Related papers: On Freiman's Theorem in a function field setting

We show that the Fr\"oberg conjecture holds in the second non-trivial degree for an ideal generated by generic forms of degree $d>2$. We also show that the conjecture is true up to degree $2d-1$ provided that the number of variables is…

For schemes X over global or local fields, or over their rings of integers, K. Kato stated several conjectures on certain complexes of Gersten-Bloch-Ogus type, generalizing the fundamental exact sequence of Brauer groups for a global field.…

Using geometric methods, we improve on the function field version of the Burgess bound, and show that, when restricted to certain special subspaces, the M\"{o}bius function over $\mathbb F_q[T]$ can be mimicked by Dirichlet characters.…

Floer field theory is a construction principle for e.g. 3-manifold invariants via decomposition in a bordism category and a functor to the symplectic category, and is conjectured to have natural 4-dimensional extensions. This survey…

In this paper, we give a new proof and an extension of the following result of B\'ezivin. Let $f:\B{N}\to K$ be a multiplicative function taking values in a field $K$ of characteristic 0 and write $F(z)=\sum_{n\geq 1} f(n)z^n\in K[[z]]$ for…

Let $\Phi$ : R n $\rightarrow$ R $\cup$ {+$\infty$} be an even convex function and L$\Phi$ be its Legendre transform. We prove the functional form of Mahler conjecture concerning the functional volume product P ($\Phi$) = e --$\Phi$ e…

We prove that the space $M(K(x,y))$ of $\mathbb R$-places of the field $K(x,y)$ of rational functions of two variables with coefficients in a totally Archimedean field $K$ has covering and integral dimensions $\dim M(K(x,y))=\dim_\IZ…

In this paper we give a counterexample to the conjecture: Let $S\in{\rm SInn}$. Then $z\cdot S$ is onto $U$.

This article contains a proof of the MDS conjecture for $k \leq 2p-2$. That is, that if $S$ is a set of vectors of ${\mathbb F}_q^k$ in which every subset of $S$ of size $k$ is a basis, where $q=p^h$, $p$ is prime and $q$ is not and $k \leq…

In 1994, Becker conjectured that if $F(z)$ is a $k$-regular power series, then there exists a $k$-regular rational function $R(z)$ such that $F(z)/R(z)$ satisfies a Mahler-type functional equation with polynomial coefficients where the…

We prove the arithmetic fundamental lemma conjecture over a general $p$-adic field with odd residue cardinality $q\geq \dim V$. Our strategy is similar to the one used by the second author during his proof of the AFL over $\mathbb{Q}_p$…

Let us consider a generalized Artin-Schreier algebraic function field extension $F$ of the rational function field $\F_{p^n}(x)$ defined over the finite field extension $K=\F_{p^n}$ of the prime field $\F_p$. We assume that $K$ is…

We prove an analogue of the classical Bateman-Horn conjecture on prime values of polynomials for the ring of polynomials over a large finite field. Namely, given non-associate, irreducible, separable and monic (in the variable $x$)…

Let K be a function field and C a non-isotrivial curve of genus g >= 2 over K. In this paper, we will show that if C has a global stable model with only geometrically irreducible fibers, then Bogomolov conjecture over function fields holds.

Kashiwara conjectured that the hard Lefshetz theorem and the semisimplicity theorem hold for any semisimple perverse sheaf M on a variety over a field of characteristic 0. He also conjectured that if you apply to such M the nearby cycle…

Let $k$ be a Brauer field, that is, a field over which every diagonal form in sufficiently many variables has a nonzero solution; for instance, $k$ could be an imaginary quadratic number field. Brauer proved that if $f_1, \ldots, f_r$ are…

We give a new proof of Fatou's theorem: {\em if an algebraic function has a power series expansion with bounded integer coefficients, then it must be a rational function.} This result is applied to show that for any non--trivial completely…

The Serre conjecture II predicts that every torsor under a semisimple, simply connected, algebraic group over a field of cohomological dimension at most 2 and of degree of imperfection at most 1 has a rational point. We generalize this…

We prove an implicit function theorem for C^k-maps from arbitrary topological vector spaces over valued fields to Banach spaces (for k at least 2). As a tool, we show the C^k-dependence of fixed points on parameters for suitable families of…

In 1980, Faltings proved, by deep local algebra methods, a local result regarding formal functions which has the following global geometric fact as a consequence. Theorem: Let k be an algebraically closed field (of any characteristic). Let…