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## Publications

Publications (85)

For an arbitrary valued field (K,v) and a given extension v(K⁎)↪Λ of ordered groups, we analyze the structure of the tree formed by all Λ-valued extensions of v to the polynomial ring K[x]. As an application, we find a model for the tree of all equivalence classes of valuations on K[x] (without fixing their value group), whose restriction to K is e...

We extend and prove a conjecture of Bengu\c{s}-Lasnier on the parametrization of valuations on a polynomial ring by certain spaces of diskoids.

In this paper we present a characterization for the defect of a simple algebraic extensions of valued fields. This characterization generalizes the known result for the henselian case, namely that the defect is the product of the relative degrees of limit augmentations. The main tool used here is the graded algebra associated to a valuation on a po...

Given a valued field $(K,v)$ and an irreducible polynomial $g\in K[x]$, we survey the ideas of Ore, Maclane, Okutsu, Montes, Vaqui\'e and Herrera-Olalla-Mahboub-Spivakovsky, leading (under certain conditions) to an algorithm to find the factorization of $g$ over a henselization of $(K,v)$.

For a certain field $K$, we construct a valuation-algebraic valuation on the polynomial ring $K[x]$, whose Maclane--Vaqui\'e chain consists of an infinite (countable) number of limit augmentations

We classify cuts in (totally) ordered abelian groups Γ and compute the coinitiality and cofinality of all cuts in case Γ is divisible, in terms of data intrinsically associated to the invariance group of the cut. We relate cuts with small extensions of Γ in a natural way, which leads to an explicit construction of a totally ordered real vector spac...

For a prime $p$, the OM algorithm finds the $p$-adic factorization of an irreducible polynomial $f\in\mathbb{Z}[x]$ in polynomial time. This may be applied to construct $p$-integral bases in the number field $K$ defined by $f$. In this paper, we adapt the OM techniques to work with a positive integer $N$ instead of $p$. As an application, we obtain...

Let $(K,v)$ be a valued field and let $(K^h,v^h)$ be the henselization determined by the choice of an extension of $v$ to an algebraic closure of $K$. Consider an embedding $v(K^*)\hookrightarrow\Lambda$ of the value group into a divisible ordered abelian group. Let $T(K,\Lambda)$, $T(K^h,\Lambda)$ be the trees formed by all $\Lambda$-valued extens...

For a henselian valued field $(K,v)$ we establish a complete parallelism between the arithmetic properties of irreducible polynomials $F\in K[x]$, encoded by their Okutsu frames, and the valuation-theoretic properties of their induced valuations $v_F$ on $K[x]$, encoded by their MacLane-Vaqui\'e chains. This parallelism was only known for defectles...

We classify cuts in (totally) ordered abelian groups $\g$ and compute the coinitiality and cofinality of all cuts in case $\g$ is divisible, in terms of data intrinsically associated to the invariance group of the cut. We relate cuts with small extensions of $\g$ in a natural way, which leads to an explicit construction of a totally ordered real ve...

For an arbitrary valued field $(K,v)$ and a given extension $v(K^*)\hookrightarrow\Lambda$ of ordered groups, we analyze the structure of the tree formed by all $\Lambda$-valued extensions of $v$ to the polynomial ring $K[x]$. As an application, we find a model for the tree of all equivalence classes of valuations on $K[x]$ (without fixing their va...

Let (K,v) be a valued field, and μ an inductive valuation on K[x] extending v. Let Gμ be the graded algebra of μ over K[x], and κ the maximal subfield of the subring of Gμ formed by the homogeneous elements of degree zero.
In this paper, we find an algorithm to compute the field κ and the residual polynomial operator Rμ:K[x]→κ[y], where y is anothe...

Let $\Gamma$ be a totally ordered group. We use Hahn's embedding theorem to construct a totally ordered set $\Gamma\subset \Gamma_{\operatorname{sme}}$ which classifies small extensions of $\Gamma$. This small-extensions closure $\Gamma_{\operatorname{sme}}$ is complete and plays a crucial role in the description of equivalence classes of valuation...

Let $\nu$ be a valuation of arbitrary rank on the polynomial ring $K[x]$ with coefficients in a field $K$. We prove comparison theorems between MacLane-Vaqui\'e key polynomials for valuations $\mu\le\nu$ and abstract key polynomials for $\nu$. Also, some results on invariants attached to limit key polynomials are obtained. In particular, if $\opera...

