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A Generalization of Baker's Theorem

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A Generalization of Baker's Theorem View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Finite Fields and Their Applications 15 (2009) 558–568 Contents lists available at ScienceDirect Finite Fields and Their Applications www.elsevier.com/locate/ffa A generalization of Baker’s theorem ∗ Peter Beelen Department of Mathematics, Technical University of Denmark, Matematiktorvet, Building 303S, DK 2800 Kgs. Lyngby, Denmark article info abstract Article history: Baker’s theorem is a theorem giving an upper-bound for the genus Received 12 March 2007 of a plane curve. It can be obtained by studying the Newton- Revised 16 February 2009 polygon of the defining equation of the curve. In this paper we Availableonline28April2009 give a different proof of Baker’s theorem not using Newton-polygon Communicated by Michael Tsfasman theory, but using elementary methods from the theory of function Keywords: fields (Theorem 2.4). Also we state a generalization to several Genus of a curve variables that can be used if a curve is defined by several bivariate Baker’s theorem polynomials that all have one variable in common (Theorem 3.3). Explicit Riemann–Roch space As a side result, we obtain a partial explicit description of certain Castelnuovo’s inequality Riemann–Roch spaces, which is useful for applications in coding theory. We give several examples and compare the bound on the genus we obtain, with the bound obtained from Castelnuovo’s inequality. © 2009 Elsevier Inc. All rights reserved. 1. Introduction When working with a function field F , it is not always a priori clear what the genus of the func- tion field is. This is especially the case if the function field is defined by (a system of) polynomial equations. Of course there are several techniques to calculate the genus in this situation. One can for example compute the singularities of the corresponding curve and compute their desingularization trees using blow-ups. Another method is to describe the function field as a finite separable cover of a simpler function field with known genus and use the Riemann–Hurwitz–Zeuthen theorem. All these methods are easily applicable when the function field is explicitly defined by equations, but the cal- culations involved can get quite messy. Therefore it can be very useful to have a priori upper bounds on the genus that are easy to calculate. This is in particular useful when one is looking for function fields defined over a finite field with many rational places [10]. In certain constructions, one obtains a family of absolutely irreducible polynomial equations having many solutions over a finite field [3,14], * Fax: +45 452881399. E-mail address: [email protected]. 1071-5797/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.ffa.2009.04.003 P. Beelen / Finite Fields and Their Applications 15 (2009) 558–568 559 but then the main difficulty is to determine the behaviour of the genus of the function fields defined by this family. A well-known example of such an upper bound is the Plücker bound, which states that the genus of a plane algebraic curve given by a polynomial equation of degree d cannot exceed (d − 1)(d − 2)/2. There exists a less well-known generalization of Plücker’s bound, namely Baker’s theorem, which states that the genus of a plane curve cannot exceed the number of interior integral points lying in the Newton polygon of its defining equation (see Theorem 2.4). This theorem has been around for over a hundred years, but is not very well known. The theorem was stated for the first time in [1] and was generalized in [16]. For both articles it is assumed that the field of definition of X is the field C of complex numbers. Baker’s theorem was proved in [17] for any characteristic using cohomological methods on toric varieties, and in [3] using blow-ups and desingularization methods. The first goal of this paper is to give a completely elementary proof of Baker’s theorem in the language of function fields. All necessary prerequisites can be found in [20]. This will hopefully make Baker’s theorem accessible for a larger audience than before. As a bonus, we can construct a large part of certain Riemann–Roch spaces, a fact which is useful when one wishes to construct algebraic geometric codes (see [11,13]). The second goal of this paper is to state and prove a generalization of Baker’s theorem, not unlike the one in [16], though applicable to a different situation. It is valid for any characteristic. The proof of this generalization is still elementary and stated in the language of function fields. The generalization applies to the situation when a function field is defined by a certain system of polynomial equations. This is useful, since many curves defined over a finite field with many rational points are defined by fiber products of suitable curves over a projective line (see for example [5,9,15,19]). It will turn out that in some cases this generalization gives a better upper bound for the genus than Castelnuovo’s inequality. 