unique factorization, J. reine angew. Math., 286/287, 1976, MR 55#2834. MASSER,D. [80] Sur les [onciions entieres, Comptes Rendus Acad. Sci. Paris, 291, 1980, AI-A4. MR 8Ii: MATLIS,E. [66] Decomposable modules, Trans. Amer. l\bth. Soc., 121, 1966, 147-li9. MR 34# [70] The two-genemtor property for ideals, Michigan J. Math., 17, 1970, MR 42#247. McQUILLAN,D.L. [73a] Modules over algebraic tnteqers, Sern. Th. Nombres, Bordeaux, 1972/73, exp.4. MR 53# [73b] Reseauz sur les alllleaux d'entiers alqebriques, Sern. Delange-Pisot-Poitou, 14, 1972/73, no.25. MR.53# [78] On the coefficients and values of polynomial rinqs, Archiv Math., 30, 1978, MR 57# [85a] On Priifer domains of polunomials, J. reine angew. Math., 358, 1985, MR 86k: [85b] On ideals in Priljer domains of polunotnials, Archiv Math., 45, 1985, MR 87c: [85c] Rings of inteqer-ualued polunomials determined by finite sets, Proc. Roy. Irish Acad., 85A, 1985, MR 87g: [91] On a theorem of R.Cilmer, J. Number Th., :39,1991, MR 92i: MORTON.P. [92] Arithmetic properties ofperiodic poiut.s ofquadratic maps, Acta Arith., 62, 1992, ; II, preprint MORTON,P., SILVERMAN,J.H. [93a] Periodic points, multiplicities and dynamical units, preprint [93b] Rational periodic points of rational [unctions, preprint MOUSSA,P. GERONIMO,J.S., BESSIS,D. [84] Ensembles de Julia et p1'oprietes de localisation des families iierees d'entiers algebriques, Compt.es Rendus Acad. Sci. Paris, 299, 1984, MR 86f: MULLEN,G.,STEVENS,H. [84] Polynomial [u net (mod m), Acta Math. Ac. Sci. Hung., 44,1984, MR 86a:II004. NAG ELL, T. [18] Einige Sot:«iiber die qanzen rationalen Fuuktionen, Nyt Tydsskrift f. Maternatik, 29B, 1918,.1: [19] Uber zahleniheoretische Polunome, Norsk Mat. Tidsskrift, I, 1919, NAKANO,N. [53] ldealtheorie ill einem speziellen unenillichen algebra/schell Zahlkorper, J. Sci. Hiroshima Univ., Ser.A., 16, 19.13, MR 15 p..1io. NARKIEWICZ W. [62] On polunomial transformations, Acta Arit.h., 7, 1962, MR 26#110; II, ibidems, 1962/63, IVIR 26#4987.

