A Composite Problem Asmita Sodhi Dalhousie University - PowerPoint PPT Presentation

Intro to IVPs IVPs over Matrix Rings The 3 × 3 Case The 4 × 4 Case A Composite Problem Asmita Sodhi Dalhousie University acsodhi@dal.ca November 2, 2018

Intro to IVPs IVPs over Matrix Rings The 3 × 3 Case The 4 × 4 Case Overview Intro to IVPs 1 The ring of integer-valued polynomials p -orderings and p -sequences IVPs over Matrix Rings 2 Moving the problem to maximal orders An analogue to p -orderings The Maximal Order ∆ n The 3 × 3 Case 3 Subsets of ∆ 3 Characteristic polynomials Towards computing ν -sequences The 4 × 4 Case 4 Structure of ∆ 4 Determining the ν -sequence of ∆ 4

Intro to IVPs IVPs over Matrix Rings The 3 × 3 Case The 4 × 4 Case The Ring of Integer-Valued Polynomials The set Int( Z ) = { f ∈ Q [ x ] : f ( Z ) ⊆ Z } of rational polynomials taking integer values over the integers forms a subring of Q [ x ] called the ring of integer-valued polynomials (IVPs). �� x � � Int( Z ) is a polynomial ring and has basis : k ∈ Z > 0 as a k Z -module, with � x � � x � � x � := x ( x − 1) · · · ( x − ( k − 1)) , = 1 , = x . k ! 0 1 k This basis is a regular basis , meaning that the basis contains exactly one polynomial of degree k for k ≥ 1.

Intro to IVPs IVPs over Matrix Rings The 3 × 3 Case The 4 × 4 Case p -orderings The study of IVPs on subsets of the integers greatly benefited from the introduction of p -orderings by Bhargava [1]. Definition Let S be a subset of Z and p be a fixed prime. A p-ordering of S is a sequence { a i } ∞ i =0 ⊆ S defined as follows: choose an element a 0 ∈ S arbitrarily. Further elements are defined inductively where, given a 0 , a 1 , . . . , a k − 1 , the element a k ∈ S is chosen so as to minimize the highest power of p dividing k − 1 � ( a k − a i ) . i =0

Intro to IVPs IVPs over Matrix Rings The 3 × 3 Case The 4 × 4 Case p -sequences The choice of a p -ordering gives a corresponding sequence: Definition The associated p-sequence of S , denoted { α S , p ( k ) } ∞ k =0 , is the sequence wherein the k th term α S , p ( k ) is the power of p minimized at the k th step of the process defining a p -ordering. More explicitly, given a p -ordering { a i } ∞ i =0 of S , � k − 1 k − 1 � � � α S , p ( k ) = ν p ( a k − a i ) = ν p ( a k − a i ) . i =0 i =0

Intro to IVPs IVPs over Matrix Rings The 3 × 3 Case The 4 × 4 Case Though the choice of a p -ordering of S is not unique, the associated p -sequence of a subset S ⊆ Z is independent of the choice of p -ordering [1]. These p -orderings can be used to define a generalization of the binomial polynomials to a specific set S ⊆ Z which serve as a basis for the integer-valued polynomials of S over Z , Int( S , Z ) = { f ∈ Q [ x ] : f ( S ) ⊆ Z } .

Intro to IVPs IVPs over Matrix Rings The 3 × 3 Case The 4 × 4 Case IVPs over Matrix Rings We are particularly interested in studying IVPs over matrix rings. We denote the set of rational polynomials mapping integer matrices to integer matrices by Int Q ( M n ( Z )) = { f ∈ Q [ x ] : f ( M ) ∈ M n ( Z ) for all M ∈ M n ( Z ) } . We know from Cahen and Chabert [2] that Int Q ( M n ( Z )) has a regular basis, but it is not easy to describe using a formula in closed form [3].

Intro to IVPs IVPs over Matrix Rings The 3 × 3 Case The 4 × 4 Case Link to Maximal Orders Finding a regular basis for Int Q ( M n ( Z )) is related to finding a regular basis for its integral closure, and we understand the latter object through studying its localizations at rational primes. If p is a fixed prime, D is a division algebra of degree n 2 over K = Q p , and ∆ n is its maximal order, then we obtain the following useful result: Proposition ([3], 2.1) The integral closure of Int Q ( M n ( Z ) ( p ) ) is Int Q (∆ n ). Thus, the problem of describing the integral closure of Int Q ( M n ( Z ) ( p ) ) is exactly that of describing Int Q (∆ n ), and so we move our attention towards studying IVPs over maximal orders.

Intro to IVPs IVPs over Matrix Rings The 3 × 3 Case The 4 × 4 Case An Analogue to p -orderings Definition-Proposition ([4], 1.1, 1.2) Let K be a local field with valuation ν , D a division algebra over K to which ν extends, ∆ the maximal order in D , and S a subset of ∆. A ν -ordering of S is a sequence { a i } ⊆ S such that for each k > 0, the element a k minimizes the quantity ν ( f k ( a 0 , . . . , a k − 1 )( a )) over a ∈ S , where f k ( a 0 , . . . , a k − 1 ( x )) is the minimal polynomial of the set { a 0 , a 1 , . . . , a k − 1 } , with the convention that f 0 = 1. We call α S = { α S ( k ) = ν ( f k ( a 0 , . . . , a k − 1 )( a k )) : k = 0 , 1 , . . . } the ν -sequence of S . Additionally, let π ∈ ∆ be a uniformizing element. Then the ν -sequence α S depends only on the set S , and not on the choice of ν -ordering. The sequence of polynomials { π − α S ( k ) f k ( a 0 , . . . , a k − 1 )( x ) : k = 0 , 1 , . . . } forms a regular ∆-basis for the ∆-algebra of polynomials which are integer-valued on S .

