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THE INTEGER-VALUED POLYNOMIALS ON LUCAS NUMBERS by Amitabh Kumer Halder Submitted in partial fulfillment of the requirements for the degree of Master of Science at Dalhousie University Halifax, Nova Scotia August 2017 c Copyright by Amitabh Kumer Halder, 2017 Dedicated to my wife and kids ii Table of Contents Abstract ...................................... v List of Abbreviations and Symbols Used .................. vi Acknowledgements ............................... viii Chapter 1 Introduction .......................... 1 Chapter 2 Algebraic Background .................... 3 2.1 Properties of Local Rings . 3 2.2 Background for the p-adic Integers . 5 2.3 Hensel's Lemma . 6 2.4 P -adic Closures . 7 2.5 The Chinese Remainder Theorem . 8 2.6 Integer-Valued Polynomials . 10 2.7 Background about Integer-Valued Polynomials on a Subset . 11 2.8 Polynomial Closures . 12 2.9 Relation between Polynomial Closures and P-adic Closures . 13 2.10 p-Orderings and the Associated p-Sequences . 14 2.11 Method for Computing p-Orderings and the Associated p-Sequences by Shuffling . 21 Chapter 3 Background on Fibonacci Numbers, Lucas Numbers and Linear Recurrence Sequences ................ 23 3.1 Golden Ratio . 23 3.2 Fibonacci Numbers . 23 3.3 Lucas Numbers . 24 iii Chapter 4 Lucas Numbers as Images of the Maps f+ and f− ... 26 4.1 Closures of Images of f+ and f− .................... 26 4.2 Regular Z-basis for L ........................... 34 Chapter 5 The General Sequences for a Given Pair of Initial Values 37 5.1 Binet's Formula for the Sequence G ................... 37 5.2 Closures of Images of F+ and F− .................... 38 5.3 Regular Z-basis for G ........................... 43 Chapter 6 Conclusion ............................ 48 Bibliography ................................... 49 Appendix ..................................... 50 iv Abstract An integer-valued polynomial on a subset, S, of the set of integers, Z, is a polynomial f(x) 2 Q[x] such that f(S) ⊆ Z. The collection, Int(S; Z), of such integer-valued polynomials forms a ring with many interesting properties. The concept of p-ordering and the associated p-sequence due to Bhargava [2] is used for finding integer-valued polynomials on any subset, S, of Z. In this thesis, we concentrate on extending the work of Keith Johnson and Kira Scheibelhut [14] for the case S = L, the Lucas numbers, where they work on integer- valued polynomials on S = F, Fibonacci numbers. We also study integer-valued polynomials on the general 3 term recursion sequence, G, of integers for a given pair of initial values with some interesting properties. The results are well-agreed with those of [14]. v List of Abbreviations and Symbols Used Int(S; Z) The set of integer-valued polynomials x k The binomial polynomial L The sequence of Lucas numbers Lk The kth Lucas number Z(p) The p-local integers fS;p;k(x) The Z(p)-basis elements bS;p;k(x) The monic polynomials in Z[x] 1 fαS;p(k)gk=0 The associated p-sequence of S νk(Z; p) p-adic valuation of any element in Z D A domain F The quotient field of D R A subring of D Dp A discrete valuation domain E A fractional subset of D ν An essential valuation of Z ordpa The highest power of p dividing an integer a > 0 j · jp The p-adic norm on Q jj · jj A nontrivial norm on Q Qp The p-adic numbers Zp The ring of p-adic integers × Zp The p-adic units !(x) An arithmetic function on a ring R d(x; y) An ultradistance on a ring R R^ The completion of a ring R P^ The p-adic closure N The positive integers S¯ The polynomial D-closure of S f A primitive polynomial vi d(S; f) The fixed divisor of f Z=nZ A factor ring L(n) First n-terms of Lucas numbers p τ The Golden ratio (1 + 5)=2 U The units of Zp p p U( 5) The units of Zp( 5) p p 0 U ( 5) The units of Zp( 5) with norm ±1 p N(z) The norm of z 2 U( 5) f+; f− The functions in the Binet's formula of L < τ > The group generated by τ G The general sequence for a given pair of integers F+;F− The functions used in the Binet's formula of G vii Acknowledgements I would like to thankfully acknowledge the fundings I received namely the Nova Scotia Graduate Scholarship, research grant from the project operated by Dr. Sara Faridi, departmental funding from the Department of Mathematics