11 120 - [63a] Remark on rational transjormations, Colloq. Mat.h., 10, 1963, MR 26# [63b] Problems 415,416, Colloq. Math., , p.1k7. - [64] On transformations by polynomials in two nariables, Colloq. Math., 12,1964, MR 29#4757; II, ibidem,13, 1964, MR 30#3K68. - [65] On polynomial transjormaiions in several variables, Acta Arith., 11, 1965, MR 32# [66] Solution to Problem 51, Wiadom. Mat., (2), 9, 1966, (Polish). - [89] Polynomial cycles in algebraic number jields, Colloq. Math., 58, 1989, MR 90k: NECAEV,A.A. [80] Polynomial transjornuitions of jinite commutative local principal ideal rings, Mat. Zametki, 27, 1980, 885-S97. I\IR 82b: (Russian) NIVEN,I.,WARREN,L.J. [57] A qeneralization of Fermat's theorem, Proc. Amer. Math. Soc., 8, 1957, 30G-:H3. I\IR 19 p.246. NOBAUER,W. [76] Compatible and conseroatiue [unctions on residue-class rings of the inieqers. Call. Math. Soc. J. Bolyai 1:3, Topics in Number Theory, Debrecen 1974, North-Holland 1976, MR 5.1# NORTHCOTT,D.G. [49a] An inequality in the theory of arithmetic on algebmic varieties, Proc. Cambridge Philos. Soc., MR II p [49b] A further inequality in the theoru of arith metic on alqebraic varteties, Proc. Cambridge Philos. Soc., 45, 1949, I\IR 11 p.: [50] Periodic points on an algebraic variety, Annals of Mat.h., 51, 19.10, MR 11 p.615. OSTROWSKI,A. [19] {Tber qanzioertiqe Poluuonie in alqebraischen Zahlkorpern, J. reine angew. Math., 149, 1919, PEZDA,T. [93] Polynomial cycles m certain local domains, Acta Arith., 66, 1994, [94a] Cycles ofpolynomial mappings I1l seoeral variables, Manuscr. Math., 83, 1994, [94b] Cycles ofpolynomials in alqebraicalh, closed ofpositive characteristics, Colloq. Math., 67, 1994, PHONG,B.M. [91] Multiplicative [unctions sati,.;j/jinq a conqruence property, Studia Sci. Math. Hungar., 26, 1991, PHONG,B.M., FEHER,J. [90] Note on uiultipiicat.ive functions satisfying a congruence property, Ann. Univ. Sci. Budapest, 3:3, MR 93d:ll008. PI SOT,C. [42] Ober qanzuiertiqe qan:«fuukt.ioncu, Jah resber. Deutsche Math. Ver., 52, 1942, I\1R 4 p [46a] Sur ies [onciions aritlunetiques analut.ique«d croiswi1lct:' erponentielle, Comptes Rendus Acad. Sci. Paris, 222, 1946, 99S-990. MR 8 p [46b] Sur les [onciions anulijtiques oritlunetiques et presque aritlunetiques. Comptes Rendus Acad. Sci. Paris, 222, 1946, MR 8 p.23.

12 121 ROSENBERG,I.G. [75] Polynomial functions olit I' finite rings, POLYA,G. [15] Uebel' ganzwertige qan :e Funkiioneu, Rend. eire. Mat. Palermo, 40,1915, [19] Uber qanzuiertiqe Polynome m alqelnvnschen Zahlkorpern, J. reine angew. Math., 149,1919, [20] Ueber qanze qanzurertuje Funkiionen, Nachr. Ges. Wiss. Gottingen, 1920, PRESTEL,A., RIPOLI,C.C. [91] Integral-ualued rational [unctions on valued fields, Manuser. Math., 73, 1991, I\IR 93b: 120J.':i. RAUSCH,U. [87] On a class of integer-valued [unctions, Archiv Math., 48, 1987, MR 88c:l1011. RAUZY,G. [67] Ensembles arithmetiquement dense, Cornptes Rendus Acad. Sci. Paris, 265, 1967, MR 36#5104. REDEI,L., SZELE, T. [47] Alqebraisclizahlentheoretisclie Beirachiunqen iiber Ringe, I, Acta Mathematiea, , MR 9 p.407: II, ibidem.82, 19.50, MR 12 p.6. ROBINSON,R.M. [71] Integer-valued entire functions. Trans. Arner. Math. Soc., 153, 1971, MR 43#.522. ROGERS,K., STRAUS,E.G. [85] Infinitely intcqer-volued polynomials Ollf'!' an algebmic number field, Pacific,J. rvlath., 118, 1985,.507-S22. rvlr 8Gi:II058. ROQUETTE,P. [71] Beuierkunqen zur Theorie del' formal p-adischen [(01'- per, Beitrage ZUI' Algebra und Geometric, 1, :3. l\ir 4S#8G43. Glasnik Mat., 10, 1975, MR.52#3.52. RUSH,D.E. [85] Generating ideals in rings of inieqer-volued polynomials, J. Algebra, 92, 1985, MR 8Gg: RUSSO,P. WALDE,R. [94] Rationol periodic points of the quadratic function Qc(x) = x 2 + c, American Math. Monthly, 101, :tH. RUZSA,I. [71] On conqruence-preservinq [unctions, Math. Lapok, 22, MR 48#2044. (Hnngarian) SALLY,J., VASCONCELOS,W. [74] Stable rings, J.Pure Applied Algebra, 4, 1974, MR 53# SATO,D., STRAUS,E.G. [64] Rate of growth of Hurwitz entire functions and integer valued entire functions, Bull. Amer. Math. Soc., ,30: MR 28#3161. SCHULTING,H.-W. [79] (Tbe,' die Erzeuqendenauzahl uiuertierbarer Ideale in Priiferrinqen. Comm. Algebra, 7, 1979, MR 8Ij: SELBERG,A. [41a] i)bf'r qanzuiertiqe qanze transeetulente Funktionen, Arch. Math. Naturvid., 44, 1941, MR 2 IdS6. - [41b] (Tber einen Sot: lion Gelfond. Arch. Mat.h.Nar.urvid.. 44, 1941, MR 4 p.6. - [41c] ['bel' qarizuiertrq«qan;e transzendent.e Funkt.ionen, II.. Arch. Math. Naturvid., 44,1941, MR 4 p.6. SHAPIRO,H.S. [57] The range of an inteqer-valued polynomial, Arnerican Math. Monthly, 64,1957, MR 19 p.115.