Intro to IVPs IVPs over Matrix Rings The 3 × 3 Case The 4 × 4 Case In order to use this proposition, we need to be able to construct a ν -ordering for the maximal order ∆ n . A recursive method for constructing ν -orderings for elements of a maximal order is based on two lemmas. Lemma (see [4], 6.2) Let { a i : i = 0 , 1 , 2 , . . . } be a ν -ordering of a subset S of ∆ n with associated ν -sequence { α S ( i ) : i = 0 , 1 , 2 , . . . } and let b be an element in the centre of ∆ n . Then: i) { a i + b : i = 0 , 1 , 2 , . . . } is a ν -ordering of S + b , and the ν -sequence of S + b is the same as that of S ii) If p is the characteristic of the residue field of K (so that ( p ) = ( π ) n in ∆ n ), then { pa i : i = 0 , 1 , 2 , . . . } is a ν -ordering for pS and the ν -sequence of pS is { α S ( i ) + in : i = 0 , 1 , 2 , . . . }

Intro to IVPs IVPs over Matrix Rings The 3 × 3 Case The 4 × 4 Case Lemma ([4], 5.2) Let S 1 and S 2 be disjoint subsets of S with the property that there is a non-negative integer k such that ν ( s 1 − s 2 ) = k for any s 1 ∈ S 1 and s 2 ∈ S 2 , and that S 1 and S 2 are each closed with respect to conjugation by elements of ∆ n . If { b i } and { c i } are ν -orderings of S 1 and S 2 respectively with associated ν -sequence { α S 1 ( i ) } and { α S 2 ( i ) } , then the ν -sequence of S 1 ∪ S 2 is the sum of the linear sequence { ki : i = 0 , 1 , 2 , . . . } with the shuffle { α S 1 ( i ) − ki } ∧ { α S 2 ( i ) − ki } , and this shuffle applied to { b i } and { c i } gives a ν -ordering of S 1 ∪ S 2 .

Intro to IVPs IVPs over Matrix Rings The 3 × 3 Case The 4 × 4 Case The theory presented in the previous slides is utilized by Evrard and Johnson [3] to construct a ν -order for ∆ 2 and establish a ν -sequence and regular basis for the IVPs on ∆ 2 when the division algebra D is over the local field Q 2 . We would like to extend these results to the general case, in order to find a regular basis for the integer-valued polynomials on ∆ n over the local field Q 2 .

Intro to IVPs IVPs over Matrix Rings The 3 × 3 Case The 4 × 4 Case Constructing ∆ n We can use these lemmas by decomposing ∆ n as a union of subsets to which the lemmas apply. Let Q 2 denote the 2-adic numbers, and let ζ be a (2 n − 1) th root of unity. Let θ be the automorphism of Q 2 ( ζ ) that maps θ ( ζ ) = ζ 2 . Define n × n matrices ω n and π n as:     ζ 0 · · · 0 0 1 · · · 0 . . ... ... 0 θ ( ζ ) · · · 0 . .     . . ω n = π n = . . .  ...    . . .     . . . 0 0 · · · 1     θ n − 1 ( ζ ) 0 0 · · · 2 0 · · · 0 The maximal order ∆ n with which we concern ourselves is ∆ n = Z 2 [ ω n , π n ] where Z 2 denotes the 2-adic integers.

Intro to IVPs IVPs over Matrix Rings The 3 × 3 Case The 4 × 4 Case ∆ n = Z 2 [ ω n , π n ]     0 · · · 0 0 1 · · · 0 ζ . . ... ... 0 θ ( ζ ) · · · 0 . .     . . ω n = π n =  . . .    ... . . .     . . . 0 0 · · · 1     θ n − 1 ( ζ ) 0 0 · · · 2 0 · · · 0 The elements ω n and π n observe the commutativity relation π n ω n = ω 2 n π n , and note also that π n n = 2 I n . An element z ∈ ∆ n can be expressed as a Z 2 -linear combination of the elements n π j { ω i n : 0 ≤ i , j ≤ n − 1 } , or else uniquely in the form z = α 0 + α 1 π + · · · + α n − 1 π n − 1 with α i ∈ Z 2 ( ζ ). n

Intro to IVPs IVPs over Matrix Rings The 3 × 3 Case The 4 × 4 Case The Maximal Order We present in particular some results for ∆ 3 = Z 2 [ ω, π ] with     ζ 0 0 0 1 0 ζ 2 ω = 0 0 π = 0 0 1     ζ 4 0 0 2 0 0 where ζ is a 7 th root of unity. In addition to the relations πω = ω 2 π and π 3 = 2 I 3 , we also work with the convention that ζ + ζ 2 + ζ 4 ≡ 0 (mod 2) ζ 3 + ζ 5 + ζ 6 ≡ 1 (mod 2) . and The valuation in ∆ 3 is described by ν ( z ) = ν 2 (det( z )) for z ∈ ∆ 3 realized as a matrix, where ν 2 denotes the 2-adic valuation.