An Introduction to Integer Valued Polynomials Marie-Andre B.Langlois - PowerPoint PPT Presentation

An Introduction to Integer Valued Polynomials Marie-Andrée B.Langlois Dalhousie University February 2018 Marie B.Langlois Dalhousie University Integer Valued Polynomials 1/ 37

What are Integer Valued Polynomials ? For this talk Definition For any subset S of Z the ring of integer valued polynomials on S is defined to be Int( S ) = { f ( x ) ∈ Q [ x ] | f ( S ) ⊆ Z } . Marie B.Langlois Dalhousie University Integer Valued Polynomials 2/ 37

What are Integer Valued Polynomials ? For this talk Definition For any subset S of Z the ring of integer valued polynomials on S is defined to be Int( S ) = { f ( x ) ∈ Q [ x ] | f ( S ) ⊆ Z } . The formal definition is for a domain D and field of fractions K . Definition For any subset S of D the ring of integer valued polynomials on S is defined to be Int( S , D ) = { f ( x ) ∈ K [ x ] | f ( S ) ⊆ D } . Marie B.Langlois Dalhousie University Integer Valued Polynomials 2/ 37

What are Integer Valued Polynomials ? Lets start with Int( Z ) and find some examples : • 25 x 5 − 13 x 3 + 7 x − 23 Marie B.Langlois Dalhousie University Integer Valued Polynomials 3/ 37

What are Integer Valued Polynomials ? Lets start with Int( Z ) and find some examples : • 25 x 5 − 13 x 3 + 7 x − 23 • is a boring example, we want non-integer coefficients ! Marie B.Langlois Dalhousie University Integer Valued Polynomials 3/ 37

What are Integer Valued Polynomials ? Lets start with Int( Z ) and find some examples : • 25 x 5 − 13 x 3 + 7 x − 23 • is a boring example, we want non-integer coefficients ! • Degree 1, can we do better than x ? Marie B.Langlois Dalhousie University Integer Valued Polynomials 3/ 37

What are Integer Valued Polynomials ? Lets start with Int( Z ) and find some examples : • 25 x 5 − 13 x 3 + 7 x − 23 • is a boring example, we want non-integer coefficients ! • Degree 1, can we do better than x ? x ( x − 1 ) • Degree 2, 2 Marie B.Langlois Dalhousie University Integer Valued Polynomials 3/ 37

What are Integer Valued Polynomials ? Lets start with Int( Z ) and find some examples : • 25 x 5 − 13 x 3 + 7 x − 23 • is a boring example, we want non-integer coefficients ! • Degree 1, can we do better than x ? x ( x − 1 ) • Degree 2, 2 x ( x − 1 )( x − 2 ) • Degree 3, 2 · 3 Marie B.Langlois Dalhousie University Integer Valued Polynomials 3/ 37

What are Integer Valued Polynomials ? In general, for degree n : � x � = x ( x − 1 ) · · · ( x − n + 1 ) n ! n Theorem A polynomial is integer valued on Z if and only if it can be written as a Z -linear combination of the polynomials � x � = x ( x − 1 ) · · · ( x − k + 1 ) , n ! n for n = 0 , 1 , 2 , . . . . Marie B.Langlois Dalhousie University Integer Valued Polynomials 4/ 37

Today’s Plan We will go over the following : • Bases and IVPs on subsets of the integers. • p -orderings, p -sequences and invariants of Int( S ) . • Multivariable and homogeneous case. Marie B.Langlois Dalhousie University Integer Valued Polynomials 5/ 37

Int( S ) is a Ring • Most of the axioms follow from Q [ x ] being a ring. • Int( S ) is also closed under addition and multiplication. • Int( S ) is a Z -module. Marie B.Langlois Dalhousie University Integer Valued Polynomials 6/ 37

Int( S ) is a Module Definition An R -module M , over the ring R consist of an abelian group ( M , +) and an operation R × M → M (scalar multiplication). For all r , s ∈ R , x , y ∈ M we have 1 r ( x + y ) = rx + ry . 2 ( r + s ) x = rx + sx . 3 ( rs )( x ) = r ( sx ) . 4 1 R x = x . Marie B.Langlois Dalhousie University Integer Valued Polynomials 7/ 37

Int( S ) is a Z -module In this case R = Z , we want for all m , n ∈ Z and f ( x ) , g ( x ) ∈ Int( S ) : 1 m ( f ( x ) + g ( x )) = m · f ( x ) + m · g ( x ) . 2 ( m + n ) f ( x ) = m · f ( x ) + n · f ( x ) . 3 ( mn ) f ( x ) = m ( n · f ( x )) . 4 1 · f ( x ) = f ( x ) . Multiply f ( x ) an IVP by an integer n will preserve its integer valued property. Marie B.Langlois Dalhousie University Integer Valued Polynomials 8/ 37

Bases Definition A basis B of the R -module B is said to be a regular basis if it is formed by one and only one polynomial of each degree. �� x � = x ( x − 1 ) ··· ( x − n + 1 ) � • A regular basis for Int( Z ) is { 1 }∪ n ≥ 1 . n n ! Marie B.Langlois Dalhousie University Integer Valued Polynomials 9/ 37

What if S ⊂ Z We will look at Int( S ) for S ⊂ Z , to motivate why we need better tools to find bases. Here are examples of sets we can consider • Even/Odd integers • Prime numbers • Fibonacci Numbers • Sum of ℓ d -th powers x = x d 1 + x d 2 + · · · + x d ℓ Marie B.Langlois Dalhousie University Integer Valued Polynomials 10/ 37

Even/Odd Integers Marie B.Langlois Dalhousie University Integer Valued Polynomials 11/ 37