13 122 SHIBATA,F., SUGATANI,T., YOSHIDA,K. [86] Note 011 rings of integralvalued polynomials, Comptes Rendus Math. Rep. Acad. Sci. Canada, 8, 1986, MR 87j: SILVERMAN,J.H. [93] Integer points, diophantine approximation and it.eration of rational maps, Duke Math. J., 71,1993, SINGMASTER,D. [74], On polynomial functions mod 111, J. Number Th., 6, 1974, MR 51#3127. SKOLEM,T, [36] Ein Sat: tiber qanzuieriiqe Polunome. Kong. Norske Vid. Selsk., 9,1936, [37a] Uber die Losbarkeit qeunsser linearer Gleichunqen im Bereiche del' ganzwertigen Polynome, Kong. Norske Vid. Selsk. Forh., 9, 1937, [37b] Siitze tiber qanzuiertiqe Polynome, Kong. Norske Vid. Selsk. Forh., 10, 1937, [40] Einige Siitze tiber Polynome, Avhandl, Norske Videnskap. Akad. Oslo, I., Mat.-Naturv. KI., 1940, No.4, MR 2 p.247. SOMAYAJULU,A. [68] On arithmetic functions with congruence property, Portugal. Math., 27, 1968, MR 41#5281. SPIRA,R. [68] Polynomial interpolation over cotn.m uiatiue rings, American Math. Monthly, 1968, MR 37# STACKEL,P. [95] Ueber arithmetisclie Eiqenschaften mwlytischer Funkt.ionen, Math. Ann., 46, 1895, STEINITZ,E. [10] Algebmische Theorie del' Korper, J. reine angew. Mat.h., 137, 1910, STRAUS,E.G. [51] On the polynomials WhOM derivatives have inteqral IJalues on the integers, Proc. Arner. Math. Soc., 2, 19.')1, l\rr 12 p [52] Functions periodic modulo each of a SH/UflICF ofint.eqers, Duke Math. J., 19, 1952, MR 14 p.21. SUBBARAO,M.V. [66] A rithmetic functions satls/i/ing a conqruenrc ]>/'Operty, Canad. Math. Bull., 9, 1966, 14:3-147.!\'IR SZEKERES,G. [52] A canonical basis for the ideals of0 polsjnonual donuun. American Math. Monthly, 59, 1952, MR 1:3 p.90:3. TROTTER,P.G. [78] Ideals in Z[x, y], Acta Math. Ac. Sci. Hung., :32, 1978, MR 81g: TURK,J. [86] The fixed divisor of a polynomial, American Math. Monthly, 93, 1986, MR 87e: UCHIDA,K. [71] Class numbers of imoqinari; abelian number Tohoku Math.J. (2),23, 1971, MR 44#2727. WAGNER,C.G. [76] Polynomials over GF(q, J:) with inieqral ualued differences, Archiv Math., 27, 1976, MR 54# WALDE,R.,RUSSO,P. [94] Rational periodic points of the quadratic function Qc(x) = x 2 + c, American Math. Monthly, 101, 1994, : WALDSCHMIDT,M. [78] P61ya's theorem by Schneider 's method. Acta Math. Ac. Sci. Hung., 31, 1978, MR.58# WALLISER,R. [69] Verallqemeinerte qan:e ganzwertige Funktiouen Exponentialtypus, J. reine angew. Math., 235, 1969, MR 39# VO/ll

14 123 - [85] Uber ganze Funktioneu, die in einer qeomeirischen Folge qanze Werte annehmen, Monatsh. Math., 100, 1985, MR 87b: WIESENBAUER,J. [82] On polynomial functions over residue class rinqs of Z, Contributions to General Algebra 2,1982,39.')-:398. MR 85b: ZANTEMA,H. [82] Integer-valued polynomials over a number field, Manuser. Math., 40, 1982, MR 84i:12005.

