Let Lambda be a numerical semigroup. Assume there exists an algebraic
function field over GF(q) in one variable which possesses a rational place that
has Lambda as its Weierstrass semigroup. We ask the question as to how many
rational places such a function field can possibly have and we derive an upper
bound in terms of the generators of Lambda and q. Our bound is an improvement
to a bound by Lewittes which takes into account only the multiplicity of Lambda
and q. From the new bound we derive significant improvements to Serre's upper
bound in the cases q=2, 3 and 4. We finally show that Lewittes' bound has
important implications to the theory of towers of function fields.Comment: 16 pages, 3 table

Garcia

Høholdt

Lewittes

Miura

Olav Geil

Pellikaan

Ryutaroh Matsumoto

Stöhr

Suzuki

English

arXiv

AbstractWe present a new bound on the number of Fq-rational places in an algebraic function field. It uses information about the generators of the Weierstrass semigroup related to a rational place. As we demonstrate, the bound has implications to the theory of towers of function fields

Geil, Olav

Matsumoto, Ryutaroh

Elsevier - Publisher Connector 

Bounding the number of Fq-rational places in algebraic function fields using Weierstrass semigroups 

Crossref

Bounding the number of <mml:math altimg="si10.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">F</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msub></mml:math>-rational places in algebraic function fields using Weierstrass semigroups

Let Lambda be a numerical semigroup. Assume there exists an algebraic function field over Fq in one variable which possesses a rational place that has Lambda as its Weierstrass semigroup. We ask the question as to how many rational places such a function field can possibly have and we derive an upper bound in terms of the generators of Lambda and q. Our bound is an improvement to Lewittes' bound in [6] which takes into account only the multiplicity of Lambda and q. From the new bound we derive significant improvements to Serre's upper bound in the cases q = 2, 3 and 4. We finally show that Lewittes' bound has important implications to the theory of towers of function fields. Udgivelsesdato: 2007Let Lambda be a numerical semigroup. Assume there exists an algebraic function field over Fq in one variable which possesses a rational place that has Lambda as its Weierstrass semigroup. We ask the question as to how many rational places such a function field can possibly have and we derive an upper bound in terms of the generators of Lambda and q. Our bound is an improvement to Lewittes' bound in [6] which takes into account only the multiplicity of Lambda and q. From the new bound we derive significant improvements to Serre's upper bound in the cases q = 2, 3 and 4. We finally show that Lewittes' bound has important implications to the theory of towers of function fields

Geil, Hans Olav

Bounding the number of rational places using Weierstrass semigroups

We present a new bound on the number of Fq-rational places in an algebraic function field. It uses information about the generators of the Weierstrass semigroup related to a rational place. As we demonstrate, the bound has implications to the theory of towers of function fields. Udgivelsesdato: JUNWe present a new bound on the number of Fq-rational places in an algebraic function field. It uses information about the generators of the Weierstrass semigroup related to a rational place. As we demonstrate, the bound has implications to the theory of towers of function fields

Bounding the number of <em>F<sub>q</sub></em>-rational places in algebraic function fields using Weierstrass semigroups

