A WEIGHTED MODULE VIEW OF INTEGRAL CLOSURES OF AFFINE DOMAINS OF TYPE I

. A type I presentation S = R/J of an aﬃne (order) domain has a weight function injective on the monomials in the footprint ∆( J ). This can be extended naturally to a presentation, R/J , of the integral closure ic ( S ). This presentation over P := F [ x n ,...,x 1 ] as an aﬃne P -algebra relative to a corresponding grevlex-over-weight monomial ordering is shown to have a minimal, reduced Gr¨obner basis (for the ideal of relations J ) consisiting only of P -quadratic relations deﬁning the multiplication of the P -module generators and possibly some P -linear relations if those generators are not independent over P . There then may be better choices for P to minimize the number of P -module generators needed. The intended coding theory application is to the description of one-point AG codes, not only from curves (with P = F [ x 1 ]) but also from higher-dimensional varieties.


Introduction
To properly describe a curve X to be used to define a one-point AG code, it is necessary to put it in special position relative to that one special point P ∞ , with variables corresponding to rational homogeneous functions modulo X , with no poles except possibly at P ∞ . Then the generator and/or parity-check functions come from the vector space L(mP ∞ ) of said functions with pole order at most m, contained in the ring L(∞P ∞ ) of all such functions. The footprint ∆(J) of the affine domain S = R/J of type I defining the curve does not usually define all of L(∞P ∞ ), but the footprint of its integral closure (in its field of fractions) does. So there is a compelling reason to study integral closures in the context of AG coding.
If one views the pole orders as corresponding to the (negatives of the) trailing exponents in the Laurent series expansions in terms of a local parameter t ∞ at P ∞ , then the obvious generalization to n-dimensional surfaces is in terms of the trailing exponent vector α ∈ N n 0 in an expansion involving n independent local parameters. The view in this paper is to start with a polynomial ring P := F[x n , . . . , x 1 ] with a monomial ordering (which can be viewed as a weight function wt P injective on the set of monomials), deal with type I integral extensions S := P [y]/ f (y) so that wt P can be naturally extended to a weight function wt S , which naturally induces a weight function on the integral closure ic(S) of S. While similar to the general theory of order domains in [5], there are two important differences here. The view there is that one starts with S, may or may not care about P , (since the weight function on S may not be a monomial ordering on every choice of P ), and the ∆-set is viewed as an infintie-dimensional vector space over F, not in terms of a finite P -module generating set.
The presentation of ic(S) = R/J desired here is relative to this P -module generating set y 0 := 1, y 1 , . . . , y s−1 , so that J has a minimal, reduced Gröbner basis with elements of the form y i y j − k c i,j,k y k , c i,j,k ∈ P defining the algebra multiplication, and possibly some elements of the form if the P -module generators are not independent over P . (This means that, in general, s is not necessarily the same as the degree of the integral extension.) This is followed by a possible minimization of such a presentation by making a more enlightened choice for P . After all, when n = 1, it is known ( [10]) that there is an F[f ρ ]-module basis, ρ being the smallest positive pole order, with one basis element of smallest positive pole order congruent to i mod ρ for each i. This leaves open the question of what the minimum number of P -module generators is when n > 1, not only because it may not be easy to see whether an example is minimized, but also because there is no longer any definite relation between the degree of the extension and that number even when the problem is given as a minimized integral extension.
But what is missing from the generic presentations of integral closures given by most current implementations is a nice description of the monomials in the footprint. Here the footprint necessarily has monomials all of the form y i x α for y i one of the P -module generators, and x α a monomial of P , whether or not a complete minimization is found. Section 2 will be primarily concerned with notation and the implications of said notation. Section 3 contains as small an example as known by the author in which the P -module generators are not independent. Section 4 gives the theorem summarizing the structure of S and ic(S). Section 5 gives some discussion of the limitations of various implementations. Section 6 contains examples of minimization and a theorem guaranteeing the the structure is not compromised by minimization. Examples here and on the website were done using an implementation of the author's qth-power algorithm [9], [11], written in MAGMA [12] and available on the http://www.dms.auburn.edu/~leonada website. [There are also examples on the website of input and output of current implementations, which may be useful in understanding the following commentary in this section, as well as other points made later on.] Before proceeding, we should note that this approach is definitely not taken by anyone else. Weight functions, fundamental to the study of order domains [5], [8], are virtually unused or actively ignored in the study of integral closures. The result is usually a generic form of integral closure presentation, with a default monomial ordering, having little to do with the original monomial order of the ring. It is perhaps more surprising, however, that there is never any coherent attempt to give a readable presentation for the integral closure, such as the one suggested here, which is, after all, along the lines of structure constant algebras defined by s 3 structure constants (from P ). (It is even more surprising given that current algorithms are based on producing sequences of rings each defined in terms of quadratic and linear relations over the previous ring.) But then again, the prevailing viewpoint, at least for n > 1, is not relative to P , and sometimes not relative to a presentation at all, but merely relative to a set of fractions generating ic(S) over S. At the very least, one might have hoped for a presentation consisting of: • even if one doesn't ask for c ∈ P , (y 0 := 1, y 1 , . . . y s−1 ) to be P -module generators, J to have the particularly nice form above, or a weight function injective on the monomials of the footprint ∆(J ).