15 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, ,31,45,46,.55,56,64 Carlitz,L. 7,11,44 Carlson,F. 15

16 125 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 ,23,37 Geronimo,S. 109 Gihner,R. 20,23,48,59,65 Goldman,O.51

17 Gramain,F. 15,24 Grazzini,F. 45 group of units 61 Gruman,L. 15 Gunji,H. 12,29,33,51, 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 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

18 127 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

19 128 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] Sally,J. 58 Sato,D. 14 Schnitzer,F.J. 15 Schulting,H.W. 59

20 129 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

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

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Rosolini (Eds.), Category Theory. Proceedings, VII, 494 pages Vol. 1489: A. Mielke, Hamiltonian and Lagrangian Flows on Center Manifolds. X, 140 pages Vol. 1490: K. Metsch, Linear Spaces with Few Lines. X11I, 196 pages Vol. 1491: E. Lluis Puebla,J. L. Loday, H. Gillet, C. Soule, V. Snaith, Higher Algebraic K Theory: an overview. IX, 164 pages Vol. 1492: K. R. Wicks. Fractals and Hyperspaces. V11I, 168 pages Vol. 1493: E. Benoit (Ed.). Dynamic Bifurcations. Proceedings, Luminy VII, 219 pages Vol. 1494: M. T. Cheng. X. W. Zhou, D. G. Deng (Eds.). Harmonic Analysis. Proceedings, IX. 226 pages Vol. 1495: J. M. Bony, G. Gruhb, L. Hormander, H. Komatsu. J. Sjostrand, Microlocal Analysis and Applications. Montecatini Terme, Editors: L. Cattabriga, L. Rodino. VII, 349 pages Vol. 1496: C. Foias, B. Francis, J. W. Helton, H. Kwakernaak, J. B. Pearson. Hm Control Theory. Como, Editors: E. Mosca, L. Pandolfi. VII. 336 pages Vol. 1497: G. T. Herman. A. K. Louis, F. Natterer (Eds.), Mathematical Methods in Tomography. Proceedings X, 268 pages Vol. 1498: R. Lang. Spectral Theory of Random Schrodinger Operators. X, 125 pages Vol. 1499: K. Taira, Boundary Value Problems and Markov Processes. IX. 132 pages Vol. 1500: J. P. Serre. Lie Algebras and Lie Groups. VII, 168 pages Vol. 1501: A. De Masi, E. Presutti, Mathematical Methods for Hydrodynamic Limits. IX, 196 pages Vol. 1502: C. Simpson, Asymptotic Behavior of Monodromy. V, 139 pages Vol. 1503: S. Shokranian, The Selberg Arthur Trace Formula (Lectures by J. Arthur). VII, 97 pages Vol. 1504: J. Cheeger, M. Gromov, C. Okonek, P. Pansu, Geometric Topology: Recent Developments. Editors: P. de Bartolomeis, F. Tricerri. VII, 197 pages Vol. 1505: K. Kajitani, T. Nishitani, The Hyperbolic. Cauchy Problem. VII, 168 pages Vol. 1506: A. Buium, Differential Algebraic Groups of Finite Dimension. XV, 145 pages Vol. 1507: K. Hulek, T. Peternell, M. Schneider, F. a. Schreyer (Eds.), Complex Algebraic Varieties. Proceedings, VII, 179 pages Vol. 1508: M. Vuorinen (Ed.), Quasiconformal Space Mappings. A Collection of Surveys IX, 148 pages