Let $(K,v)$ be a valued field. We review some results of MacLane and Vaqui\'e on extensions of $v$ to valuations on the polynomial ring $K[x]$. We introduce certain MacLane-Vaqui\'e chains of residually transcendental valuations, and we prove that every valuation $\mu$ on $K[x]$ is a limit of a finite or countably infinite MacLane-Vaqui\'e chain. T...

Let (K,v) be a henselian valued field. Let Pdless⊂K[x] be the set of monic, irreducible polynomials which are defectless and have degree greater than one. For a certain equivalence relation ≈ on Pdless, we establish a canonical bijection M→Pdless/≈, where M is a discrete MacLane space, constructed in terms of inductive valuations on K[x] extending...

Let $(K,v)$ be a henselian valued field. Let $\mathbb{P}^{dless}\subset K[x]$ be the set of monic, irreducible polynomials which are defectless and have degree greater than one. For a certain equivalence relation $\,\approx\,$ on $\,\mathbb{P}^{dless}$, we establish a canonical bijection $\mathbb{M}\to \mathbb{P}^{dless}/\!\!\approx$, where $\mathb...

Let $(K,v)$ be a valued field, and $\mu$ an inductive valuation on $K[x]$ extending $v$. Let $G_\mu$ be the graded algebra of $\mu$ over $K[x]$, and $\kappa$ the maximal subfield of the subring of $G_\mu$ formed by the homogeneous elements of degree zero. In this paper, we find an algorithm to compute the field $\kappa$ and the residual polynomial...

Let K be a field. For a given valuation on K[x], we determine the structure of its graded algebra and describe its set of key polynomials, in terms of any given key polynomial of minimal degree. We also characterize valuations not admitting key polynomials.

We discuss Weber’s formula which gives the quotient of two Thetanullwerte for a plane smooth quartic in terms of the bitangents. In particular, we show how it can easily be derived from the Riemann-Jacobi formula.

We introduce a canonical form for reduced bases of integral closures of discrete valuation rings, and we describe an algorithm for computing a basis in reduced normal form. This normal form has the same applications as the Hermite normal form: identification of isomorphic objects, construction of global bases by patching local ones, etc. but in add...

We adapt an old local-to-global technique of Ore to compute, under certain
mild assumptions, an integral basis of a number field without a previous
factorization of the discriminant of the defining polynomial. In a first phase,
the method yields as a by-product successive splittings of the discriminant.
When this phase concludes, it requires a squa...

Let $K$ be a field equipped with a discrete valuation $v$. In a pioneering
work, S. MacLane determined all valuations on $K(x)$ extending $v$. His work
was recently reviewed and generalized by M. Vaqui\'e, by using the graded
algebra of a valuation. We extend Vaqui\'e's approach by studying residual
ideals of the graded algebra of a valuation as an...

We discuss Weber's formula which gives the quotient of two Thetanullwerte for
a plane smooth quartic in terms of the bitangents. In particular, we show how
it can easily be derived from the Riemann-Jacobi formula.

Let $A$ be a Dedekind domain whose field of fractions $K$ is a global field. Let $p$ be a non-zero prime ideal of $A$, and $K_p$ the completion of $K$ at $p$. The Montes algorithm factorizes a monic irreducible separable polynomial $f(x)\in A[x]$ over $K_p$, and it provides essential arithmetic information about the finite extensions of $K_p$ deter...

Types over a discrete valued field $(K,v)$ are computational objects that
parameterize certain families of monic irreducible polynomials in $K_v[x]$,
where $K_v$ is the completion of $K$ at $v$. Two types are considered to be
equivalent if they encode the same family of prime polynomials. In this paper,
we characterize the equivalence of types in t...

Let $K$ be the number field determined by a monic irreducible polynomial $f(x)$ with integer coefficients. In previous papers we parameterized the prime ideals of $K$ in terms of certain invariants attached to Newton polygons of higher order of the defining equation $f(x)$. In this paper we show how to carry out the basic operations on fractional i...

Let $(K,v)$ be a discrete valued field with valuation ring $\oo$, and let
$\oo_v$ be the completion of $\oo$ with respect to the $v$-adic topology. In
this paper we discuss the advantages of manipulating polynomials in $\oo_v[x]$
in a computer by means of OM representations of prime (monic and irreducible)
polynomials. An OM representation supports...

