Unirational fields of transcendence degree one and functional decomposition

In this paper we present an algorithm to compute all unirational fields of transcendence degree one containing a given finite set of multivariate rational functions. In particular, we provide an algorithm to decompose a multivariate rational function f of the form f=g(h), where g is a univariate rational function and h a multivariate one.Comment: 17 pages (single column

Computation of unirational fields (extended abstract)

In this paper we present an algorithm for computing all algebraic intermediate subfields in a separably generated unirational field extension (which in particular includes the zero characteristic case). One of the main tools is Groebner bases theory. Our algorithm also requires computing computing primitive elements and factoring over algebraic extensions. Moreover, the method can be extended to finitely generated K-algebras.Comment: 6 pages, 2 images (pictures of authors

Aplicacion de la descomposicion racional univariada a monstrous moonshine (in Spanish)

This paper shows how to use Computational Algebra techniques, namely the decomposition of rational functions in one variable, to explore a certain set of modular functions, called replicable functions, that arise in Monstrous Moonshine. In particular, we have computed all the rational relations with coefficients in Z between pairs of replicable functions. ----- En este articulo mostramos como usar tecnicas de Algebra Computacional, concretamente la descomposcion de funciones racionales univariadas, para estudiar un cierto conjunto de funciones modulares, llamadas funciones replicables, que aparecen en Monstrous Moonshine. En concreto, hemos calculado todas las relaciones racionales con coeficientes en Z entre pares de funciones replicables.Comment: 6 pages, In Spanis

Computing the fixing group of a rational function

Let G=Aut_K (K(x)) be the Galois group of the transcendental degree one pure field extension K(x)/K. In this paper we describe polynomial time algorithms for computing the field Fix(H) fixed by a subgroup H < G and for computing the fixing group G_f of a rational function f in K(x).Comment: 8 page

A recombination algorithm for the decomposition of multivariate rational functions

International audienceIn this paper we show how we can compute in a deterministic way the decomposition of a multivariate rational function with a recombination strategy. The key point of our recombination strategy is the used of Darboux polynomials. We study the complexity of this strategy and we show that this method improves the previous ones. In appendix, we explain how the strategy proposed recently by J. Berthomieu and G. Lecerf for the sparse factorization can be used in the decomposition setting. Then we deduce a decomposition algorithm in the sparse bivariate case and we give its complexit

Komplexe Analysis - Algebraicity and Transcendence (hybrid meeting)

This is the report of the Oberwolfach workshop Komplexe Analysis 2020. It was mainly devoted to the transcendental methods of complex algebraic geometry and featured eighteen talks about recent important developments in Hodge theory, moduli spaces, hyperbolicity, Fano varieties, algebraic foliations, algebraicity theorems for subvarieties and their applications to transcendence proofs for numbers. Two talks were more algebraic in nature and devoted to non-commutative deformations and syzygies of secant varieties

Building counterexamples to generalizations for rational functions of Ritt's decomposition theorem

The classical Ritt's Theorems state several properties of univariate polynomial decomposition. In this paper we present new counterexamples to Ritt's first theorem, which states the equality of length of decomposition chains of a polynomial, in the case of rational functions. Namely, we provide an explicit example of a rational function with coefficients in Q and two decompositions of different length. Another aspect is the use of some techniques that could allow for other counterexamples, namely, relating groups and decompositions and using the fact that the alternating group A_4 has two subgroup chains of different lengths; and we provide more information about the generalizations of another property of polynomial decomposition: the stability of the base field. We also present an algorithm for computing the fixing group of a rational function providing the complexity over Q.Comment: 17 page

Homogeneous spaces and equivariant embeddings

This is a draft of a monograph to appear in the Springer series "Encyclopaedia of Mathematical Sciences", subseries "Invariant Theory and Algebraic Transformation Groups". The subject is homogeneous spaces of algebraic groups and their equivariant embeddings. The style of exposition is intermediate between survey and detailed monograph: some results are supplied with detailed proofs, while the other are cited without proofs but with references to the original papers. The content is briefly as follows. Starting with basic properties of algebraic homogeneous spaces and related objects, such as induced representations, we focus the attention on homogeneous spaces of reductive groups and introduce two important invariants, called complexity and rank. For the embedding theory it is important that homogeneous spaces of small complexity admit a transparent combinatorial description of their equivariant embeddings. We consider the Luna-Vust theory of equivariant embeddings, paying special attention to the case of complexity not greater than one. A special chapter is devoted to spherical varieties (= embeddings of homogeneous spaces of complexity zero), due to their particular importance and ubiquity. A relation between equivariant embedding theory and equivariant symplectic geometry is also discussed. The book contains several classification results (homogeneous spaces of small complexity, etc). The text presented here is not in a final form, and the author will be very grateful to any interested reader for his comments and/or remarks, which may be sent to the author by email.Comment: Monograph-survey, draft version, 250 pages, 219 references, requires AmSLaTeX with style packages `longtable', `verbatim', and Washington cyrillic font

Felix Klein's "About the Solution of the General Equations of Fifth and Sixth Degree (Excerpt from a letter to Mr. K. Hensel)"

This is an English translation of Felix Klein's classical paper "\"Uber die Aufl\"osung der allgemeinen Gleichungen f\"unften und sechsten Grades (Auszug aus einem Schreiben an Herrn K. Hensel)" from 1905 and is put in modern notation. The original work first appeared in the Journal for Pure and Applied Mathematics (Volume 129) and then was reprinted in Mathematische Annalen (Volume 61, Issue 1). Klein's work (including his "Lectures on the Icosahedron and the Solution of Equations of Fifth Degree") lies at the heart of the 19th and 20th work on solving generic polynomials. In this paper, Klein summarizes his approach to solving the generic quintic and sextic polynomials. He also lays the foundation for the modern framework of resolvent degree.Comment: 26 pages, includes a modern bibliography of the referenced works in Appendix

CRM lectures on curves and Jacobians over function fields

These are notes related to a 12-hour course of lectures given at the Centre de Recerca Mathem\`atica near Barcelona in February, 2010. The aim of the course was to explain results on curves and their Jacobians over function fields, with emphasis on the group of rational points of the Jacobian, and to explain various constructions of Jacobians with large Mordell-Weil rank. They may be viewed as a continuation of my Park City notes (arXiv:1101.1939). In those notes, the focus was on elliptic curves and finite constant fields, whereas here we discuss curves of higher genera and results over more general base fields.Comment: v2: small corrections and additions. To appear in "Advanced Courses at the CRM" by Birkhaeuse