Journal of Pure and Applied Algebra 213 (2009) 1152–1156
Contents lists available at ScienceDirect
Journal of Pure and Applied Algebra
journal homepage: www.elsevier.com/locate/jpaa
Bounding the number of Fq-rational places in algebraic function fields
using Weierstrass semigroups
Olav Geil a,∗, Ryutaroh Matsumoto b
a Department of Mathematical Sciences, Aalborg University, 9220, Denmark
b Department of Communications and Integrated Systems, Tokyo Institute of Technology, 152-8550, Japan
a r t i c l e i n f o
Article history:
Received 12 December 2007
Received in revised form 25 September
2008
Available online 19 December 2008
Communicated by J. Walker
MSC:
Primary: 14G15
11G20
14H05
14H25
a b s t r a c t
We present a new bound on the number of Fq-rational places in an algebraic function field.
It uses information about the generators of theWeierstrass semigroup related to a rational
place. As we demonstrate, the bound has implications to the theory of towers of function
fields.
© 2008 Elsevier B.V. All rights reserved.
1. Introduction
Throughout this paper by a function fieldwewill alwaysmean an algebraic function field of one variable. Given a function
field F/Fq, we denote by N(F) the number of rational places and we denote by g(F) the genus. We will always assume that
Fq is the full constant field of F. For applications in coding theory it is desirable to have N(F)/g(F) as high as possible as
this allows for the construction of codes with good parameters. The above observation has led to extensive research on the
problem of deciding, given a constant field Fq and a number g , what is the highest number Nq(g) such that a function field
F/Fq exists with N(F) = Nq(g) and g(F) = g .
Recall that, for any rational place the number of gaps in the correspondingWeierstrass semigroupΛ equals the genus g of
the corresponding function field. This suggests that in some cases aWeierstrass semigroupΛ for a rational placemight hold
more information about the number of rational places of the function field than does the genus alone. This themewas firstly
explored by Lewittes in [4], though the bound by Stöhr and Voloch ([9, pp. 14–15]) induces a bound in terms of aWeierstrass
semigroup under certain conditions. The smallest non-zero element in a numerical semigroup Λ is called the multiplicity
of Λ and we denote it by λ1. Lewittes showed that if λ1 is the multiplicity of a Weierstrass semigroup corresponding to a
rational place of F/Fq then N(F) ≤ qλ1 + 1 holds. In the present paper we derive an improved upper bound on N(F) as we
take into account not only the multiplicity but also all the other elements in a generating set ofΛ.
2. The bounds
In the following Λ is always a numerical semigroup with finitely many gaps and {λ1, . . . , λm} is a generating set for Λ
with 0 < λ1 < · · · < λm.
∗ Corresponding author.
E-mail addresses: olav@math.aau.dk (O. Geil), ryutaroh@rmatsumoto.org (R. Matsumoto).
URLs: http://www.math.aau.dk/∼olav/publications.html (O. Geil), http://www.rmatsumoto.org/research.html (R. Matsumoto).
0022-4049/$ – see front matter© 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.jpaa.2008.11.013
O. Geil, R. Matsumoto / Journal of Pure and Applied Algebra 213 (2009) 1152–1156 1153
Definition 1. Let Λ be fixed. If there exist function fields over Fq having a rational place whose Weierstrass semigroup is
equal toΛ then we define
Nq(Λ) = max{N(F) | F is a function field over Fq having a rational place which Weierstrass semigroup equalsΛ}.
If such function fields do not exist, we define Nq(Λ) = 0.
Theorem 1.
Nq(Λ) ≤ #
(
Λ \ m∪
i=1(qλi +Λ)
)
+ 1 (1)
which implies
Nq(Λ) ≤ #(Λ \ (qλ1 +Λ))+ 1 = qλ1 + 1. (2)
Here, γ +Λmeans {γ + λ | λ ∈ Λ}.
Proof. Let F/Fq be a function field. Let its rational places be P1, . . . ,PN−1,P and assume that the Weierstrass semigroup
corresponding toP isΛ. DefineL = ∪∞s=0L(sP ) and letLt = L(tP ) for t ∈ N0 ∪ {−1}. In particularL−1 = {0}. It is well
known that
