Related papers: The uniform Martin's conjecture for many-one degre…

We prove a global uniform Artin-Rees lemma type theorem for sections of ample line bundles over smooth projective varieties. This result is used to prove an Artin-Rees lemma for the polynomial ring with uniform degree bounds. The proof is…

In 1993 one of the authors formulated some conjectures on monotonicity of ratios for exponential series sections. They lead to more general conjecture on monotonicity of ratios of Kummer hypergeometric functions and was not proved from…

Different questions lead to the same class of functions from natural integers to integers: those which have integral difference ratios, i.e. verifying $f(a)-f(b)\equiv0 \pmod {(a-b)}$ for all $a>b$. We characterize this class of functions…

We provide an introduction of some basic facts of uniformly almost periodic functions, such as Fourier series representations. A result is then proved about Fourier coefficients which is a generalization of the purely periodic case. We then…

We consider some variations on the classical method of Runge for effectively determining integral points on certain curves. We first prove a version of Runge's theorem valid for higher-dimensional varieties, generalizing a uniform version…

Functions with singularities are notoriously difficult to approximate with conventional approximation schemes. In computational applications, they are often resolved with low-order piecewise polynomials, multilevel schemes, or other types…

This article describes a sequence of rational functions which converges locally uniformly to the zeta function. The numerators (and denominators) of these rational functions can be expressed as characteristic polynomials of matrices that…

We define a class of computable functions over real numbers using functional schemes similar to the class of primitive and partial recursive functions defined by G\"odel and Kleene. We show that this class of functions can also be…

We introduce certain special polynomials in an arbitrary number of indeterminates over a finite field. These polynomials generalize the special polynomials associated to the Goss zeta function and Goss-Dirichlet $L$-functions over the ring…

We investigate the role of continuous reductions and continuous relativisation in the context of higher randomness. We define a higher analogue of Turing reducibility and show that it interacts well with higher randomness, for example with…

We provide formulas for invariants defined on a tensor product of defining representations of unitary groups, under the action of the product group. This situation has a physical interpretation, as it is related to the quantum mechanical…

In this paper we give simple proofs for the bounds (some of them sharp) of the difference of the moduli of the second and the first logarithmic coefficient for the general class of univalent functions and for the class of convex univalent…

In this paper, we give fundamental solutions of some $q$-difference equations satisfied by the universal mock theta functions and the higher level Appell functions. As an application, we provide an alternative proof of the representation…

Following D. Ramakrishnan, we explain how L. Lafforgue's modularity theorem and an analytic theorem of H. Jacquet and J. Shalika can be applied to prove the following result related to the Tate Conjecture: for a smooth, projective,…

For a fixed polynomial $\Delta$, we study the number of polynomials $f$ of degree $n$ over $\mathbb F_q$ such that $f$ and $f+\Delta$ are both irreducible, an $\mathbb F_q[T]$-analogue of the twin primes problem. In the large-$q$ limit, we…

It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential…

A deep approximation is an approximating function defined by composing more than one layer of simple functions. We study deep approximations of functions of one variable using layers consisting of low-degree polynomials or simple conformal…

We estimate the covariance in counts of almost-primes in $\mathbb{F}_q[T]$, weighted by higher-order von Mangoldt functions. The answer takes a pleasant algebraic form. This generalizes recent work of Keating and Rudnick that estimates the…

Reynold's parametricity theory captures the property that parametrically polymorphic functions behave uniformly: they produce related results on related instantiations. In dependently-typed programming languages, such relations and…

An enumerative invariant theory in Algebraic Geometry, Differential Geometry, or Representation Theory, is the study of invariants which 'count' $\tau$-(semi)stable objects $E$ with fixed topological invariants $[E]=\alpha$ in some…