Semigroups and recurrences Ivan Martino∗, Luca Martino† June 30, 2012
Abstract In this work, we prove that the set the set of M -order linear recurrences with a non-trivial solution vanishing at least in all the points of the finite set I ⊂ N is non empty if I is a subset of the gaps of a numerical semigroup S finitely generated by a1 < a2 < · · · < aN and M = aN . We also obtain a power series associated to a numerical semigroup using a promising probabilistic approach.
1
Introduction
Numerical semigroups have been studied since the 18-th century and they appear naturally in combinatorics [16, 17], in commutative algebra (as the set of valuation of a C-algebra [1]) and in other branches of mathematics like probability and harmonic analysis. In this work, we connect the numerical semigroup theory with a problem posed by R. Fr¨ oberg and B. Shapiro in [8]. Inspired by the Skolem-Mahler-Lech Theorem [11], they have defined the variety V(M ;I) , the set of all M -order linear recurrences with a non-trivial solution vanishing at least in all the points of the finite set I ⊂ N. They have related the study of a particular open subvariety of V(M ;I) to the Shur function ideals [12]. They also stated certain open problems. For instance, one open issue is to understand for which pairs (M ; I) the variety V(M ;I) is empty or not. Here, we prove that the variety V(M ;I) is not empty when I is a subset of the gaps of a numerical semigroup S finitely generated by a1 < a2 < · · · < aN and M = aN . Indeed, given the probability mass wi , we construct a suitable stochastic process [10, 15] (specifically, a Markov chain with a countable state space N [5, 9], i.e., a random walk on N) associated to S, i.e., a1 with probability w1 , Xt = Xt−1 + ... aN with probability wN . The probabilities {gk }k∈N of visiting a state k in a generic time step t are connected together by a recurrence equation uS , uS : gk = w1 gk−a1 + ... + wN gk−aN , ∀ k ≥ 0, ∗ Stockholm
University, Department of Mathematics, email:
[email protected] Carlos III de Madrid, Department of Signal theory and Communications, email:
[email protected] † Universidad
1
given g0 = 1 and other initial conditions. They are zero if and only if these states k coincide exactly with the gaps of the numerical semigroup S. From an algebraic point of view, the probability generating function (PGF) [5, 9] associated to {gk }k∈N , G(z) =
1 − w1
z a1
1 , − ... − wN z aN
resembles the Hilbert series HN (z) of a non-commutative polynomial ring in N variables, Khx1 , x2 , . . . , xN i, with fine grading deg(xi ) = ai and char(K) = 0 [4]. However, to obtain HN (z) from G(z), we should force wi = 1, i = 1, ..., N , so that they lose any probability meaning (for further considerations, see Section 4). The rest of the paper is organized as follows. In Section 2, we first recall some background material on numerical semigroups. In Section 3, we describe the variety of linear recurrence equations and we state the open question we want to tackle. Section 4 is devoted to introduce the construction of a random walk associated to a given numerical semigroup. Finally, we present new results in Section 5.
2
Numerical semigroups
A semigroup (M, ◦) is a set M equipped with an associative operation ◦. When it has also a neutral element it is called monoid. An affine semigroup is a submonoid of (Nr , +) such that 0 ∈ S. If r = 1, the semigroup is called numerical semigroup. Any numerical semigroup S is finitely generated [3], that is there exist a1 , a2 , . . . , aN ∈ N such that (N ) X S = ha1 , a2 , . . . , aN i = ni ai : ni ∈ N i=1
If a natural number does not belong to S is called gap of S. We denote by ∆(a1 , . . . , aN ) = N \ S, the set of the gaps of S and it is well known that |∆(a1 , . . . , aN )| < ∞ if and only if gcd(a1 , a2 , . . . , aN ) = 1. We will always assume that this condition holds for S, because we can always reduce to this case [7]. The gaps of a numerical semigroup S are a very well studied [17] and are strongly connected, for instance, with the Frobenius number’s problem [6, 19] and the Hilbert function’s problem [16]. In this paper, we use essentially numerical semigroup, but the properties that we discuss are generalized for any semigroup [13].