2. Baker’s theorem In this section we will state and prove Baker’s theorem using elementary tools from the theory of function fields. Overall in this paper, we denote by F a field and by F its algebraic closure. We impose no restriction on F. Given a polynomial F (X, Y ) ∈ F[X, Y ], we say that F (X, Y ) is absolutely irreducible, if F (X, Y ) is irreducible as an element of F[X, Y ]. Given an absolutely irreducible poly- nomial F (X, Y ),wedenotebyF(x, y) the function field with constant field F and defining relation F (x, y) = 0. We also say that F(x, y) is the function field defined by F (X, Y ). = i j F Definition 2.1. Let F (X, Y ) (i, j)∈A αi, j X Y be a bivariate polynomial over the field ,with 2 A ⊂ N some finite index set such that αi, j ∈ F \{0} for all (i, j) ∈ A.WedefinetheNewtonpolygon (F ) to be the convex hull of the set A. Givensuchapolygon(F ), we mean by an interior integral point of (F ) an element of the set (F ) ∩ Z2 lying in the interior of (F ) (using the conventional topology of R2). Knowledge of the boundary of (F ) suffices in general to calculate the number of interior integral points as we will see in the next lemma. ∈ F[ ] = Lemma 2.2. Let F (X, Y ) X, Y be a bivariate polynomial such that degX F a and such that neither X nor Y divides F (X, Y ). Further suppose that (F ) is not a line segment. For 0 r adefine αr := inf b (r, b) ∈ (F ) and βr := sup b (r, b) ∈ (F ) . 560 P. Beelen / Finite Fields and Their Applications 15 (2009) 558–568 Then the number of interior integral points of (F ) equals: a−1 βr −αr −1 . r=1 Proof. This follows by noting that the expression βr −αr −1 is equal to the number of interior integral points of (F ) with first coordinate equal to r. 2 The condition that F (X, Y ) cannot be divided by X nor Y is not essential, since multiplication by X or Y only gives rise to a translation of the Newton polygon. The condition that (F ) is not a line segment can be weakened. First of all note that (F ) is a line segment if and only if there exist relative prime numbers n and m and a polynomial f (T ) such that F (X, Y ) = Y m deg f f (Xn/Y m). Clearly, the polynomial F (X, Y ) is absolutely irreducible if and only if f (T ) is absolutely irreducible. This can only happen if deg f = 1. Since n and m are relatively prime, this means that the only integral points on (F ) are the vertices (n, 0) and (0,m). Now note that the only case where the expression βr −αr −1(with1 r a − 1) does not count the number of interior integral points with first coordinate r, is when αr = βr and αr ,βr ∈ Z. This can only happen when (F ) is a line segment which has more integral points one it than just its vertices. Thus by the above reasoning we see the polynomial F (X, Y ) is reducible in this case. All this shows that the above lemma holds for any absolutely irreducible polynomial F (X, Y ). Before proceeding to Baker’s theorem, we give a constructive lemma. ∈ F[ ] = Lemma 2.3.Let F (X, Y ) X, Y be an absolutely irreducible polynomial with degX F a. We write = a s F F (X, Y ) s=0 ps(Y )X ,wheretheps(y) are suitable polynomials, and denote by (x, y) the function field defined by F (X, Y ).Alsowedefinefor2 r a the function a s−r fr := x ps(y)x . s=r Then we have the following inequality of divisors: ( fr ) αr−1(y)0 − βr−1(y)∞, with 2 r aandαr−1 and βr−1 as in the previous lemma. Proof. Given a function f ∈ F(x, y),wedenoteby( f ), the divisor of f and by ( f )0 (resp. ( f )∞)the zero-divisor (resp. pole-divisor) of f .First,wenotethatbythedefinitionofαs and βs and the strict triangle inequality, we have ps(y) αs(y)0 − βs(y)∞. (1) We will prove the lemma with descending induction on r. In order to do this, we will use the following equations, which follow from the convexity of (F ) and the definition of αr and βr : αs + (r − s − 1)αr (r − s)αr−1 (2) and βs + (r − s − 1)βr (r − s)βr−1.
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