Valuations in algebraic field extensions

Let K be a field of characteristic p with q elements and FE in L[X,Y] be a polynomial with p> deg_Y(F) and total degree d. In [40], we showed that rational Puiseux series of F above X=0 could be computed with an expected number of O~(d5+d3log q) arithmetic operations in L. In this paper, we reduce this bound to O~(d4+d2log q) using Hensel lifting and changes of variables in the Newton-Puiseux algorithm that give a better control of the number of steps. The only asymptotically fast algorithm required is polynomial multiplication over finite fields. This approach also allows to test the irreducibility of F in L[[X]][Y] with O(d3) operations in K. Finally, we describe a method based on structured bivariate multiplication [34] that may speed up computations for some input.

/pdf/improving-complexity-bounds-for-the-computation-of-puiseux-443yn7zv9w.pdf

Improving Complexity Bounds for the Computation of Puiseux Series over Finite Fields

This paper is a step in our program for proving the Piece-Birkhoff Conjecture for regular rings of any dimension (this would contain, in particular, the classical Pierce-Birkhoff conjecture which deals with polynomial rings over a real closed field). We first recall the Connectedness and the Definable Connectedness conjectures, both of which imply the Pierce - Birkhoff conjecture. Then we introduce the notion of a system of approximate roots of a valuation v on a ring A (that is, a collection Q of elements of A such that every v-ideal is generated by products of elements of Q). We use approximate roots to give explicit formulae for sets in the real spectrum of A which we strongly believe to satisfy the conclusion of the Definable Connectedness conjecture. We prove this claim in the special case of dimension 2. This proves the Pierce-Birkhoff conjecture for arbitrary regular 2-dimensional rings.

/pdf/approximate-roots-of-a-valuation-and-the-pierce-birkhoff-3kalhr6o66.pdf

Approximate roots of a valuation and the Pierce-Birkhoff Conjecture

The problem of resolution of singularities and its solution in various contexts can be traced back to I. Newton and B. Riemann. This paper is an attempt to give a survey of the subject starting with Newton till the modern times, as well as to discuss some of the main open problems that remain to be solved. The main topics covered are the early days of resolution (fields of characteristic zero and dimension up to three), Zariski’s approach via valuations, Hironaka’s celebrated result in characteristic zero and all dimensions and its subsequent strengthenings and simplifications, existing results in positive characteristic (mostly up to dimension three), de Jong’s approach via semi-stable reduction, Nash and higher Nash blowing up, as well as reduction of singularities of vector fields and foliations. In many places, we have tried to summarize the main ideas of proofs of various results without getting too much into technical details.

/pdf/resolution-of-singularities-an-introduction-1kyojetumq.pdf

Resolution of Singularities: An Introduction

The purpose of this paper is to give an algorithm for deciding the irreducibility of reduced algebroid curves over an algebraically closed field of arbitrary characteristic. To do this, we introduce a new notion of local tropical variety which is a straightforward extension of tropism introduced by Maurer, and then give irreducibility criterion for algebroid curves in terms local tropical varieties. We also give an algorithm for computing the value-semigroups of irreducible algebroid curves. Combining the irreducibility criterion and the algorithm for computing the value-semigroups, we obtain an algorithm for deciding the irreducibility of algebroid curves.

/pdf/irreducibility-criterion-for-algebroid-curves-4kyqo34dyo.pdf

Irreducibility criterion for algebroid curves

The well-known Tschirnhausen transformation, , eliminates the second term of the polynomial xn + axn-l + …. By a mere repeated application of this transformation, one can decide whether a given element of k[[x,y]] is prime (irreducible) or not. Here K is an algebraically closed field of characteristic 0. A generalised version of Hensel's Lemma is developed for the proofs. The entire paper can be understood by undergraduate students.

/pdf/a-simple-algorithm-for-deciding-primes-in-k-x-y-51anc2xaei.pdf

A Simple Algorithm for Deciding Primes in K[[x,y]]

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A Simple Algorithm for Deciding Primes in K[[x,y]]