Lt = Lt−1 if t ∈ N0 \Λ and dim(Lt) = dim(Lt−1)+ 1 if t ∈ Λ. (3)
Here dim denotes the dimension as a vector space over Fq. Let ϕ : L → FN−1q be the map ϕ(f ) = (f (P1), . . . , f (PN−1))
and define Et = ϕ(Lt) for t ∈ N0 ∪ {−1}. From Eq. (3) we observe that dim(E−1) = 0 and that dim(Et) = dim(Et−1) for
all t ∈ N0 \ Λ. For t ∈ Λ we can either have dim(Et) = dim(Et−1) or dim(Et) = dim(Et−1) + 1. The map ϕ is surjective
meaning that for t large enough dim(Et) = N − 1. Hence, if we can give an upper bound on the number of t ∈ Λ for
which dim(Et) = dim(Et − 1)+ 1 holds then this upper bound will also be an upper bound on the number N − 1. To prove
Eq. (1) we therefore only need to show that dim(Et) = dim(Et−1)+ 1 cannot happen when t ∈ qλi +Λ for some i. For this
purpose let for i = 1, . . . ,m, xi ∈ L be an element with−vP (xi) = λi. Here vP is the valuation corresponding to P . Given
t = qλi + λwith λ ∈ Λ choose f ∈ Lλ \ Lλ−1. We have xqi f ∈ Lt \ Lt−1 and xif ∈ Lt−1. Clearly, ϕ(xqi f ) = ϕ(xif ) and the
proof of Eq. (1) is complete. The left part of Eq. (2) is an immediate consequence of Eq. (1) and the right part corresponds
to [3, Lemma 5.15]. 
The Serre bound implies that ifΛ is of genus g then
Nq(Λ) ≤ gb2√qc + q+ 1 (4)
holds. We observe that Lewittes’ bound (2) is better than the bound (4) if and only if (λ1 − 1)/g < b2√qc/q holds. As a
consequence the bound (2) is always better than the bound (4) when q ≤ 4.
Example 1. In Table 1we consider a collection of 3 semigroups.We apply the bounds to a number of fields of characteristics
2 and 3. Restricting to characteristics 2 and 3 allows us to get information on the number Nq(g) from van der Geer and van
der Vlugt’s table in [2]. An entry x/y in the row named ‘‘bounds’’ indicates that Lewittes’ bound produces x and that the new
bound produces y. An interval in the row named Nq(g) means that Nq(g) is known to be in this interval. By 〈λ1, . . . , λn〉
we mean the semigroup generated by λ1, . . . , λn. Table 1 illustrates that the new bound can be quite an improvement to
Lewittes’ bound and that it can be much smaller than Nq(g) also when Lewittes’ bound is not. We get the most significant
results for small q.
Example 2. From [2] we have N2(8) = 11, N3(8) ∈ {17, 18} and N4(8) ∈ {21, 22, 23, 24}. If one applies the bounds (1) and
(2) to the 67 different semigroups of genus 8 (these semigroups can be found at [5]) one gets the following picture. Lewittes’
bound tells us that in a function field over F2 of genus 8 and with N2(8) = 11 rational places 13 semigroups are not allowed
as Weierstrass semigroups of a rational place. The new bound gives us that 33 semigroups are not allowed. Assuming that
N3(8) = 18 and N4(8) = 24 Lewittes’ bound excludes in both cases 26 semigroups whereas the new bound excludes in
both cases 31 semigroups.
The following Proposition gives us some information on how good or bad the bound in Eq. (1) can possibly be.
Proposition 1. We have
qλ1 + 1− g ≤ #
(
Λ \
(
m∪
i=1(qλi +Λ)
))
+ 1 ≤ min{qλ1 + 1, qm + 1}.
1154 O. Geil, R. Matsumoto / Journal of Pure and Applied Algebra 213 (2009) 1152–1156
Table 1
Semigroups from Example 1.
Λ = 〈8, 9, 20〉 g = 20
q 2 3 4 8 9 16
Bounds 17/9 25/16 33/25 65/65 73/73 129/129
Nq(g) 19–21 30–34 40–45 76–83 70–91 127–139
Λ = 〈13, 14, 20〉 g = 42
q 2 3 4 8 9 16
Bounds 27/9 40/17 53/33 105/95 118/102 209/195
Nq(g) 33–35 52–59 75–80 129–147 122–161 209–254
Λ = 〈10, 11, 20, 22〉 g = 45
q 2 3 4 8 9 16
Bounds 21/5 31/10 41/17 81/65 91/82 161/141
Nq(g) 33–37 54–62 80–84 144–156 136–170 242–268
Proof. To see the first inequality observe that there are at least qλ1 − g elements inΛ that are smaller than qλ1, and these
elements must belong toΛ \ ∪mi=1(qλi+Λ). Regarding the last inequality the upper bound λ1q+ 1 comes from Theorem 1.
To see the upper bound qm + 1 we note that all λ ∈ Λ can be written as a1λ1 + · · · + amλm for some a1, . . . , am ∈ N0. If
λ ∈ Λ \ (∪mi=1(qλi +Λ)) then necessarily a1, . . . , am < qmust hold. 
