About the algebraic closure of the field of power series in several variables in characteristic zero

We construct algebraically closed fields containing an algebraic closure of the field of power series in several variables over a characteristic zero field. Each of these fields depends on the choice of an Abhyankar valuation and are constructed via the Newton-Puiseux method. Then we study more carefully the case of monomial valuations and we give a result generalizing the Abhyankar-Jung Theorem for monic polynomials whose discriminant is weighted homogeneous.Comment: final versio

Conic Optimization Theory: Convexification Techniques and Numerical Algorithms

Optimization is at the core of control theory and appears in several areas of this field, such as optimal control, distributed control, system identification, robust control, state estimation, model predictive control and dynamic programming. The recent advances in various topics of modern optimization have also been revamping the area of machine learning. Motivated by the crucial role of optimization theory in the design, analysis, control and operation of real-world systems, this tutorial paper offers a detailed overview of some major advances in this area, namely conic optimization and its emerging applications. First, we discuss the importance of conic optimization in different areas. Then, we explain seminal results on the design of hierarchies of convex relaxations for a wide range of nonconvex problems. Finally, we study different numerical algorithms for large-scale conic optimization problems.Comment: 18 page

Geometries in perturbative quantum field theory

In perturbative quantum field theory one encounters certain, very specific geometries over the integers. These `perturbative quantum geometries' determine the number contents of the amplitude considered. In the article `Modular forms in quantum field theory' F. Brown and the author report on a first list of perturbative quantum geometries using the `$c_2$-invariant' in $\phi^4$ theory. A main tool was `denominator reduction' which allowed the authors to examine graphs up to loop order (first Betti number) 10. We introduce an improved `quadratic denominator reduction' which makes it possible to extend the previous results to loop order 11 (and partially orders 12 and 13). For comparison, also 'non-$\phi^4$' graphs are investigated. Here, we were able to extend the results from loop order 9 to 10. The new database of 4801 unique $c_2$-invariants (previously 157)---while being consistent with all major $c_2$-conjectures---leads to a more refined picture of perturbative quantum geometries.Comment: 35 page

Groups elementarily equivalent to a free nilpotent group of finite rank

In this paper we give a complete algebraic description of groups elementarily equivalent to a given free nilpotent group of finite rank

Continuity of the Green function in meromorphic families of polynomials

We prove that along any marked point the Green function of a meromorphic family of polynomials parameterized by the punctured unit disk explodes exponentially fast near the origin with a continuous error term.Comment: Modified references. Added a corollary about the adelic metric associated with an algebraic family endowed with a marked poin