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 Motivation 1 Cyclotomic Numerical Semigroups 2 Main Questions 3 Symmetric Numerical Semigroups 4 Gluings of Numerical Semigroups 5 Gluings of Numerical Semigroups Complete Intersection Numerical Semigroups More Examples Semigroup Polynomial Divisors of x n − 1 6 Polynomially Related Numerical Semigroups 7 Gluings Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 2 / 29
Motivation s ∈ S x s , its Hilbert To a numerical semigroup S we associate H S ( x ) = � series, and P S ( x ) = (1 − x ) H S ( x ), its semigroup polynomial. It is known that, if S = � a , b � is a numerical semigroup, then 1 − x ab P S ( x ) = (1 − x )(1 − x ab ) H S ( x ) = and (1 − x a )(1 − x b ) . (1 − x a )(1 − x b ) It is also known that, if Φ n ( x ) denotes the n -th cyclotomic polynomial and p � = q are primes, then Φ pq ( x ) = (1 − x )(1 − x pq ) (1 − x p )(1 − x q ) . Therefore, if p � = q are primes and S = � p , q � , then P S ( x ) = Φ pq ( 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 Φ pq ( x ) = P � p , q � ( x ), one can prove results about the coefficients of Φ pq by appealing entirely to numerical semigroups). Various authors have studied the coefficients of cyclotomic polynomials or of divisors of x n − 1 . Note that if such a polynomial f ( x ) is of the form P S ( 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, P S 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 P S (0) � = 0 and the well-known property that 0 if n = 1 if n = p m Φ n (1) = p 1 otherwise we deduce the following Theorem 2 Suppose that S is a cyclotomic numerical semigroup. Then we have � Φ d ( x ) e d , P S ( x ) = d ∈D where D = { d 1 , . . . , d n } is a set of composite positive integers and the e d 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 P S ( x ) | ( x m − 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 P S ( x ) | ( x d − 1) h , where both d and h are chosen minimally, that is, P S ( x ) does not divide ( x n − 1) h − 1 for any n and it does not divide ( x d 1 − 1) h for any d 1 < d . Lemma 2 Let S be a cyclotomic numerical semigroup. If n � Φ d i ( x ) e i , P S ( x ) = i =1 then S is of depth d = lcm( d 1 , . . . , d n ) and height h = max { e 1 , . . . , e n } . 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 P S 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 P S 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 P S .) 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 P S 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 P S .) 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 P S is selfreciprocal. Proof. Φ n is selfreciprocal for n > 1. Theorem 4 If S is a cyclotomic numerical semigroup, then it must be symmetric. Proof. Use that S symmetric ⇔ P S 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 = � am 1 , am 2 , bm 1 + cm 2 � with a , b , c , m 1 , m 2 ∈ N such that m 1 , m 2 , a , b + c ≥ 2 and gcd( m 1 , m 2 ) = gcd( a , bm 1 + cm 2 ) = 1. The semigroup polynomial can be easily computed: P S ( x ) = (1 − x )(1 − x am 1 m 2 )(1 − x a ( bm 1 + cm 2 ) ) (1 − x bm 1 + cm 2 )(1 − x am 1 )(1 − x am 2 ) . 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 P S ( x ) = x 10 − x 9 + x 5 − 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 ≥ 2 q + 3 and let S = � m , m + 1 , qm + 2 q + 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 ≥ 2 q + 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 , T 1 and T 2 be submonoids of N . We say that T is the gluing of T 1 and T 2 if 1 T = T 1 + T 2 , 2 lcm( d 1 , d 2 ) ∈ T 1 ∩ T 2 , with d i = gcd( T i ) for i ∈ { 1 , 2 } . We denote this fact by T = T 1 + d T 2 , with d = lcm( d 1 , d 2 ). If T = S is a numerical semigroup, and S = T 1 + d T 2 , then T i = d i S i , with S i = T i / d i numerical semigroups, gcd( d 1 , d 2 ) = 1, lcm( d 1 , d 2 ) = d 1 d 2 , and d i ∈ S j 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 or the gluing of two complete intersection numerical semigroups. Let S = � a 1 , . . . , a t � be a minimally generated complete intersection numerical semigroup. Proceeding recursively, we find positive integers g 1 , . . . , g t − 1 such that S = a 1 N + g 1 · · · + g t − 1 a t N . By a Theorem of Assi et al., we then obtain t − 1 t � (1 − x g i ) � (1 − x a i ) − 1 , H S ( x ) = i =1 i =1 and t − 1 t � (1 − x g i ) � (1 − x a i ) − 1 . P S ( x ) = (1 − x ) i =1 i =1 Ciolan, Garc´ ıa-S´ anchez, Moree Cyclotomic Numerical Semigroups Cortona, September 11, 2014 14 / 29
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