Genus computation of global function fields

Abstract

In this paper we present an algorithm that computes the genus of a global function field. Let F/k be function field over a field k, and let k 0 be the full constant field of F/k. By using lattices over subrings of F , we can express the genus g of F in terms of [k 0 : k] and the indices of certain orders of the finite and infinite maximal orders of F . If k is a finite field, the Montes algorithm computes the latter indices as a by-product. This leads us to a fast computation of the genus of global function fields. Our algorithm does not require the computation of any basis, neither of finite nor infinite maximal order.

Residual ideals of MacLane valuations

“…Otherwise, N = S + S ′ is the Newton polygon whose left end point is the vector sum of the two left end points of S and S ′ , and whose sides are the join of S and S ′ , considered with increasing slopes from left to right (see Figure 4). The two first equalities are a consequence of equation (2) and Lemma 2.6. In order to prove the third, let g = 0≤s a s φ s , h = 0≤t b t φ t be the φ-expansions of g, h, respectively.…”

“…Clearly, conditions (2) and (3) are equivalent. Let us show that (1) implies (2). Suppose that g is µ ′ -minimal; write g = G + H, where G is the sum of all monomials a s φ s with µ ′ (a s φ s ) = µ ′ (g) and H is the sum of all…”

“…These operators make the whole theory constructive and well-suited to computational applications. These ideas led to the design of several fast algorithms to perform arithmetic tasks in global fields [2,4,6,7,9,13].…”

Genetics of polynomials over local fields

“…J.-D. Bauch has developed a Magma package +Divisors.m where divisors are manipulated as such pairs D = (I, I ∞ ), and ideals are handled as OM representations. This approach leads to fast OM routines to compute the genus of a curve [1] and the divisor of a function. Also, it yields an acceleration of the classical methods to compute k-bases of the Riemann-Roch spaces of divisors defined over k.…”