We now present some corollaries to Theorem 1.
Corollary 1. Define t = #{λ ∈ Λ | λ ∈ [λ1 + 1, λ1 + dλ1/qe − 1]}. We have Nq(Λ) ≤ qλ1 − t + 1.
Proof. For λ ∈ Λwith λ ∈ [λ1 + 1, λ1 + dλ1/qe − 1]we have qλ 6= qλ1 + η for any η ∈ Λ as there are no non-zero η ∈ Λ
with η < λ1. This implies qλ ∈ (∪mi=1(qλi + Λ)) \ (qλ1 + Λ). Therefore the number on the right side of Eq. (1) is at least t
smaller than the number on the right side of Eq. (2). 
Example 3. Consider the case λ1 = g + 1. That is, the case Λ = {0, g + 1, g + 2, . . .}. The number t from Corollary 1
becomes equal to d(g + 1)/qe − 1. Hence
Nq(Λ) ≤ q(g + 1)+ 2− d(g + 1)/qe (5)
holds. Given λ > λ1 we have qλ 6∈ qλ1 +Λ if and only if λ ∈ [λ1 + 1, λ1 + dλ1/qe − 1] and qλ+ η ∈ qλ1 +Λ holds for all
η ∈ Λ \ {0}. Hence, for the particular semigroup in the present example, we have
#
(
Λ \
(
m∪
i=1(λi +Λ)
))
+ 1 = qλ1 − t + 1 = q(g + 1)+ 2− d(g + 1)/qe.
Remark 1. The conductor of a semigroupΛ ⊆ N0 with finitely many gaps is the smallest number c such that there are no
gaps greater or equal to c. The conductor is known to be smaller or equal to 2g ([3, Proposition 5.7]). If qλ1 + c ≤ qλ2 then
it is clear that the number on the right side of Eq. (1) is the same as the number on the right side of Eq. (2). In particular the
numbers are the same if qλ1 + 2g ≤ qλ2.
3. Bounds on Nq(g)
From Lewittes’ bound (2) we immediately get Nq(g) ≤ q(g + 1)+ 1 as the multiplicity of a semigroup with g gaps can
be at most g+1. This fact is not stressed in [4] as the paper contains slightly better bounds on Nq(g) namely Nq(g) ≤ qg+2
([4, Theorem 1, part (a)]) and N2(g) ≤ 2g − 2 ([4, Eq. (19)]). We now investigate the implication of the new result in Eq. (1)
for establishing bounds on Nq(g). We get the following proposition.
Proposition 2.
Nq(g) ≤
(
q− 1
q
)
g + q+ 2− 1
q
. (6)
Proof. The proof uses Corollary 1. An estimate of the number t in Corollary 1 can be given in terms of λ1 and g alone. We
have
t ≥ dλ1/qe − 1− (g − (λ1 − 1)) ≥ λ1q + λ1 − g − 2
O. Geil, R. Matsumoto / Journal of Pure and Applied Algebra 213 (2009) 1152–1156 1155
as there are g − (λ1 − 1) gaps greater than λ1. Hence,
Nq(g) ≤ max
{
qλ1 −
(
λ1
q
+ λ1 − g − 2
)
+ 1 | 2 ≤ λ1 ≤ g + 1
}
. 
Observe, that the bound (6) was obtained by showing that the semigroup considered in Example 3 is the worst case.
Proposition 2 implies
N2(g) ≤ 112g + 3
1
2
, N3(g) ≤ 223g + 4
2
3
, N4(g) ≤ 334g + 5
3
4
(7)
which is much better than Serre’s upper bound. It should be mentioned that the bounds in Eqs. (7) compete with Ihara’s
bound only for small values of g .
4. Towers of function fields
Recall, that a sequence of function fields (F (1)/Fq, F (2)/Fq, · · ·) is called a tower if F (i) ⊆ F (i+1) holds for all i ≥ 1. Given a
tower of function fields we write N (i) = N(F (i)), g(i) = g(F (i)) and we say that the tower is asymptotically good if g(i) →∞
for i → ∞ and lim infi→∞(N (i)/g(i)) = κ holds for some κ > 0. Eq. (8) in the following corollary is a consequence of
Lewittes’ bound (2). Eq. (9) seems a well-known fact but it also immediately follows from the last part of Proposition 1.
Corollary 2. Assume a tower of function fields is given with g(i) → ∞ for i → ∞ and lim infi→∞(N(i)g(i) ) = κ > 0. Let
(P (1),P (2), . . .) be any sequence such that P (i) is a rational place of F (i) for i = 1, 2, . . .. Let λ(i)1 be the multiplicity of the
Weierstrass semigroupΛ(i) related to P (i) and let mi be the number of generators in some description of Λ(i). We have
lim inf
i→∞ (λ
(i)
1 /g
(i)) ≥ κ/q, (8)
mi →∞ for i→∞. (9)
Example 4. In [1] Garcia and Stichtenoth introduced a tower of function fields over Fq2 which satisfies g(i) →∞ for i→∞
and limi→∞(N (i)/g(i)) = q − 1. This tower was further studied in [8] where Pellikaan, Stichtenoth and Torres found the
generators of a sequence of Weierstrass semigroups related to it. Using the results in [8] one finds that limi→∞(λ(i)1 /g(i)) =
1/q holds. For comparison Eq. (8) reads lim infi→∞(λ(i)1 /g(i)) ≥ (1− (1/q))/q.
