The intersection of valuation rings
The intersection theorem for orderings of higher level in rings.
The inverse Galois problem and rational points on moduli spaces.
The Joly–Becker theorem for –orderings
We prove the –version of the Joly–Becker theorem: a skew field admits a –ordering of level iff it admits a –ordering of level for some (resp. all) odd . For skew fields with an imaginary unit and fields stronger results are given: a skew field with imaginary unit that admits a –ordering of higher level also admits a –ordering of level . Every field that admits a –ordering of higher level admits a –ordering of level or
The Lamé family of connections on the projective line
This paper deals with rank two connections on the projective line having four simple poles with prescribed local exponents 1/4 and . This Lamé family of connections has been extensively studied in the literature. The differential Galois group of a Lamé connection is never maximal : it is either dihedral (finite or infinite) or reducible. We provide an explicit moduli space of those connections having a free underlying vector bundle and compute the algebraic locus of those reducible connections....
The least admissible value of the parameter in Hilbert's Irreducibility Theorem
The Lehmer constants of an annulus
Let be the Mahler measure of an algebraic number , and be an open subset of . Then its Lehmer constant is inf , the infimum being over all non-zero non-cyclotomic lying with its conjugates outside . We evaluate when is any annulus centered at . We do the same for a variant of , which we call the transfinite Lehmer constant .Also, we prove the converse to Langevin’s Theorem, which states that if contains a point of modulus . We prove the corresponding result for .
The level of cyclic division algebras.
The limit behaviour of exponential terms
The local lifting problem for actions of finite groups on curves
Let be an algebraically closed field of characteristic . We study obstructions to lifting to characteristic the faithful continuous action of a finite group on . To each such a theorem of Katz and Gabber associates an action of on a smooth projective curve over . We say that the KGB obstruction of vanishes if acts on a smooth projective curve in characteristic in such a way that and have the same genus for all subgroups . We determine for which the KGB obstruction...
The logic of Rumely's local-global principle.
The Lojasiewicz exponent at infinity for overdetermined polynomial mappings
We prove that the study of the Łojasiewicz exponent at infinity of overdetermined polynomial mappings , m > n, can be reduced to the one when m = n.
The maximum number of real roots of a multihomogeneous system of polynomial equations.
The model theory of generalized real closed fields.
The moduli space of n tropically collinear points in Rd.
The tropical semiring (R, min, +) has enjoyed a recent renaissance, owing to its connections to mathematical biology as well as optimization and algebraic geometry. In this paper, we investigate the space of labeled n-point configurations lying on a tropical line in d-space, which is interpretable as the space of n-species phylogenetic trees. This is equivalent to the space of n x d matrices of tropical rank two, a simplicial complex. We prove that this simplicial complex is shellable for dimension...
The Newton polygon of a product of power series.
The Noether-Lefschetz theorem and sums of 4 squares in the rational function field
The non-archimedean Corona problem
The norm of uniform convergence on the -algebraic maximal spectrum of an algebra over a non-archimedean valuation field
The number of zeros of polynomials in valuation rings of complete discretely valued fields