Nonunique Factorization in the Ring of Integer-Valued Polynomials - PowerPoint PPT Presentation

Nonunique Factorization in the Ring of Integer-Valued Polynomials Paul Baginski Fairfield University March 22, 2019 Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

2016 Fairfield University REU project. Gregory Knapp, Case Western Reserve University Jad Salem, Oberlin College Gabrielle Scullard, University of Rochester Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

The ring of integer-valued polynomials is Int( Z ) = { f ( x ) ∈ Q [ x ] | ∀ n ∈ Z f ( n ) ∈ Z } Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

The ring of integer-valued polynomials is Int( Z ) = { f ( x ) ∈ Q [ x ] | ∀ n ∈ Z f ( n ) ∈ Z } Z [ x ] � Int( Z ) � Q [ x ] because x 2 / ∈ Int( Z ) x ( x − 1) ∈ Int( Z ) but 2 and Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

The ring of integer-valued polynomials is Int( Z ) = { f ( x ) ∈ Q [ x ] | ∀ n ∈ Z f ( n ) ∈ Z } Z [ x ] � Int( Z ) � Q [ x ] because x 2 / ∈ Int( Z ) x ( x − 1) ∈ Int( Z ) but 2 � x � = x ( x − 1)( x − 2) · · · ( x − ( n − 1)) ∈ Int( Z ) and n n ! Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

Int( Z ) is non-Noetherian. Irreducible elements: primes p ∈ Z ; linear polynomials ax + b in Z [ x ] with a � = 0 and gcd( a , b ) = 1; � x � binomial polynomials ; n many other polynomials. Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

Int( Z ) is non-Noetherian. Irreducible elements: primes p ∈ Z ; linear polynomials ax + b in Z [ x ] with a � = 0 and gcd( a , b ) = 1; � x � binomial polynomials ; n many other polynomials. Int( Z ) also has nonunique factorization: � x � x ( x − 1)( x − 2)( x − 3)( x − 4) = · 2 · 2 · 2 · 3 · 5 5 = x ( x − 2)( x − 4) ( x − 1)( x − 3) · 3 3 Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

If f ∈ Int( Z ), the set of factorizations is Z ( f ) = { f = g 1 · · · g k | k ∈ N , g i ∈ Int( Z ) irreducible } which can be graded into factorizations of length k Z k ( f ) = { f = g 1 · · · g k | g i ∈ Int( Z ) irreducible } The multiplicity of a length k is | Z k ( f ) | , the number of factorizations of f of that length k . Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

The set of lengths for f is L( f ) = { k ∈ N | f = g 1 g 2 · · · g k , g i ∈ Int( Z ) irreducible } = { k ∈ N | Z k ( f ) � = ∅} Since � x � x ( x − 1)( x − 2)( x − 3)( x − 4) = ∗ 2 ∗ 2 ∗ 2 ∗ 3 ∗ 5 5 = x ( x − 2)( x − 4) ( x − 1)( x − 3) ∗ 3 3 we have { 4 , 5 , 6 } ⊆ L( x ( x − 1)( x − 2)( x − 3)( x − 4)). Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

The set of lengths for f is L( f ) = { k ∈ N | f = g 1 g 2 · · · g k , g i ∈ Int( Z ) irreducible } = { k ∈ N | Z k ( f ) � = ∅} For f = x ( x − 1)( x − 2)( x − 3)( x − 4), we have L ( f ) = { 4 , 5 , 6 } | Z 4 ( f ) | = | Z 5 ( f ) | = | Z 6 ( f ) | = 1 = Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

The set of lengths for f is L( f ) = { k ∈ N | f = g 1 g 2 · · · g k , g i ∈ Int( Z ) irreducible } = { k ∈ N | Z k ( f ) � = ∅} For f = x ( x − 1)( x − 2)( x − 3)( x − 4), we have L ( f ) = { 4 , 5 , 6 } | Z 4 ( f ) | = 3 | Z 5 ( f ) | = | Z 6 ( f ) | = 1 = Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

The set of lengths for f is L( f ) = { k ∈ N | f = g 1 g 2 · · · g k , g i ∈ Int( Z ) irreducible } = { k ∈ N | Z k ( f ) � = ∅} For f = x ( x − 1)( x − 2)( x − 3)( x − 4), we have L ( f ) = { 4 , 5 , 6 } | Z 4 ( f ) | = 3 | Z 5 ( f ) | = 18 | Z 6 ( f ) | = 1 | Z ( f ) | = 22 Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

Theorem (Frisch 2013) For any finite nonempty subset L ⊆ N ≥ 2 and any function µ : L → N ≥ 1 , there exists f ∈ Int( Z ) with ∀ k ∈ L( f ) µ ( k ) = | Z k ( f ) | L( f ) = L and Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

Theorem (Frisch 2013) For any finite nonempty subset L ⊆ N ≥ 2 and any function µ : L → N ≥ 1 , there exists f ∈ Int( Z ) with L( f ) = L ∀ k ∈ L( f ) µ ( k ) = | Z k ( f ) | and Recursive construction and the degree of the polynomials grows quickly. Question: How bad is factorization if we restrict the polynomial degree n ? Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