3
The variety of linear recurrence
In this section we present the open question, stated in [8], we are going to deal with. First of all, consider that to any M -tuple of complex numbers α = (α1 , . . . , αM ) we can associate the linear recurrence equation U(α) defined as U(α) : gk + α1 gk−1 + · · · + αi gk−i + · · · + αM gk−M = 0. 2
(1)
If αM 6= 0, then U(α) is of order M . The solutions of a linear homogeneous recurrence equation with constant coefficients are well-known [2, 18] and depend on the roots of the characteristic polynomial of U(α), pα (y) : y M + α1 y M −1 + · · · + αi y i + · · · + αM = 0. Let {ρ1 , . . . , ρM } be the set of the roots of pα (y) (called characteristic roots or poles). If the roots are distinct, ρi 6= ρj for all i, j, a generic solution of the recurrence equation has the following analytic form gk = c1 ρk1 + c2 ρk2 + · · · + cM ρkM ,
(2)
where the coefficients ci , i = 1, .., M are determined depending on the initial conditions of the difference equation. With multiple roots and complex roots, other functional forms appear in the solutions as cosine and sine functions [2,18]. Definition 3.1. Let M ∈ N and let I be a finite subset of N. The open linear recurrence variety associated to the pair (M ; I), V(M ;I) , is the set of all linear recurrences of order exactly M having a non-trivial solution vanishing at least in all the points of I. The bijection U(−) implies that we can always assume the set of all linear recurrences (of order at most M ) as the affine space AM C . Being of order exactly M means that they belong to the affine principal open set AM C \ {αM 6= 0}. In [8], it is stated the following questions: Question. For which pairs (M ; I) the variety V(M ;I) is empty/not empty? If V(M ;I) 6= ∅ are there recurrence vanishing in a finite number of points? In what follows we are going to show that to each numerical semigroup S = ha1 , a2 , . . . , aN i it is possible to associate a recurrence uS of order aN vanishing on its gaps ∆(a1 , . . . , aN ) so that V(aN ;∆(a1 ,...,aN )) 6= ∅.
4
Random walk on a numerical semigroup
In this section we assume that the reader is familiar with basis concepts of stochastic processes and elements of probability theory (for further informations see, for instance, [5, 9, 10, 15]). Let us consider the stochastic process {Xt }t∈N [10, 15], where Xt ∈ N. For each t ∈ N, let Rt be a random variable (RV) that takes values in the set PN {a1 , ....aN } with probability mass wi , i = 1, ..., N (clearly, i=1 wi = 1). We assume that ai ∈ N\{0}, i = 1, ..., N and they are sorted in ascending order, that is a1 < ... < aN . Furthermore, let Xt = Xt−1 + Rt−1 be the relationship among the sequence of RVs {Xt }t∈N . In other words, a1 with probability w1 Xt = Xt−1 + ... aN with probability wN . 3
Equivalently, we could also write Xt = R0 + R1 + ... + Rt−1 =
t−1 X
Ri .
(3)
i=0
We denote PGF associated to the discrete RV Xt , that is P∞ by Fi t (z),Pthe ∞ i Ft (z) = f z = Prob{X t = i}z . ft,i represents the probability i=0 t,i i=0 that the Markov chain [10] passes through k exactly at the time t. It is well F
(i)
(0)
known that ft,i = t i! [14] (indeed, ft,i can be seen as the residue at 0 of the (i) complex function Ftz(z) (z) is the i-th derivative of Ft (z). It is an i ), where Ft easy task to see that, assuming X0 = 0, F0 (z)
=
F1 (z)
=
1, N X
wi z ai = w1 z a1 + w2 z a2 + ... + wN z aN ,
i=1
and more in general, associated to the RV Xt , "N #t X t ai Ft (z) = [F1 (z)] = wi z ,
(4)
i=1
because Xt can be expressed as sum of independent RVs as in (3) (see [5,9,10]). Moreover, let us consider the following probabilities gk = Prob{X0 = k ∨ X1 = k ∨ ...Xt = k ∨ Xt+1 = k ∨ . . . }, gk = Prob{Xt = k for some t}.