Even/Odd Integers � x / 2 � • For S = 2 Z , a basis for Int( S ) is made of the polynomials n � � 1 , x 2 , x ( x − 2 ) , x ( x − 2 )( x − 4 ) , . . . 8 48 Marie B.Langlois Dalhousie University Integer Valued Polynomials 11/ 37

Even/Odd Integers � x / 2 � • For S = 2 Z , a basis for Int( S ) is made of the polynomials n � � 1 , x 2 , x ( x − 2 ) , x ( x − 2 )( x − 4 ) , . . . 8 48 • For S = 1 + 2 Z , a basis for Int( S ) is made of the polynomials � ( x − 1 ) / 2 � n � 1 , ( x − 1 ) , ( x − 1 )( x − 3 ) , ( x − 1 )( x − 3 )( x − 5 ) � , . . . 2 8 48 Marie B.Langlois Dalhousie University Integer Valued Polynomials 11/ 37

Prime Numbers The beginning of a basis for Int( P ) : f 0 = 1 , f 1 = ( x − 1 ) , f 2 = ( x − 1 )( x − 2 ) , 2 , f 4 = ( x − 1 ) 2 ( x − 2 )( x − 3 ) f 3 = ( x − 1 )( x − 2 )( x − 3 ) , 24 48 f 5 = ( x − 1 )( x − 2 )( x − 3 )( x − 5 )( x − 79 ) , . . . 5760 and f 3 ( 4 ) = 1 4 , f 4 ( 4 ) = 3 4 and f 5 ( 4 ) = 5 64 . Marie B.Langlois Dalhousie University Integer Valued Polynomials 12/ 37

Manjul Bhargava We will go over some work that Bhargava did during his undergraduate degree and define • p -orderings • p -sequences. For the next part of the presentation we will work locally Source of Image : https://opc.mfo.de/ at a prime p . detail?photo_id=7108 Marie B.Langlois Dalhousie University Integer Valued Polynomials 13/ 37

A Game Called p -orderings Fix a prime p . Definition (Bhargava) A p -ordering of S a subset of Z is a sequence ( a n ) n ≥ 0 , such that a 0 is arbitrarily chosen and for each n > 0 , a n ∈ S is chosen to minimize ν p (( a 0 − a n ) · · · ( a n − 1 − a n )) . where � max { ν ∈ N : p ν | n } n ≥ 0 ν p ( n ) = ∞ n = 0 Marie B.Langlois Dalhousie University Integer Valued Polynomials 14/ 37

Lets play ! Let p = 2 and S = { 0 , 1 , 2 , 3 , 4 } Marie B.Langlois Dalhousie University Integer Valued Polynomials 15/ 37

Lets play ! Let p = 2 and S = { 0 , 1 , 2 , 3 , 4 } • a 0 = 0 Marie B.Langlois Dalhousie University Integer Valued Polynomials 15/ 37

Lets play ! Let p = 2 and S = { 0 , 1 , 2 , 3 , 4 } • a 0 = 0 • for a 1 we want to minimize the power of 2 dividing ( 0 − a 1 ) take any odd number, a 1 = 1 Marie B.Langlois Dalhousie University Integer Valued Polynomials 15/ 37

Lets play ! Let p = 2 and S = { 0 , 1 , 2 , 3 , 4 } • a 0 = 0 • for a 1 we want to minimize the power of 2 dividing ( 0 − a 1 ) take any odd number, a 1 = 1 • a 2 ( 0 − a 2 )( 1 − a 2 ) pick 2 or 3, a 2 = 2 Marie B.Langlois Dalhousie University Integer Valued Polynomials 15/ 37

Lets play ! Let p = 2 and S = { 0 , 1 , 2 , 3 , 4 } • a 0 = 0 • for a 1 we want to minimize the power of 2 dividing ( 0 − a 1 ) take any odd number, a 1 = 1 • a 2 ( 0 − a 2 )( 1 − a 2 ) pick 2 or 3, a 2 = 2 • a 3 ( 0 − a 3 )( 1 − a 3 )( 2 − a 3 ) , need to pick an odd value a 3 = 3 Marie B.Langlois Dalhousie University Integer Valued Polynomials 15/ 37

Lets play ! Let p = 2 and S = { 0 , 1 , 2 , 3 , 4 } • a 0 = 0 • for a 1 we want to minimize the power of 2 dividing ( 0 − a 1 ) take any odd number, a 1 = 1 • a 2 ( 0 − a 2 )( 1 − a 2 ) pick 2 or 3, a 2 = 2 • a 3 ( 0 − a 3 )( 1 − a 3 )( 2 − a 3 ) , need to pick an odd value a 3 = 3 • a 4 = 4. Our p -ordering of S is { 0 , 1 , 2 , 3 , 4 } . Marie B.Langlois Dalhousie University Integer Valued Polynomials 15/ 37

In General Proposition (Bhargava) The natural ordering of Z ≥ 0 with a i = i is a p -ordering of Z for all primes p . Are p -orderings unique ? Marie B.Langlois Dalhousie University Integer Valued Polynomials 16/ 37

In General Proposition (Bhargava) The natural ordering of Z ≥ 0 with a i = i is a p -ordering of Z for all primes p . Are p -orderings unique ? No, we made some choices in the previous example. Marie B.Langlois Dalhousie University Integer Valued Polynomials 16/ 37

p -sequences Definition (Bhargava) Given ( a n ) n ≥ 0 a p -ordering of S a subset of Z with α 0 = 0 and α n ( S , p ) = ν p (( a 0 − a n ) · · · ( a n − 1 − a n )) , { α n ( S , p ) } is the associated p -sequence of S . A nice property for a set S is that the p -sequence is independent of the choice of p -ordering. Example : { α n ( Z , p ) } = { ν p ( n !) } . Marie B.Langlois Dalhousie University Integer Valued Polynomials 17/ 37