26 Vol. 1509: J. Aguade, M. Castellet, F. R. Cohen (Eds.), Algebraic Topology - Homotopy and Group Cohomology. Proceedings, X, 330 pages Vol. 1510: P. P. Kulish (Ed.), Quantum Groups. Proceedings, XII, 398 pages Vol. 1511: B. S. Yadav, D. Singh (Eds.), Functional Analysis and Operator Theory. Proceedings, VIII, 223 pages Vol. 1512: L. M. Adleman, M.-D. A. Huang, Primality Testing and Abelian Varieties Over Finite Fields. VII, 142 pages Vol. 1513: L. S. Block, W. A. Coppel, Dynamics in One Dimension. VIII, 249 pages Vol. 1514: U. Krengel, K. Richter, V. Warstat (Eds.), Ergodic Theory and Related Topics III, Proceedings, VIII, 236 pages Vol. 1515: E. Ballico, F. Catanese, C. Ciliberto (Eds.), Classification of Irregular Varieties. Proceedings, VII, 149 pages Vol. 1516: R. A. Lorentz, Multivariate Birkhoff Interpolation. IX, 192 pages Vol. 1517: K. Keimel, W. Roth, Ordered Cones and Approximation. VI, 134 pages Vol. 1518: H. Stichtenoth, M. A. Tsfasman (Eds.), Coding Theory and Algebraic Geometry. Proceedings, VIII, 223 pages Vol. 1519: M. W. Short, The Primitive Soluble Permutation Groups of Degree less than 256. IX, 145 pages Vol. 1520: Yu. G. Borisovich, Yu. E. Gliklikh (Eds.), Global Analysis - Studies and Applications V. VII, 284 pages Vol. 1521: S. Busenberg, B. Forte, H. K. Kuiken, Mathematical Modelling of Industrial Process. Bari, Editors: V. Capasso, A. Fasano. VII, 162 pages Vol. 1522: J.-M. Delort, F. B. I. Transformation. VII, 101 pages Vol. 1523: W. Xue, Rings with Morita Duality. X, 168 pages Vol. 1524: M. Coste, L. Maht!, M.-F. Roy (Eds.), Real Algebraic Geometry. Proceedings, VIII, 418 pages Vol. 1525: C. Casacuberta, M. Castellet (Eds.), Mathematical Research Today and Tomorrow. VII, 112 pages. I992. Vol. 1526: J. Azerna, P. A. Meyer, M. Yor(Eds.), Seminaire de Probabilites XXVI. X, 633 pages Vol. 1527: M. I. Freidlin, J.-F. Le Gall, Ecole d'ete de Probabilites de Saint-Flour XX Editor: P. L. Hennequin. VIII, 244 pages Vol. 1528: G. Isac, Complementarity Problems. VI, 297 pages Vol. 1529: J. van Neerven, The Adjoint of a Semigroup of Linear Operators. X, 195 pages Vol. 1530: J. G. Heywood, K. Masuda, R. Rautmann, S. A. Solonnikov (Eds.), The Navier-Stokes Equations 11 - Theory and Numerical Methods. IX, 322 pages Vol. 1531: M. Stoer, Design of Survivable Networks. IV, 206 pages Vol. 1532: J. F. Colombeau, Multiplication of Distributions. X, 184 pages Vol. 1533: P. Jipsen, H. Rose, Varieties of Lattices. X, 