Notation
Let F be a field (which here and in the intended applications is either a finite field or the algebraic closure of a finite field) and P := F[x n , . . . , x 1 ] a multivariate polynomial ring over F in n independent variables.
Consider the following adaptation of more standard definitions of weight functions in order domains [5], written in terms of the ring R and ideal J instead of implicitly in terms of the quotient ring S = R/J.
Let A P be a non-singular n × n matrix over N 0 defining the (global) monomial ordering on P (with the default here being the grevlex ordering, x n ≻ · · · ≻ x 1 ), and hence a weight function given by wt P (x α ) := A P α t , with distinct monomials obviously having distinct weights (since A P defines a total order on monomials, or equivalently since it is non-singular). Let be a monic, absolutely irreducible polynomial of degree d. Use it to define an integral extension, the affine domain S := P [y]/J for J := f (y) . To extend wt P to a weight function wt S on S, define wt S (x α ) := d · wt P ((x α ), and wt S (y) := max{ wtS(fi) If that max is taken on at only one value of i, that value is i = 0, LM (f 0 ) := x α , and gcd{d, gcd{α i : 0 ≤ i < d}} = 1, then S is said to be type I. (This terminology probably dates back to [4], but with a more rudimentary definition. The current definiton can be found at least as early as [11].) The standard grevlex order is defined by the s × s (0, 1) matrix G (s) , with G (s) i,j := 1 iff 1 ≤ i + j ≤ s + 1. Regardless of monomial order, N F (g, I) will always mean the normal form of g, meaning the remainder after division by elements of I, LC(g), LM (g), and LT (g) = LC(g)LM (g), the leading coefficient, leading monomial, and leading term of g (relative to the given ordering). SP (f, g) will denote the spolynomial of f and g.
For W ind a non-singular n × n matrix over N 0 , and W dep some n × s matrix over N 0 , we introduce grevlex-over-weight order defined by the matrix and weight-over-grevlex orders defined by the matrix with the former emphasizing the P -module structure, the latter the weights. In either case, define the weight of the monomial y γ x α as W dep γ t +W ind α t . The monomial ordering on the extension S of P above, is given by The following short lemma is included here only because it explains the use of the gcd condition in the definition of type I above. Proof. Suppose, for some 0 ≤ j < i < d, for these two distinct monomials in the footprint ∆(J). Then But because of the gcd condition, this forces either d|i − j or α = β. Example 1. The Klein quartic is most often given in terms of the affine model (with 2 points at infinity) F 2 [y, x]/ y 3 + x 3 y + x , probably because this has nice symmetry in its homogeneuos form. It is already integrally closed. If one were dealing with 2-point codes here, it would be possible to write this as ; While one can try to define a weight function with wt(y) := 3, and wt(x) := 2, both y 2 = y 2 and x 3 = x 3 2 are standard monomials with the same weight, 6. This happened because wt( The one-point description: gotten by using f 5 := yx and f 3 := y, is not integrally closed since  (Note that the common module orderings generally referred to as TOP (Term-Over-Position) and POT (Position-Over-Term), at least in the standard text [1] in the section on Gröbner bases for modules, assume no interplay between module positions and terms in those positions, whereas we have a monomial ordering on a ring viewed as a module. There are many places where block orders are implemented and used, but not so for orderings of the type suggested here, which can only be used in various computational algebra packages by defining the whole matrix.)

Weighted basis theorem
To define the P -linear relation hypothesis, it is necessary to introduce nonstandard notation, to write LM (f ) = x α LM P (f ), to split LM (f ) over P into a "coefficient" x α from P and a "monomial" LM P (f ). (These correspond to a leading coefficient and a leading monomial only if one is working with P as the coefficient ring and allows a weight function to deal with all the variables that occur, whether or not they are variables defined over P or variables in P itself. This is not the view taken currently in any computational algebra packages.) Proof. Since ic(S) is a ring, the minimal, reduced Gröbner basis will have to contain elements of the form w i w j = k c i,j,k w k =: N F (w i w j , J) for some structure constants c i,j,k ∈ P . Any other elements in a Gröbner basis for ic(S) must be P -linear combinations of the w's. Since SP (w i , w j ) = k b i,j,k w k =: N F (SP (w i , w j ), J), when LM P (w i ) = LM P (w j ), the minimal P -linear combinations will be of the form SP (w i , w j ) − N F (SP (w i , w j ), J), with LM P (w i ) = LM P (w j ).

Integral closure algorithms
Integral closure algorithms( [2], [3]) are based on some version of S ⊆ ic(S) ⊆ M = ∆ −1 S for some ∆, not necessarily in P . Most are based on finding an increasing sequence of rings with the elements in R i+1 of the form f i /d i for f i ∈ R i and d 1 · · · d i |∆. The qth power algorithm [11] on the other hand is based on finding a sequence of P -modules (This works best when all the coefficients involved are in F q , since it is then a linear algorithm with major cost in computing the normal forms involved.) But other than choosing ∆ ∈ P it is not so important which type of algorithm is used. What is more important is choosing to find a large enough set of variables, as above, so that the leading terms are all of degree at most 2 in the dependent variables. And secondarily, the monomial order of the base ring should extend to the integral closure. (The algorithm used to produce output for this paper is, however, based on the qth-power algorithm.) [There is an example on the website which is a thinly disguised example of a simple Hermitian curve worked out in MAGMA, SINGULAR, and MACAULAY 2, showing the limitations of each implementation. This example was used to show that there were bugs in the stopping criteria in the latter two implementations.] In the example given, the normal.lib function of Singular . though the last 4 are not given explicitly, and the first 5 are only given explicitly in that the embedding map from S maps z, y 1 , y 2 , x 2 , and x 1 onto T (1), T (2), T (3), T (4), and T (5) respectively. The Gröbner basis elements are an unpredictable mess, since the order is grevlex on (T (1), . . . , T (9)). Our suggested form would be ic(S) = F 2 [z 9 , z 8 , z 7 , z 6 , z 5 , z 4 , z 3 , z 2 , z 1 ; x 2 , x 1 ]/I with weight function given by the weight matrix M := ( 34 34 31 29 26 23 19 12 15 9 9 30 21 21 24 24 15 12 9 9 9 0 ) and order gotten by completing this to a grevlex-over-weight order.
But in either case the weight of the first variable is not really dependent on the weight of the third. So maybe using weights won't be the best way to see that the latter gives a smaller presentation in terms of the F[y, x]-module basis (1, z) than the former, with F[y, z]-module basis (1, x, x 2 , x 3 ).
But there are times when it is possible to use the weights to determine a minimal presentation, including the important case n = 1. Proof.
for some non-negative integers a i and b i , and some positive integers d 1 and d 2 . But then This is clearly a dependence among independent weights, so b n a i + d 1 b i = 0 for 1 ≤ i < n. But this forces b i = 0 for 1 ≤ i < n, and d 2 wt S (x n ) = b n wt S (y).