For the construction of one-point geometric Goppa codes with efficient decoding algorithms [3], we need a basis
{f1, f2, . . .} of L(iP ) such that vP (fi) > vP (fi+1) and have to compute fi(Pj), which are generally difficult even when a
set of defining equations is explicitly provided. Miura [6] and Pellikaan [7] independently and simultaneously proposed a
standard form of defining equations for affine algebraic curves which renders that the subsequent finding of the required
fi’s and the computing of fi(Pj) is straightforward. The number of equations in that standard form becomes the minimum if
and only if theWeierstrass semigroupΛ ofP is telescopic [10]. Therefore, it is desirable to find asymptotically good towers
of function fields with telescopic Weierstrass semigroups. We will show that we cannot find such a tower.
Definition 2. Let (a1, . . . , ak) be a sequence of positive integers. Define di = gcd(a1, . . . , ai), i = 1, . . . , k. If dk = 1 and
ai/di ∈ 〈a1/di−1, . . . , ai−1/di−1〉 for i = 2, . . . , k, then the sequence (a1, . . . , ak) is called telescopic. A semigroup is called
telescopic if it is generated by a telescopic sequence.
We will need the following result corresponding to [3, Lem. 5.34].
Lemma 1. If (a1, . . . , ak) is telescopic then for any λ ∈ 〈a1, . . . , ak〉 there exist (uniquely determined) non-negative integers
x1, . . . , xk such that 0 ≤ xj < dj−1/dj for 2 ≤ j ≤ k and λ =∑kj=1 xjaj.
Proposition 3. Let (F (1)/Fq, F (2)/Fq, . . .) be a tower of function fields such that for infinitely many i the following holds: F (i)
possesses a rational place P (i) having a telescopic Weierstrass semigroupΛ(i). Then the tower is asymptotically bad.
Proof. Let (λ(i)1 , . . . , λ
(i)
mi) be a telescopic sequence generatingΛ
(i)withmi being the smallest possible. By [6, pp. 1420–1421],
{λ(i)1 , . . . , λ(i)mi} is a minimum generating set for Λ(i). Write d(i)j = gcd(λ(i)1 , . . . , λ(i)j ) for 1 ≤ j ≤ mi. Clearly, d(i)j | d(i)j−1 for
j ≥ 2 and by minimality of mi, Lemma 1 implies d(i)j−1 ≥ 2d(i)j . The genus g(i) is given by the following expression (see
[3, Pro. 5.35])
g(i) =
(
1+
mi∑
j=2
(
dj−1
dj
− 1
)
λ
(i)
j
)/
2,
and therefore g(i) ≥ mi−12 λ(i)1 holds. From Eq. (8) we see that the only hope for the tower to be asymptotically good is that
the sequence ofmi’s is bounded above. Eq. (9) tells us the opposite. 
1156 O. Geil, R. Matsumoto / Journal of Pure and Applied Algebra 213 (2009) 1152–1156
Acknowledgments
The authors would like to thank Peter Beelen, Tom Høholdt, Massimiliano Sala, Ruud Pellikaan and Masaaki Homma for
pleasant discussions.
References
[1] A. Garcia, H. Stichtenoth, On the asymptotic behaviour of some towers of function fields over finite fields, J. Number Theory 61 (1996) 248–273.
[2] G. van der Geer, M. van der Vlugt, Tables of curves withmany points, (November 17, 2007) http://www.science.uva.nl/~geer/tables-mathcomp18.pdf.
[3] T. Høholdt, J. van Lint, R. Pellikaan, Algebraic Geometry Codes, in: V.S. Pless, W.C. Huffman (Eds.), Handbook of Coding Theory, vol. 1, Elsevier,
Amsterdam, 1998, pp. 871–961. (Chapter 10).
[4] J. Lewittes, Places of degree one in function fields over finite fields, J. Pure Appl. Algebra 69 (1990) 177–183.
[5] Nivaldo Medeiros, homepage http://w3.impa.br/nivaldo/algebra/semigroups/ (September 26, 2008).
[6] S. Miura, Linear codes on affine algebraic curves, Trans. IEICE J81-A (1998) 1398–1421 (in Japanese).
[7] R. Pellikaan, On the existence of order functions, J. Statist. Plann. Inference 94 (2001) 287–301.
[8] R. Pellikaan, H. Stichtenoth, F. Torres, Weierstrass semigroups in an asymptotically good tower of function fields, Finite Fields Appl. 4 (1998) 381–392.
[9] K.O. Stöhr, J.F. Voloch, Weierstrass points and curves over finite fields, Proc. London Math. Soc. (3) 52 (1986) 1–19.
[10] J. Suzuki, Miura conjecture on affine curves, Osaka J. Math 44 (2007) 187–196.


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