(5)
In other words, gk represents the probability that the Markov chain visits the state k in some time step t. Basic statistical considerations [5, 9, 10] lead us to write the associated PGF G(z) for the probabilities gk , G(z) =
∞ X
gk z k =
∞ X
Ft (z).
t=1
k=0 t
Since Ft (z) = [F1 (z)] , we have G(z) =
∞ X
Ft (z) =
∞ X t=0
t=0
t
[F1 (z)] =
1 , 1 − F1 (z)
hence finally G(z) can be expressed as G(z) =
1 − w1
z a1
1 . − ... − wN z aN
(6)
Observation. The Hilbert function of the non-commutative polynomial ring A = Khx1 , x2 , . . . , xN i, with fine grading deg(xi ) = ai and char(K) = 0 is HA (z) =
1 . 1 − z a1 − ... − z aN
To obtain it from (6), we can fix wi = 1, i = 1, ...N , losing the probability meaning. The construction, in next section, still holds. However, the connection between HA (z) and G(z) is in the non-commutative denumerands [13]. 4
4.1
A recurrence equation associated to a semigroup
We observe that we can rewrite (6) as G(z)(1 − w1 z a1 − ... − wN z aN ) = 1, and then G(z) = 1 + w1 z a1 G(z) + ... + wN z aN G(z). Explicitly, we get +∞ X k=0 +∞ X
gk z k gk z k
=
1 + w1
+∞ X
gk z k+a1 + ... + wN
k=0 +∞ X
= z 0 + w1
gk z k+aN ,
k=0 +∞ X
gj−a1 z j + ... + wN
j=a1
k=0
+∞ X
gi−aN z i .
i=aN
Algebraically, the equality holds if the coefficients associated to the powers of z k are equal in both sides, then clearly one has uS : gk = w1 gk−a1 + ... + wN gk−aN , ∀ k ≥ 0
(7)
with g0 = 1 and gk = 0 for k < 0. The previous equation denoted as uS is the recurrence equation associated to the random walk on the semigroup S = ha1 , a2 , . . . , aN i with weight wi , i = 1, ..., N .
5
The main result
In this section we link the previuos results with the variety V(M ;I) proving that V(M ;I) is not empty when I is a subset of the gaps of a numerical semigroup S finitely generated by a1 < a2 < · · · < aN and M = aN . More precisely, in the following theorem, we provide a set of recurrences in V(aN ;I) having only a finite number of zero contained in I. Theorem 5.1. Let S be a numerical semigroup finitely generated by a1 < a2 < · · · < aN with gcd(a1 , a2 , . . . , aN ) = 1 and let ∆(a1 , a2 , . . . , aN ) be the set of its gaps. Let I ⊆ ∆(a1 , a2 , . . . , aN ). Then, for every choice of the weight {wi }N i=1 (with wN 6= 0), the recurrence equation uS in (7) associated to a random walk on S belongs to V(aN ;I) . Proof. Comparing (1) and (7) we note that ( wk j = ak αj = 0 otherwise Requiring wN 6= 0 then αaN 6= 0. Hence uS is a recurrence equation of order aN . The solutions {gk }k∈N of the recurrence (7) are described in Section 3. This sequence {gk }k∈N is zero if and only if k 6∈ S. Corollary 5.1. Let S be a numerical semigroup finitely generated by a1 < a2 < · · · < aN with gcd(a1 , a2 , . . . , aN ) = 1 and let ∆(a1 , a2 , . . . , aN ) be the set of its gaps. If I ⊆ ∆(a1 , a2 , . . . , aN ), then V(aN ;I) 6= ∅. 5
Finally, we can easily gather the following result. Corollary 5.2. Let S be a numerical semigroup finitely generated by a1 < a2 < · · · < aN with gcd(a1 , a2 , . . . , aN ) = 1 and let ∆(a1 , a2 , . . . , aN ) be the set of its gaps. Let I ⊆ ∆(a1 , a2 , . . . , aN ). Then the krull dimension of V(aN ;I) , dimC (V(aN ;I) ), is at least N . Proof. V(aN ;I) is an open algebraic variety (indeed defined by polynomial equation and by αaN 6= 0). The set of points (i.e., the recurrences) shown in the previous theorem is a non-algebraic (because we have constrains 0 ≤ wi ≤ 1 and wi ∈ R) and it has dimension N . Thus V(aN ;I) has to contains a complex subvariety of dimension N .
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