162 pages Vol. 1534: C. Greither, Cyclic Galois Extensions of Commutative Rings. X, 145 pages Vol. 1535: A. B. Evans, Orthomorphism Graphs of Groups. VIII, 114 pages Vol. 1536: M. K. Kwong, A. zeui. Norm Inequalities for Derivatives and Differences. VII, 150 pages Vol. 1537: P. Fitzpatrick, M. Martelli, J. Mawhin, R. Nussbaum, Topological Methods for Ordinary Differential Equations. Montecatini Terme, Editors: M. Furi, P. Zecca. VII, 218 pages Vol. 1538: P.-A. Meyer, Quantum Probability for Probabilists. X, 287 pages Vol. 1539: M. Coornaert, A. Papadopoulos, Symbolic Dynamics and Hyperbolic Groups. VIII, 138 pages Vol. 1540: H. Komatsu (Ed.), Functional Analysis and Related Topics, Proceedings. XXI, 413 pages Vol. 1541: D. A. Dawson, B. Maisonneuve, J. Spencer, Ecole d" Ete de Probabilites de Saint-Flour XXI Editor: P. L. Hennequin. VIII, 356 pages Vol. 1542: J.Frohlich, Th.Kerler, Quantum Groups, Quantum Categories and Quantum Field Theory. VII, 431 pages Vol. 1543: A. L. Dontchev, T. Zolezzi, Well-Posed Optimization Problems. XII, 421 pages Vol. 1544: M.Schllrmann, White Noise on Bialgebras. VII, 146 pages Vol. 1545: J. Morgan, K. O'Grady, Differential Topology of Complex Surfaces. VIII, 224 pages Vol. 1546: V. V. Kalashnikov, V. M. Zolotarev (Eds.), Stability Problems for Stochastic Models. Proceedings, VIII, 229 pages Vol. 1547: P. Harmand, D. Werner, W. Werner, M-ideals in Banach Spaces and Banach Algebras. VIII, 387 pages Vol. 1548: T. Urabe, Dynkin Graphs and Quadrilateral Singularities. VI, 233 pages Vol. 1549: G. Vainikko, Multidimensional Weakly Singular Integral Equations. XI, 159 pages Vol. 1550: A. A. Gonchar, E. B. Saff (Eds.), Methods of Approximation Theory in Complex Analysis and Mathematical Physics IV, 222 pages, Vol. 1551: L. Arkeryd, P. L. Lions, P.A. Markowich, S.R. S. Varadhan. Nonequilibrium Problems in Many-Particle Systems. Montecatini, Editors: C. Cercignani, M. Pulvirenti. VII, 158 pages Vol. 1552:J. Hilgert, K.-H. Neeb, Lie Semigroups and their Applications. XII, 315 pages Vol. 1553: J.-L- Colliot-Thelene, J. Kato, P. Vojta. Arithmetic Algebraic Geometry. Trento, Editor: E. Ballico. VII, 223 pages Vol. 1554: A. K. Lenstra, H. W. Lenstra, Jr. (Eds.), The Development of the Number Field Sieve. VIII, 131 pages Vol. 1555: O. Liess, Conical Refraction and Higher Microlocalization. X, 389 pages Vol. 1556: S. B. Kuksin, Nearly Integrable Infinite-Dimensional Hamiltonian Systems. XXVII, 101 pages

