Cyclotomic Numerical Semigroups Alexandru Ciolan Rheinische - - PowerPoint PPT Presentation

Cyclotomic Numerical Semigroups

Alexandru Ciolan Rheinische Friedrich-Wilhelms-Universit¨ at Bonn Joint work with Pedro A. Garc´ ıa-S´ anchez and Pieter Moree Cortona, September 11, 2014

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 1 / 29

Overview

1

Motivation

2

Cyclotomic Numerical Semigroups

3

Main Questions

4

Symmetric Numerical Semigroups

5

Gluings of Numerical Semigroups Gluings of Numerical Semigroups Complete Intersection Numerical Semigroups More Examples

6

Semigroup Polynomial Divisors of xn − 1

7

Polynomially Related Numerical Semigroups Gluings

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 2 / 29

Motivation

To a numerical semigroup S we associate HS(x) =

s∈S xs, its Hilbert

series, and PS(x) = (1 − x)HS(x), its semigroup polynomial. It is known that, if S = a, b is a numerical semigroup, then HS(x) = 1 − xab (1 − xa)(1 − xb) and PS(x) = (1 − x)(1 − xab) (1 − xa)(1 − xb). It is also known that, if Φn(x) denotes the n-th cyclotomic polynomial and p = q are primes, then Φpq(x) = (1 − x)(1 − xpq) (1 − xp)(1 − xq). Therefore, if p = q are primes and S = p, q, then PS(x) = Φpq(x).

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 3 / 29

Motivation

There seem to be hidden connections between cyclotomic polynomials and numerical semigroups (for e.g., using that Φpq(x) = Pp,q(x),

• ne can prove results about the coefficients of Φpq by appealing

entirely to numerical semigroups). Various authors have studied the coefficients of cyclotomic polynomials or of divisors of xn − 1. Note that if such a polynomial f (x) is of the form PS(x) for some numerical semigroup S, then we know that its non-zero coefficients alternate between 1 and −1. Otherwise, given a cyclotomic polynomial, or a product of cyclotomic polynomials, it is hard to draw conclusions regarding its coefficients. This being the case, PS has only roots on the unit circle; it turns out that certain semigroup polynomials have indeed this property.

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 4 / 29

Cyclotomic Numerical Semigroups

This motivated P. Moree to introduce the concept of cyclotomic numerical semigroup.

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 5 / 29

Cyclotomic Numerical Semigroups

This motivated P. Moree to introduce the concept of cyclotomic numerical semigroup.

Definition 1

We call a numerical semigroup cyclotomic if its semigroup polynomial is Kronecker, that is, a monic polynomial with integer coefficients having its roots in the unit disc.

Lemma 1 (Kronecker, 1857)

If f is a Kronecker polynomial with f (0) = 0, then all roots of f are on the unit circle and f factorizes as a product of cyclotomic polynomials.

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 5 / 29

Cyclotomic Numerical Semigroups

As a consequence of this Lemma, the fact that PS(0) = 0 and the well-known property that Φn(1) =      if n = 1 p if n = pm 1

• therwise

we deduce the following

Theorem 2

Suppose that S is a cyclotomic numerical semigroup. Then we have PS(x) =

• d∈D

Φd(x)ed, where D = {d1, . . . , dn} is a set of composite positive integers and the ed are positive integers too.

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 6 / 29

Cyclotomic Numerical Semigroups

If S is a cyclotomic numerical semigroup, then PS(x)|(xm − 1)e for some positive integers m and e.

Definition 3

We say that a numerical semigroup S is cyclotomic of depth d and height h if PS(x)|(xd − 1)h, where both d and h are chosen minimally, that is, PS(x) does not divide (xn − 1)h−1 for any n and it does not divide (xd1 − 1)h for any d1 < d.

Lemma 2

Let S be a cyclotomic numerical semigroup. If PS(x) =

n

• i=1

Φdi(x)ei, then S is of depth d = lcm(d1, . . . , dn) and height h = max {e1, . . . , en}.

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 7 / 29

Main Questions

Problem 1

Find an intrinsic characterization of the numerical semigroup S for which it is cyclotomic, that is, a characterization that does not involve PS or its roots in any way.

