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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) but x(x − 1) 2 ∈ Int(Z) 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) but x(x − 1) 2 ∈ Int(Z) and x n

• = x(x − 1)(x − 2) · · · (x − (n − 1))

n! ∈ Int(Z)

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; binomial polynomials x

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; binomial polynomials x

n

• ;

many other polynomials. Int(Z) also has nonunique factorization: x(x − 1)(x − 2)(x − 3)(x − 4) = x 5

• · 2 · 2 · 2 · 3 · 5

= x(x − 2)(x − 4) 3 (x − 1)(x − 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 = g1 · · · gk | k ∈ N, gi ∈ Int(Z) irreducible} which can be graded into factorizations of length k Zk(f ) = {f = g1 · · · gk | gi ∈ Int(Z) irreducible} The multiplicity of a length k is |Zk(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 = g1g2 · · · gk, gi ∈ Int(Z) irreducible} = {k ∈ N | Zk(f ) = ∅} Since x(x − 1)(x − 2)(x − 3)(x − 4) = x 5

• ∗ 2 ∗ 2 ∗ 2 ∗ 3 ∗ 5

= x(x − 2)(x − 4) 3 (x − 1)(x − 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 = g1g2 · · · gk, gi ∈ Int(Z) irreducible} = {k ∈ N | Zk(f ) = ∅} For f = x(x − 1)(x − 2)(x − 3)(x − 4), we have L(f ) = {4, 5, 6} |Z4(f )| = |Z5(f )| = |Z6(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 = g1g2 · · · gk, gi ∈ Int(Z) irreducible} = {k ∈ N | Zk(f ) = ∅} For f = x(x − 1)(x − 2)(x − 3)(x − 4), we have L(f ) = {4, 5, 6} |Z4(f )| = 3 |Z5(f )| = |Z6(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 = g1g2 · · · gk, gi ∈ Int(Z) irreducible} = {k ∈ N | Zk(f ) = ∅} For f = x(x − 1)(x − 2)(x − 3)(x − 4), we have L(f ) = {4, 5, 6} |Z4(f )| = 3 |Z5(f )| = 18 |Z6(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 L(f ) = L and ∀k ∈ L(f ) µ(k) = |Zk(f )|

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 and ∀k ∈ L(f ) µ(k) = |Zk(f )| 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} |Z4(f )| = 3 |Z5(f )| = 18 |Z6(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 = z0, z1, . . . , zk = z′, such that for all 0 ≤ i ≤ k − 1, d(zi, zi+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 ) = ZZ(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 2 with 1 ≤ r − s ≤ Ω(n!) − 1 ∃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

• r

2 ≤ cat(f ) ≤ Ω(n!) + 1

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

• r

2 ≤ cat(f ) ≤ Ω(n!) + 1 Conversely, given c = 0

• r

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 t + k , c

• c ∈ [3, Ω(n!) + 1], k ≥ 0, s ∈ [c, Ω(n!) + 1], t ∈ [2, s]
• A ⊇ {(1, 0), (1, 2)}∪

u + k k , c

• c ∈ [3, Ω(n!) + 1], k ≥ 2, and u|c − 2
• Paul Baginski

Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

Previous theorems use high-degree irreducibles of Z[x] in

• constructions. By contrast:

Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

Previous theorems use high-degree irreducibles of Z[x] in

• constructions. By contrast:

Theorem If f ∈ Z[x] has degree n ≥ 2 and f factors in Z[x] as a product of linear polynomials and constants, i.e. f = c1c2 . . . ck(a1x − b1) · · · (anx − bn) then f considered in Int(Z) will satisfy max L(f ) ≤ k + Ω(n!) + 1 1 ≤ ρ(f ) ≤ k + Ω(n!) + 1 k + 2 cat(f ) ≤ Ω(n!) + 1

Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

Previous theorems use high-degree irreducibles of Z[x] in

• constructions. By contrast:

Theorem If f ∈ Z[x] has degree n ≥ 2 and f factors in Z[x] as a product of linear polynomials and constants, i.e. f = c1c2 . . . ck(a1x − b1) · · · (anx − bn) then f considered in Int(Z) will satisfy max L(f ) ≤ k + Ω(n!) + 1 1 ≤ ρ(f ) ≤ k + Ω(n!) + 1 k + 2 cat(f ) ≤ n

Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

Thank you.

Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials

References

• P. Baginski, G. Knapp, J. Salem, G. Scullard, Elasticity and

catenary degree in the ring of integer-valued polynomials, in preparation.

• S. Frisch, A construction of integer-valued polynomials with

prescribed sets of lengths of factorizations Monatsh. Math. (2013) 171 341–350. P.-J. Cahen, J.-L. Chabert, What you should know about integer-valued polynomials, Amer. Math. Monthly (2016) 123, no. 4, 311–337.

Paul Baginski Fairfield University Nonunique Factorization in the Ring of Integer-Valued Polynomials