27 Vol. 1557:J. Azema, P. A. Meyer, M. Yor(Eds.), Seminaire de Probabilites XXVII. VI, 327 pages Vol. 1558: T. J. Bridges, J. E. Furter, Singularity Theory and Equivariant Symplectic Maps. VI, 226 pages Vol. 1559: V. G. Sprindzuk, Classical Diophantine Equations. XII, 228 pages Vol. 1560: T. Bartsch, Topological Methods for Variational Problems with Symmetries. X, 152 pages Vol. 1561: I. S. Molchanov, Limit Theorems for Unions of Random Closed Sets. X, 157 pages Vol. 1562: G. Harder, Eisensteinkohomologie und die Konstruktion gemischter Motive. XX, 184 pages Vol. 1563: E. Fabes, M. Fukushima, L. Gross, C. Kenig, M. Reckner, D. W. Stroock, Dirichlet Forms. Varenna, Editors: G. Dell'Antonio, U. Mosco. VII, 245 pages Vol. 1564: J. Jorgenson, S. Lang, Basic Analysis of Regularized Series and Products. IX, 122 pages Vol. 1565: L. Boutet de Monvel, C. De Concini, C. Procesi, P. Schapira, M. Vergne. D modules, Representation Theory, and Quantum Groups. Venezia, Editors: G. Zampieri, A. D'Agnolo. VII, 217 pages Vol. 1566: B. Edixhoven, J. H. Evertse (Eds.), Diophantine Approximation and Abelian Varieties. XIII, 127 pages Vol. 1567: R. L. Dobrushin, S. Kusuoka, Statistical Mechanics and Fractals. VII, 98 pages Vol. 1568: F. Weisz, Martingale Hardy Spaces and their Application in Fourier Analysis. VIII, 217 pages Vol. 1569: V. Totik, Weighted Approximation with Varying Weight. VI, 117 pages Vol. 1570: R. delaubenfels, Existence Families, Functional Calculi and Evolution Equations. XV, 234 pages Vol. 1571: S. Yu. Pilyugin, The Space of Dynamical Systems with the CO Topology. X, 188 pages Vol. 1572: L. Gottsche, Hilbert Schemes of Zero Dimensional Subschemes of Smooth Varieties. IX, 196 pages Vol. 1573: V. P. Havin, N. K. Nikolski (Eds.), Linear and Complex Analysis Problem Book 3 Part I. XXII, 489 pages Vol. 1574: V. P. Havin, N. K. Nikolski (Eds.), Linear and Complex Analysis Problem Book 3 Part II. XXII, 507 pages Vol. 1575: M. Mitrea, Clifford Wavelets, Singular Integrals, and Hardy Spaces. XI, 116 pages Vol. 1576: K. Kitahara, Spaces of Approximating Functions with Haar Like Conditions. X, 110 pages Vol. 1577: N. Obata, White Noise Calculus and Fock Space. X, 183 pages Vol. 1578: J. Bernstein, V. Lunts, EquivariantSheaves and Functors. V, 139 pages Vol. 1579: N. Kazamaki, Continuous Exponential Martingales and BMO. VII, 91 pages Vol. 1580: M. Milman, Extrapolation and Optimal Decompositions with Applications to Analysis. XI, 161 pages Vol. 1581: D. Bakry, R. D. Gill, S. A. Molchanov, Lectures on Probability Theory. Editor: P. Bernard. VIII, 420 pages Vol. 1582: W. Balser, From Divergent Power Series to Analytic Functions. X, 108 pages Vol. 1583: J. Azerna, P. A. Meyer, M. Yor(Eds.), Seminaire de Probabilites XXVIII. VI, 334 pages Vol. 1584: M. Brokate, N. Kenmochi, I. MillJer, J. F. Rodriguez, C. Verdi, Phase Transitions and Hysteresis. Montecatini Terme, Editor: A. Visintin. VII. 291 pages Vol. 1585: G. Frey (Ed.), On Artin's Conjecture for Odd 2 dimensional Representations. VIII, 148 pages Vol. 1586: R. Nillsen, Difference Spaces and Invariant Linear Forms. XII, 186 pages Vol. 1587: N. Xi, Representations of Affine Hecke Algebras. VIII, 137 pages Vol. 1588: C. Scheiderer, Real and Etale Cohomology. XXIV, 273 pages Vol. 1589: J. Bellissard, M. Degli Esposti, G. Forni, S. Graffi, S. Isola, J. N. Mather, Transition to Chaos in Classical and Quantum Mechanics. Montecatini, Editor: S. Graffi. VII, 192 pages Vol. 1590: P. M. Soardi, Potential Theory on Infinite Networks. VIII, 187 pages Vol. 1591: M. Abate, G. Patrizio, FinslerMetrics AGlobal Approach. IX, 180 pages Vol. 1592: K. W. Breitung, Asymptotic Approximations for Probability Integrals. IX, 146 pages Vol. 1593: J. Jorgenson & S. Lang, D. Goldfeld, Explicit Formulas for Regularized Products and Series. VIII, 154 pages Vol. 1594: M. Green, J. Murre, C. Voisin, Algebraic Cycles and Hodge Theory. Torino, Editors: A. Albano, F. Bardelli. VII, 275 pages Vol. 1595: R.D.M. Accola, Topics in the Theory of Riemann Surfaces. IX, 105 pages Vol. 1596: L. Heindorf, L. B. Shapiro, Nearly Projective Boolean Algebras. X, 202 pages Vol. 1597: B. Herzog, Kodaira Spencer Maps in Local Algebra. XVII, 176 pages Vol. 1598: J. Berndt, F. Tricerri, L. Vanhecke, Generalized Heisenberg Groups and Damek Ricci Harmonic Spaces. VIII, 125 pages Vol. 1599: K. Johannson, Topology and Combinatorics of 3 Manifolds. XVIII, 446 pages Vol. 1600: W. Narkiewicz, Polynomial Mappings. VII, 130 pages Vol. 1601: A. Pott, Finite Geometry and CharacterTheory. VII, 181 pages Vol. 1602: J. Winkelmann, The Classification of Threedimensional Homogeneous Complex Manifolds. XI, 230 pages Vol. 1604: A. Huber, Mixed Motives and their Realization in Derived Categories. XV, 207 pages