Let p be a prime number. In this paper we use an old technique of Ore, based on Newton polygons, to construct in an efficient way p-integral bases of number fields defined by a p-regular equation. To illustrate the potential applications of this construction, we show how this result yields a computation of a p-integral basis of an arbitrary quartic...

We obtain several results on the computation of different and discriminant
ideals of finite extensions of local fields. As an application, we deduce
routines to compute the $\p$-adic valuation of the discriminant $\dsc(f)$, and
the resultant $\res(f,g)$, for polynomials $f(x),g(x)\in A[x]$, where $A$ is a
Dedekind domain and $\p$ is a non-zero prim...

Let $k$ be a locally compact complete field with respect to a discrete
valuation $v$. Let $\oo$ be the valuation ring, $\m$ the maximal ideal and
$F(x)\in\oo[x]$ a monic separable polynomial of degree $n$. Let
$\delta=v(\dsc(F))$. The Montes algorithm computes an OM factorization of $F$.
The single-factor lifting algorithm derives from this data a...

We develop a theory of arithmetic Newton polygons of higher order, that provides the factorization of a separable polynomial over a $p$-adic field, together with relevant arithmetic information about the fields generated by the irreducible factors. This carries out a program suggested by \O{}. Ore. As an application, we obtain fast algorithms to co...

This is a survey on Okutsu-Montes representations of prime ideals
of certain one-dimensional integral closures. These representa-
tions facilitate the computational resolution of several arithmetic
tasks concerning prime ideals of global fields.

Let $f(x)$ be a separable polynomial over a local field. Montes algorithm
computes certain approximations to the different irreducible factors of $f(x)$,
with strong arithmetic properties. In this paper we develop an algorithm to
improve any one of these approximations, till a prescribed precision is
attained. The most natural application of this "...

We present an algorithm for computing discriminants and prime ideal decomposition in number fields. The algorithm is a refinement of a p-adic factorization method based on Newton polygons of higher order. The running-time and memory requirements of the algorithm appear to be very good: for a given prime number p, it computes the p-valuation of the...

We introduce our package '+Ideals' for Magma, designed to perform the basic tasks related to ideals in number fields without pre-computing integral bases. It is based on Montes algorithm and a number of local techniques that we have developed in a series of papers in the last years.

Let K be a local field of characteristic zero, O its ring of integers and F(x) a monic irreducible polynomial with coefficients in O. K. Okutsu attached to F(x) certain primitive divisor polynomials F_1(x),..., F_r(x), that are specially close to F(x) with respect to their degree. In this paper we characterize the Okutsu families [F_1,..., F_r] in...

Let k=F_q be a finite field of characteristic 2. A genus 3 curve C/k has many involutions if the group of k-automorphisms admits a C_2\times C_2 subgroup H (not containing the hyperelliptic involution if C is hyperelliptic). Then C is an Artin-Schreier cover of the three elliptic curves obtained as the quotient of C by the nontrivial involutions of...

We determine the isogeny classes of supersingular abelian threefolds over Fn2 containing the Jacobian of a genus 3 curve. In particular, we prove that for even n>6 there always exist a maximal and a minimal curves of genus 3 over Fn2. The methods provide an explicit construction of supersingular curves of genus 3 with Jacobian in a prescribed isoge...

Let k=Fq be a finite field. We enumerate k-rational n-sets of (unordered) points in a projective space PN over k, and we compute the generating function for the numbers of PGLN+1(k)-orbits of these n-sets. For N=1,2 we obtain a formula for these numbers of orbits as a polynomial in q with integer coefficients.

Let be a finite field of odd characteristic. We find a closed formula for the number of k-isomorphism classes of pointed, and non-pointed, hyperelliptic curves of genus g over k, admitting a Koblitz model. These numbers are expressed as a polynomial in q with integer coefficients (for pointed curves) and rational coefficients (for non-pointed curve...

We compute in a direct (not algorithmic) way the zeta function of all supersingular curves of genus 2 over a finite field k, with many geometric automorphisms. We display these computations in an appendix where we select a family of representatives of all these curves up to \({\overline{k}}\)-isomorphism and we exhibit equations and the zeta functi...

We find a closed formula for the number hyp(g) of hyperelliptic curves of genus g over a finite field k=Fq of odd characteristic. These numbers hyp(g) are expressed as a polynomial in q with integer coefficients that depend on g and the set of divisors of q−1 and q+1. As a by-product we obtain a closed formula for the number of self-dual curves of...

