Related papers: Counting rational points on hypersurfaces and high…

Let $R_s(n)$ denote the number of representations of the positive number $n$ as the sum of two squares and $s$ biquadrates. When $s=3$ or $4$, it is established that the anticipated asymptotic formula for $R_s(n)$ holds for all $n\le X$…

The number of rational points of a plane non-singular algebraic curve X defined over a finite field is computed, provided that the generic point of X is not an inflexion and that X is Frobenius non-classical with respect to conics.

In this paper, we derive a new representation for the Anger--Weber function, employing the reformulation of the method of steepest descents by C. J. Howls (Howls, Proc. R. Soc. Lond. A 439 (1992) 373--396). As a consequence of this…

We prove an asymptotic formula for the number of representations of squares by nonsingular cubic forms in six or more variables. The main ingredients of the proof are Heath-Brown's form of the Circle Method and various exponential sum…

Let $\sum_{\beta\in\nats^d} F_\beta x^\beta$ be a multivariate power series. For example $\sum F_\beta x^\beta$ could be a generating function for a combinatorial class. Assume that in a neighbourhood of the origin this series represents a…

Surprisingly, apart from some special cases, simple asymptotic expansions for the associated Legendre functions $P_\nu ^\mu (z)$ and $Q_\nu ^\mu (z)$ for large degree $\nu$ or large order $\mu$ are not available in the literature. The main…

We derive the first exact, rigorous but practical, globally valid remainder terms for asymptotic expansions about saddles and contour endpoints of arbitrary order degeneracy derived from the method of steepest descents. The exact remainder…

In this paper we describe an algorithm for implicitizing rational hypersurfaces in case there exists at most a finite number of base points. It is based on a technique exposed in math.AG/0210096, where implicit equations are obtained as…

The algebraic properties of formal power series, whose coefficients show factorial growth and admit a certain well-behaved asymptotic expansion, are discussed. It is shown that these series form a subring of $\mathbb{R}[[x]]$. This subring…

In this paper we obtain a series of asymptotic formulae in the sum--product phenomena over the prime field $\mathbf{F}_p$. In the proofs we use usual incidence theorems in $\mathbf{F}_p$, as well as the growth result in ${\rm SL}_2…

The inverse problem for representation functions takes as input a triple (X,f,L), where X is a countable semigroup, f : X --> N_0 \cup {\infty} a function, L : a_1 x_1 + ... + a_h x_h an X-linear form and asks for a subset A \subseteq X…

Gaussian quadrature rules are a classical tool for the numerical approximation of integrals with smooth integrands and positive weight functions. We derive and expicitly list asymptotic expressions for the points and weights of Gaussian…

Given a multivariate generating function F, we determine asymptotics for the coefficients. Our approach is to use Cauchy's integral formula near singular points of F, resulting in a tractable oscillating integral. This paper treats the case…

Let f(1)=1, and let f(n+1)=2^{2^{f(n)}} for every positive integer n. We conjecture that if a system S \subseteq {x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} \cup {x_i+1=x_k: i,k \in {1,...,n}} has only finitely many solutions in non-negative…

Let $f_1, ..., f_R$ be rational forms of degree $d \ge 2$ in $n > \sigma + R(R+1)(d-1)2^{d-1}$ variables, where $\sigma$ is the dimension of the affine variety cut out by the condition $\mathrm{rank}(\nabla f_k)_{k=1}^R < R$. Assume that…

We consider the spherical integral of real symmetric or Hermitian matrices when the rank of one matrix is one. We prove the existence of the full asymptotic expansions of these spherical integrals and derive the first and the second term in…

The circle method has been successfully used over the last century to study rational points on hypersurfaces. More recently, a version of the method over function fields, combined with spreading out techniques, has led to a range of results…

We consider a three-dimensional Fourier integral in which the exponent in the exponential factor is the product of some phase function and a large parameter. The asymptotics of this integral is sought when the large parameter tends to…

Caro and Pasten gave an explicit upper bound on the number of rational points on a hyperbolic surface that is embedded in an abelian variety of rank at most one. We show how to use their method to produce a refined bound on the number of…

Let $\{a_\rr : \rr \in (\Z^+)^d \}$ be a $d$-dimensional array of numbers, for which the generating function $F(\zz) := \sum_\rr a_\rr \zz^\rr$ is meromorphic in a neighborhood of the origin. For example, $F$ may be a rational multivariate…