Used two measures: 1 Elasticity ρ ( f ) = max L( f ) min L( f ) 2 Catenary degree cat( f ), measures globally how similar factorizations are, paying attention to individual factors. For f = x ( x − 1)( x − 2)( x − 3)( x − 4), we have L ( f ) = { 4 , 5 , 6 } | Z 4 ( f ) | = 3 | Z 5 ( f ) | = 18 | Z 6 ( f ) | = 1 Catenary degree asks: can we run through these 22 factorizations using just a few swaps? Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

Exact definition of catenary degree For factorizations z , z ′ of f with gcd( z , z ′ ) = z ′′ , the distance is d( z , z ′ ) = max {| z / z ′′ | , | z ′ / z ′′ |} An N - chain from z to z ′ are factorizations z = z 0 , z 1 , . . . , z k = z ′ , such that for all 0 ≤ i ≤ k − 1, d( z i , z i +1 ) ≤ N . The catenary degree of f is cat( f ) = min { N ∈ N | ∀ z , z ′ ∈ Z ( f ) z and z ′ can be connected by an N -chain } Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

Problem: Fix n ∈ N . Consider all f ∈ Int( Z ) with deg( f ) = n . What possible values do we get for ρ ( f ) and cat( f )? Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

Problem: Fix n ∈ N . Consider all f ∈ Int( Z ) with deg( f ) = n . What possible values do we get for ρ ( f ) and cat( f )? Results involve Ω( k ) = number of prime factors of k ∈ Z , counting multiplicity. E.g. Ω(20) = Ω(2 ∗ 2 ∗ 5) = 3. Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

Problem: Fix n ∈ N . Consider all f ∈ Int( Z ) with deg( f ) = n . What possible values do we get for ρ ( f ) and cat( f )? Results involve Ω( k ) = number of prime factors of k ∈ Z , counting multiplicity. E.g. Ω(20) = Ω(2 ∗ 2 ∗ 5) = 3. For n = 0 or n = 1, get ρ ( f ) = 1 and cat( f ) = 0 for all f because we have unique factorization. So let n ≥ 2. Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

Slight simplification: Lemma. Each f ∈ Int( Z ) can be written uniquely as f = af ∗ / b , where a , b ∈ N and f ∗ ∈ Z [ x ] is primitive (i.e., gcd of its coefficients is 1). Furthermore, we have Z ( f ) = Z Z ( a ) + Z ( f ∗ / b ) Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

Elasticity: Theorem If f = af ∗ / b ∈ Int( Z ) and deg( f ) = n ≥ 2 , then 1 max L( f ∗ / b ) ≤ Ω( n !) + 1 2 0 ≤ max L( f ) − min L( f ) ≤ Ω( n !) − 1 3 If max L( f ) � = min L( f ) then 1 < ρ ( f ) ≤ Ω( n !) + 1 2 Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

Elasticity: Theorem If f = af ∗ / b ∈ Int( Z ) and deg( f ) = n ≥ 2 , then 1 max L( f ∗ / b ) ≤ Ω( n !) + 1 2 0 ≤ max L( f ) − min L( f ) ≤ Ω( n !) − 1 3 If max L( f ) � = min L( f ) then 1 < ρ ( f ) ≤ Ω( n !) + 1 2 Conversely, given 1 < r s ≤ Ω( n !) + 1 with 1 ≤ r − s ≤ Ω( n !) − 1 2 ∃ f ∈ Int( Z ) with deg( f ) = n and ρ ( f ) = r / s. Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

Catenary degree: Theorem If f = af ∗ / b ∈ Int( Z ) and deg( f ) = n ≥ 2 , then cat( f ) = 0 2 ≤ cat ( f ) ≤ Ω( n !) + 1 or Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

Catenary degree: Theorem If f = af ∗ / b ∈ Int( Z ) and deg( f ) = n ≥ 2 , then 2 ≤ cat ( f ) ≤ Ω( n !) + 1 cat( f ) = 0 or Conversely, given c = 0 or 2 ≤ c ≤ Ω( n !) + 1 ∃ f ∈ Int( Z ) with deg( f ) = n and cat( f ) = c. Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

Both together: Theorem. Fix n ≥ 0 and set A = { ( ρ ( f ) , cat( f )) | f ∈ Int( Z ) , deg( f ) = n } . If n = 0 or n = 1, then A = { (1 , 0) } ; If n = 2 then A = { (1 , 0) , (1 , 2) } ; If n ≥ 3, then A ⊆ { (1 , 0) , (1 , 2) }∪ � � �� s + k � � � c ∈ [3 , Ω( n !) + 1] , k ≥ 0 , s ∈ [ c , Ω( n !) + 1] , t ∈ [2 , s ] t + k , c � A ⊇ { (1 , 0) , (1 , 2) }∪ �� u + k � � � � , c � c ∈ [3 , Ω( n !) + 1] , k ≥ 2 , and u | c − 2 � k Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials