Symbolic Hamburger-Noether expressions of plane curves and applications to AG codes

Abstract

Abstract. In this paper, we consider some practical applications of the symbolic Hamburger-Noether expressions for plane curves, which are introduced as a symbolic version of the so-called Hamburger-Noether expansions. More precisely, we give and develop in symbolic terms algorithms to compute the resolution tree of a plane curve (and the adjunction divisor, in particular), rational parametrizations for the branches of such a curve, special adjoints with assigned conditions (connected with different problems, like the so-called Brill-Noether algorithm), and the Weierstrass semigroup at P together with functions for each value in this semigroup, provided P is a rational branch of a singular plane model for the curve. Some other computational problems related to algebraic curves over perfect fields can be treated symbolically by means of such expressions, but we deal just with those connected with the effective construction and decoding of algebraic geometry codes.

δ-sequences and evaluation codes defined by plane valuations at infinity

“…Thinking of blowing-up centers, we shall say that a center p i is proximate to other p j whenever p i is on any strict transform of the divisor created after blowing-up at p j . The complete information of the resolution process can be described by means of the so-called Hamburger-Noether expansion (HNE for short) [7,9], being this expansion specially useful when the characteristic of the field k is positive. More explicitly, let {u ′ , v ′ } be local coordinates of the local ring O C,p ; in those coordinates the HNE of C at p has the form…”

Evaluation codes and plane valuations

“…The parametric equations of algorithm A2 will allow us to explicitly compute the order functions of the above Proposition. The last row of the Hamburger-Noether expansions of valuations of type B can be infinite, however we can consider the so-called symbolic Hamburger-Noether expansion for them (see [2]). This expansion contains as a last row an implicit equation in w s g and w s g−1 , which allows us to explicitly obtain as many elements of this last row as we want.…”

“…The input of our algorithm A2 is what we call the symbolic Hamburger-Noether expansion of a plane valuation ν. When ν is a valuation of type B, we consider an algebraic curve given by f ∈ R such that it is analytically irreducible and compute its symbolic Hamburger-Noether expansion as in [2] which will be the same of ν. That algorithm allows us to decide if a polynomial corresponds to an analytically irreducible curve.…”

“…The function o(= −ν) defined over the k-algebra gr o T, T := k[{q −1 i } i∈I ] ⊆ K, is a weight function whose value semigroup is S, the value semigroup of ν (2). The graded algebra associated with ν (relative to R) and that associated with o are isomorphic and both are isomorphic to the k-algebra of the semigroup S, k[S].…”

“…To do it, we must compute the Hamburger-Noether expansion of the branches of f and g (the algorithm in [2] computes it) and choose the Hamburger-Noether expansion of a divisorial valuation ν i converging to ν (gotten by a piece of the Hamburger-Noether expansion of ν) which have the largest coincidence in values a ij , h j , k j , s i to the branches of f and g (always using the same basis). Then o(f /g) = −ν(f /g) = −ν i (f /g),ν i being the normalization of the valuation ν i .…”