and Statistics in Dal- housie University, and Lett Bursary from the Lett family. I would like to express thanks to several people without whose help this thesis would not have been com- pleted. In particular, I am very grateful to: * my supervisor, Dr. Keith Johnson, for his cordial help, potential guidance, patience, passion, insight, and valuable suggestions; * my readers, Dr. Robert Pare and Dr. Dorette Pronk, for their interest and sugges- tions; * Dr. David Iron, for his academic help and sympathy; * Dr. Sara Faridi, for her support and valuable advices that helped me to get an opportunity to study at the Department of Mathematics and Statistics in Dalhousie University; * Dr. Keith Taylor and Dr. Karl Dilcher, for their help and suggestions; * the faculty, staff, and students of the Department of Mathematics and Statistics in Dalhousie University for their help and assistance; * my parents, wife, and kids, for their love, mental support, and sacrifice; * and, my friends, especially Kunpeng Wang, Tom Potter, Giuseppe Pasqualino, and Bassemah Alhulaimi, for their homework help and company. viii Chapter 1 Introduction Let S be a subset of Z, the set of integers. The ring of integer-valued polynomials on S, is given by Int(S; Z) = Int(S) = ff(x) 2 Q[x]: f(S) ⊆ Zg. This is a subring of Q[x] preserving many interesting properties. The study of the ring of integer-valued polynomials on the subsets of Z and its generalizations has been continuing for more than 100 years, as for examples [2] and [20]. The updated results up to 1997 are in [5]. In this thesis, we are interested in concentrating on the development of the work of Keith Johnson and Kira Scheibelhut [14] for the case S = L, the Lucas numbers. In general, for the description of the Z-module, Int(S; Z), we wish to find a regular Z-basis having polynomials ffk(x): k = 0; 1; 2; :::g in Int(S; Z) as basis elements such that fk(x) is a polynomial of degree k. Every element f(x) 2 Int(S) can be expressed uniquely as a Z-linear combination of the polynomials ffk(x): k = 0; 1; 2; :::g. We may consider certain suitable choices for S to form regular Z-bases. For example, every element in Int(Z) is a Z-linear combination of the binomial poly- x Qk−1 x−i nomials ffk(x): k = 0; 1; 2; :::g = f k g = f i=0 k−i g. Due to Pascal's triangle it is easy to show that the polynomials fk are in Int(Z). The polynomials fk(x) take values 0 at x = 0; 1; 2; :::; k − 1 and 1 at x = k. A similar result can be observed for S = fi2 : i 2 Zg due to [2] and S = fp : p 2 Z and p is positive prime g due to [6]. But no one has studied this construction for S = L, the Lucas numbers. The set L = fLk : k = 0; 1; 2; :::g of Lucas numbers may be defined by the recurrence relation Lk+1 = Lk + Lk−1 for k ≥ 1 starting with L0 = 2 and L1 = 1. We are interested in concentrating on the localization of the structure with respect to prime numbers and then deducing the general case from the localized structures, i.e., we require to first study Int(S)(p) = ff(x) 2 Q[x]: f(S) ⊆ Z(p)g, 1 2 a where Z(p) = f b 2 Q : gcd(p; b) = 1g is the ring of p-local integers, and then to find the regular Z(p)-basis so that every element of Int(S) can be written as a Z(p)-linear combination. We want to find the regular Z(p)-basis for all primes, p, with the polyno- (k) αS ;p mials of the form fS;p;k(x) = bS;p;k(x)=p , where bS;p;k(x)'s are monic polynomials (k) in Z[x] and so fS;p;k(x)'s are in Int(S; Z) and αS; p is nonzero for fixed k and for (k) only finitely many primes. As for example, when S = Z, we can see that pαS ;p is (k) exactly the power of p in the prime factorization of k!. In this case, αZ; p can be extracted by the well-known formulas X i X νk(Z; p) = [k=p ] = (k − ki)=(p − 1); i≥1 i≥0 P i where k is a p-adic integer, i.e., k = i≥0 kip . For such given regular Z(p)-basis for each prime, p, the Chinese Remainder Theorem ensures that for each k there exists a Z-linear combination fS;k(x), say, of fS;p;k(x)'s (k) Q αS ;p so that we have fS;k(x) = bS;k(x)= p with monic polynomials bS;k(x) in Z[x] (k) for all primes such that αS; p is positive. This shows that fS;k(x) 2 Int(S; Z) and fS;k(x)'s form regular Z(p)-basis of Int(S; Z) and so form a regular Z-basis.
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