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 8 / 29

Main Questions

Problem 1

Find an intrinsic characterization of the numerical semigroup S for which it is cyclotomic, that is, a characterization that does not involve PS or its roots in any way. For instance: symmetry (Recall that S is symmetric if S ∪ (F(S) − S) = Z. This does not involve the roots of PS.)

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 8 / 29

Main Questions

Problem 1

Find an intrinsic characterization of the numerical semigroup S for which it is cyclotomic, that is, a characterization that does not involve PS or its roots in any way. For instance: symmetry (Recall that S is symmetric if S ∪ (F(S) − S) = Z. This does not involve the roots of PS.)

Problem 2

Classify the cyclotomic numerical semigroups with a prescribed depth and height.

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 8 / 29

Symmetric Numerical Semigroups

Lemma 3

If S is a cyclotomic numerical semigroup, then PS is selfreciprocal.

• Proof. Φn is selfreciprocal for n > 1.

Theorem 4

If S is a cyclotomic numerical semigroup, then it must be symmetric.

• Proof. Use that S symmetric ⇔ PS is selfreciprocal and Lemma 3.

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 9 / 29

Symmetric Numerical Semigroups

Theorem 5

If e(S) ≤ 3, then S is cyclotomic iff S is symmetric.

• Proof. “⇒” from Theorem 4.

“⇐” Clear for e(S) = 2. If S is symmetric with e(S) = 3, then S = am1, am2, bm1 + cm2 with a, b, c, m1, m2 ∈ N such that m1, m2, a, b + c ≥ 2 and gcd(m1, m2) = gcd(a, bm1 + cm2) = 1. The semigroup polynomial can be easily computed: PS(x) = (1 − x)(1 − xam1m2)(1 − xa(bm1+cm2)) (1 − xbm1+cm2)(1 − xam1)(1 − xam2) .

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 10 / 29

Symmetric Numerical Semigroups

However, symmetric numerical semigroups are not always cyclotomic!

Example 1

S = 5, 6, 7, 8, with PS(x) = x10 − x9 + x5 − x + 1 and F(S) = 9, is the symmetric numerical semigroup, with the smallest Frobenius number, that is not cyclotomic.

Example 2

Two symmetric numerical semigroups that are not cyclotomic, S = 5, 7, 8, 9 and S = 6, 7, 8, 9, 10, both with F(S) = 11.

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 11 / 29

Symmetric Numerical Semigroups

We suspect the following two families of symmetric numerical semigroups are not cyclotomic for dimensions greater than 3. Using GAP, P.A. Garc´ ıa-S´ anchez verified the hypothesis up to multiplicity 30.

Example 3

Let m and q be positive integers such that m ≥ 2q + 3 and let S = m, m + 1, qm + 2q + 2, . . . , qm + (m − 1). Taking m = 6 and q = 1 gives S = 6, 7, 10, 11, which is not cyclotomic.

Example 4

Let m and q be non-negative integers such that m ≥ 2q + 4 and let S = m, m + 1, (q + 1)m + q + 2, . . . , (q + 1)m + m − q − 2. Taking m = 8 and q = 1 gives S = 8, 9, 19, 20, 21, which is not cyclotomic.

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 12 / 29

Gluings of Numerical Semigroups

Let T, T1 and T2 be submonoids of N. We say that T is the gluing of T1 and T2 if

1 T = T1 + T2, 2 lcm(d1, d2) ∈ T1 ∩ T2, with di = gcd(Ti) for i ∈ {1, 2}.

We denote this fact by T = T1 +d T2, with d = lcm(d1, d2). If T = S is a numerical semigroup, and S = T1 +d T2, then Ti = diSi, with Si = Ti/di numerical semigroups, gcd(d1, d2) = 1, lcm(d1, d2) = d1d2, and di ∈ Sj for {i, j} = {1, 2}.

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 13 / 29

Complete Intersection Numerical Semigroups

We know from Delorme that S is a complete intersection iff S is either N

• r the gluing of two complete intersection numerical semigroups.