We give a complete answer to the question of which polynomials occur as the characteristic polynomials of Frobenius for genus-2 curves over finite fields. Comment: LaTeX, 38 pages. Intermediate results in Section 13 have been generalized. Minor changes elsewhere

Let A be an isogeny class of abelian surfaces over F_q with Weil polynomial x^4 + ax^3 + bx^2 + aqx + q^2. We show that A does not contain a surface that has a principal polarization if and only if a^2 - b = q and b < 0 and all prime divisors of b are congruent to 1 modulo 3. We use this result in a forthcoming paper in which we determine which iso...

Let k be a finite field of even characteristic. We obtain in this paper a complete classification, up to k-isomorphism, of non-singular quartic plane curves defined over k. We find explicit rational models and closed formulas for the total number of k-isomorphism classes. We deduce from these computations the number of k-rational points of the diff...

We determine what isogeny classes of supersingular abelian surfaces over a finite field k of characteristic 2 contain jacobians. We deal with this problem in a direct way by computing explicitly the zeta function of all supersingular curves of genus 2. Our procedure is constructive, so that we are able to exhibit curves with prescribed zeta functio...

In this paper we classify hyperelliptic curves of genus 3 defined over a finite field k of even characteristic. We consider rational models representing all k-isomorphy classes of curves with a given arithmetic structure for the ramification divisor and we find necessary and sufficient conditions for two models of the same type to be k-isomorphic....

In this paper we classify curves of genus two over a perfect field k of characteristic two. We find rational models of curves with a given arithmetic structure for the ramification divisor and we give necessary and sufficient conditions for two models of the same type to be k-isomorphic. As a consequence, we obtain an explicit formula for the numbe...

For any finite field k we count the number of orbits of galois invariant n-sets of P^1(k@?) under the action of PGL"2(k). For k of odd characteristic, this counts the number of k-points of the moduli space of hyperelliptic curves of genus g over k. We get in this way an explicit formula for the number of hyperelliptic curves over k of genus g, up t...

For any finite field {\small $k=\fq$}, we explicitly describe the k-isogeny classes of abelian surfaces defined over k and their behavior under finite field extension. In particular, we determine the absolutely simple abelian surfaces. Then, we analyze numerically what surfaces are k-isogenous to the Jacobian of a smooth projective curve of genus 2...

For any finite field k = Fq, we explicitly describe the k-isogeny classes of abelian surfaces defined over k and their behavior under finite field extension. In particular, we determine the absolutely simple abelian surfaces. Then, we analyze numerically what surfaces are k-isogenous to the Jacobian of a smooth projective curve of genus 2 defined o...

Consider the natural action of PGL3(q) on the projective plane PG2(q) over a finite field GF(q). In this paper we split a set of representatives of conjugacy classes of PGL3(q) into a disjoint union of subfamilies gathering the elements that have the same cycle type as a permutation of the points of PG2(q). Also, we count the number of elements in...

Ø. Ore (Math. Ann. 99, 1928, 84-117) developed a method for obtaining the absolute discriminant and the prime-ideal decomposition of the rational primes in a number field $K$. The method, based on Newton's polygon techniques, worked only when certain polynomials $f_S(Y)$, attached to any side $S$ of the polygon, had no multiple factors. These resul...

For any prime number p > 3 we compute the formal completion of the Neron model of J0(p) in terms of the action of the Hecke algebra on the Z-module of all cusp forms (of weight 2 with respect to G0(p)) with integral Fourier development at infinity.

For any prime number p > 3 we compute the formal completion of the Néron model of J0(p) in terms of the action of the Hecke algebra on the Z-module of all cusp forms (of weight 2 with respect to G0(p)) with integral Fourier development at infinity.

Arithmetic invariants are found which determine the index i(K) of a number field K. They are used to obtain an explicit formula under certain restrictions on K. They provide also a complete explanation of a phenomenon conjectured by Ore [8] and showed by Engstrom in a particular case [2].

The decomposition of the rational primes in a cubic field K is determined in terms of the coefficients of a defining polynomial of K. As a consequence, the discriminant D of K is straightforwardly computed and the cubic fields with index i)K) = 2 are easily characterized.

Criteria are given for polynomials of the type Xn + aX3 + bX2 + cX + d, to have Galois group over any finite number field isomorphic to An. We use them to construct, for every n, infinitely many polynomials with absolute Galois group isomorphic to An, covering so, the case n even, 4 ∤ n, for which explicit equations were not known.