Let S = a1, . . . , at be a minimally generated complete intersection numerical semigroup. Proceeding recursively, we find positive integers g1, . . . , gt−1 such that S = a1N +g1 · · · +gt−1 atN. By a Theorem of Assi et al., we then obtain HS(x) =

t−1

• i=1

(1 − xgi)

t

• i=1

(1 − xai)−1, and PS(x) = (1 − x)

t−1

• i=1

(1 − xgi)

t

• i=1

(1 − xai)−1.

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 14 / 29

Complete Intersection Numerical Semigroups

Therefore, we get

Theorem 6

Every complete intersection numerical semigroup is cyclotomic. and, based on computer evidence, we assume the converse also holds true.

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 15 / 29

Complete Intersection Numerical Semigroups

Therefore, we get

Theorem 6

Every complete intersection numerical semigroup is cyclotomic. and, based on computer evidence, we assume the converse also holds true.

Conjecture 1

Every cyclotomic numerical semigroup is a complete intersection. The hypothesis was tested with GAP for all the symmetric numerical semigroups with Frobenius number up to 70.

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 15 / 29

Complete Intersection Numerical Semigroups

Attempts at proving the conjecture: According to Sz´ ekely and Wormald, if S = n1, . . . , ne is minimally generated, then K(x) = HS(x)

e

• i=1

(1 − xni) is a polynomial, whose only non-zero terms are those of degrees n ∈ S such that the Euler characteristic of the shaded set of n, i.e. ∆n =

• L ⊂ {n1, . . . , ne} : n −

s∈L s ∈ S

• , is not zero, that is

χS(n) =

L∈∆n(−1)#L = 0.

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 16 / 29

Complete Intersection Numerical Semigroups

Attempts at proving the conjecture: According to Sz´ ekely and Wormald, if S = n1, . . . , ne is minimally generated, then K(x) = HS(x)

e

• i=1

(1 − xni) is a polynomial, whose only non-zero terms are those of degrees n ∈ S such that the Euler characteristic of the shaded set of n, i.e. ∆n =

• L ⊂ {n1, . . . , ne} : n −

s∈L s ∈ S

• , is not zero, that is

χS(n) =

L∈∆n(−1)#L = 0.

If S is cyclotomic, does K(x) factorize as

b∈Betti(S)(1 − xb)?

(Betti(S) is the set of elements for which the underlying graph of ∆n is not connected.)

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 16 / 29

Complete Intersection Numerical Semigroups

One might wonder whether an expression like PS(x) = (1 − x)

t−1

• i=1

(1 − xgi)

t

• i=1

(1 − xai)−1 is unique. In fact, more is true:

Lemma 4 (Moree, 2000)

Let f (x) = 1 + a1x + · · · + adxd ∈ Z[x] be of degree d and α1, . . . , αd its

• roots. Put sf (k) = α−k

1

+ · · · + α−k

d . Then sf (k) ∈ Z and

sf (k) + a1sf (k − 1) + · · · + ak−1sf (1) + kak = 0, with ak = 0 for k > d. Put bf (k) = 1

k

• d|k sf (d)µ

k

d

• . Then bf (k) ∈ Z

and 1 + a1x + · · · + adxd =

∞

• j=1

(1 − xj)bf (j).

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 17 / 29

Complete Intersection Numerical Semigroups

As a consequence, given a numerical semigroup S, there are unique integers ǫ1, ǫ2, . . . such that PS(x) =

∞

• j=1

(1 − xj)ǫj.

Definition 7

We call ǫ = {ǫ1, ǫ2, . . .} the cyclotomic exponent sequence of S.

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 18 / 29

Complete Intersection Numerical Semigroups

As a consequence, given a numerical semigroup S, there are unique integers ǫ1, ǫ2, . . . such that PS(x) =

∞

• j=1

(1 − xj)ǫj.

Definition 7

We call ǫ = {ǫ1, ǫ2, . . .} the cyclotomic exponent sequence of S.

Problem 3

Relate the properties of S to its cyclotomic exponent sequence.

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 18 / 29

Complete Intersection Numerical Semigroups

As a consequence, given a numerical semigroup S, there are unique integers ǫ1, ǫ2, . . . such that PS(x) =

∞

• j=1

(1 − xj)ǫj.

Definition 7

We call ǫ = {ǫ1, ǫ2, . . .} the cyclotomic exponent sequence of S.

Problem 3

Relate the properties of S to its cyclotomic exponent sequence.

Lemma 5

A numerical semigroup S has a cyclotomic exponent sequence with finitely many non-zero terms iff S is cyclotomic.

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 18 / 29

Complete Intersection Numerical Semigroups

Let µ = min {n > 1 : χS(n) = 0} and let d(n) be the denumerant of n.

Lemma 6

Let S = n1, . . . , ne be minimally generated and cyclotomic. Then a) We have ǫ1 = 1. If µ > ne, then ǫn1 = ǫn2 = · · · = ǫne = −1. Further ǫµ = −χS(µ). If 1 ≤ j ≤ µ and j / ∈ {1, n1, . . . , ne, µ}, then ǫj = 0. b) If, in addition, there is some s ∈ S, s ≤ ne with d(s) ≥ 2, then ǫj ≥ 0 for all j > ne. Therefore, under some assumptions (which we hope to relax), if S = n1, . . . , ne is minimally generated, then there exist k ∈ N, 1 < δ1 < δ2 < · · · < δk and ǫi ≥ 1, i = 1, . . . , k such that HS(x) = (1 − xδ1)ǫ1 · · · (1 − xδk)ǫk (1 − xn1) · · · (1 − xne) .

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 19 / 29

Complete Intersection Numerical Semigroups

As a first step confirming our hypothesis, we have the following

Lemma 7

Suppose S = n1, . . . , ne is cyclotomic, minimally generated and that HS(x) = (1 − xδ1)ǫ1 · · · (1 − xδk)ǫk (1 − xn1) · · · (1 − xne) . (1) Then δi ∈ S, d(δi) ≥ 2 for every 1 ≤ i ≤ k, and δ1 = min {s ∈ Betti(S)}.

• Proof. Rewrite (1) as
• s∈S

xs = (1−ǫ1xδ1 +· · · )

• s∈S

d(s)xs =

• s∈S

s<δ1

d(s)xs +(d(δ1)−ǫ1)xδ1 +· · · , so δ1 is the first s ∈ S with d(s) ≥ 2, i.e. δ1 = min {s ∈ Betti(S)}.

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 20 / 29

More Examples

Example 5 (Free semigroups)

Let S = n1, . . . , nt. We say that S is free if either S = N or it is the gluing of the free semigroup n1, . . . , nt−1 and nt (the order is important). S is free iff S is generated by a smooth sequence. Let n ≥ 2 and (a1, a2, . . . , an) be a sequence of relatively prime positive

• integers. For every k = 1, . . . , n, let dk = gcd(a1, . . . , ak). For

k = 2, . . . , n let ck = dk−1/dk. Let Sk be the semigroup generated by {a1, . . . , ak}. We say that the sequence (a1, a2, . . . , an) is smooth if ckak ∈ Sk−1 for every k = 2, . . . , n. If S = a1, a2, . . . , an then, according to Leher’s Ph.D. thesis, PS(x) = (1 − x)

n

• i=2

(1 − xciai)

n

• i=1

(1 − xai)−1.

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 21 / 29

More Examples

Example 6 (Binomial semigroups)

Consider Bn(a, b) := an, ban−1, . . . , abn−1, bn, where a, b > 1 are

• coprime. Putting ak = an−kbk for k = 0, . . . , n, the sequence (a0, . . . , an)

is smooth. We have PBn(a,b)(x) = (1 − x)

n

• k=1

(1 − xan+1−kbk)

n

• k=0

(1 − xan−kbk)−1. In particular, if p = q primes, we can compute PBn(p,q)(x) =

n+1

• l=2
• i+j=l

1≤i,j≤l

Φpiqj, so that Bn(p, q) is of depth d = pn+1qn+1 and height h = 1.

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 22 / 29

Semigroup Polynomial Divisors of xn − 1

Theorem 8

Let p, q and r be distinct primes. Suppose S is cyclotomic of depth d = pqr and height h = 1. Then S = pq, r or one of its cyclic permutations.

• Proof. Suppose that PS(x)|xpqr − 1 for some S. This means that

PS = Φk1

pqΦk2 qrΦk3 prΦk4 pqr, for some 0 ≤ ki ≤ 1. By symmetry, assume

k1 ≥ k2 ≥ k3. Note that, modulo x2, f (x) = 1 + (k4 − k1 − k2 − k3)x. Since PS(x) ≡ 1 − x (mod x2), we deduce that (k1, k2, k3, k4) ∈ {(1, 0, 0, 0), (1, 1, 0, 1)} and exclude the first case, as it leads to a depth d = pq. Indeed, ΦpqΦqrΦpqr = Ppr,q.

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 23 / 29

Semigroup Polynomial Divisors of xn − 1

Theorem 9

Let p, q be distinct primes. Suppose S is cyclotomic of depth d = pnq and height h = 1. Then S = pn, q.

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 24 / 29

Semigroup Polynomial Divisors of xn − 1

Theorem 9

Let p, q be distinct primes. Suppose S is cyclotomic of depth d = pnq and height h = 1. Then S = pn, q. Trying to solve for d = pnqn, n ≥ 2 and h = 1 we obtain S = pn, qn and Bn−1(p, q) = pn−1, pn−2q, . . . , pqn−2, qn−1. We do not know whether these are all...

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 24 / 29

Polynomially Related Numerical Semigroups

Definition 10

We say that two numerical semigroups S and T are polynomially related, and denote it S ≤P T, if there exists f (x) ∈ Z[x] and an integer w ≥ 1 such that HS(xw)f (x) = HT(x) (2) (Or, equivalently, PS(xw)f (x) = PT(x)(1 + x + · · · + xw−1).)

Example 7

a) pa, qb ≤P pm, qn if 1 ≤ a ≤ m and 1 ≤ b ≤ n. b) pa, qb ≤P Bn(p, q) if a, b ≥ 1 and 2 ≤ a + b ≤ n + 1.

Problem 4

Find necessary and sufficient conditions such that S ≤P T.

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 25 / 29

Polynomially Related Numerical Semigroups

Lemma 8

Suppose that (2) holds with S, T numerical semigroups. Then a) f (0) = 1. b) f (1) = w. c) f ′(1) = w(N(T) − wN(S) + (w − 1)/2). (N(S) = # of gaps in S.) d) F(T) = wF(S) + deg(f ).

Lemma 9

Suppose that S and T are numerical semigroups. Then HS(xw)f (x) = HT(x) for some integer w ≥ 1 and f ∈ N[x] iff there are 0 = e1 < e2 < · · · < ew such that f (x) = w

i=1 xew and every t ∈ T can

be written in a unique way as t = ei + s · w, 1 ≤ i ≤ w, s ∈ S.

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 26 / 29

An Application

Theorem 11

Let p = q be primes and m, n positive integers. The quotient Q(x) := Ppm,qn(x)/Φpmqn(x) is in Z[x], is monic and has constant coefficient 1. Its non-zero coefficients alternate between 1 and −1. In fact, a more general result holds:

Theorem 12

Suppose that S and T are numerical semigroups with HS(xw)f (x) = HT(x) for some w ≥ 1 and f ∈ N[x]. Put Q(x) = PT(x)/PS(xw). Then Q(0) = 1 and Q(x) is a monic polynomial having non-zero coefficients that alternate between 1 and −1.

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 27 / 29

Gluings

Theorem 13

Assume that S = a1S1 +a1a2 a2S2. Then S1 ≤P S and S2 ≤P S with a relating polynomial f ∈ N[x].

Theorem 14

Assume that S = a1S1 +a1a2 a2S2. Put Qi(x) = PS(x)/PSi(xai) for i = 1, 2. Then Qi(0) = 1 and Q(x) is a monic polynomial having non-zero coefficients that alternate between 1 and −1.

Theorem 15

Let S and T be two numerical semigroups and HS(xw)f (x) = HT(x) for some polynomial f ∈ N[x] and w ∈ N. Let u be the gcd of the exponents

• f f . If u ∈ S, then there exists a numerical semigroup U such that

T = uU +uw wS.

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 28 / 29

Thank you!

Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 29 / 29