Related papers: Approximate computations with modular curves
Contours may be viewed as the 2D outline of the image of an object. This type of data arises in medical imaging as well as in computer vision and can be modeled as data on a manifold and can be studied using statistical shape analysis.…
Consider a rational family of planar rational curves in a certain region of interest. We are interested in finding an approximation to the implicit representation of the envelope. Since exact implicitization methods tend to be very costly,…
In the present paper we are concerned with a numerical algorithm for the approximation of the two-dimensional neural field equation with delay. We consider three numerical examples that have been analysed before by other authors and are…
The action of ring automorphisms of the polynomial ring in two variables over the real numbers on real plane curves is considered. The orbits containing degree-three polynomials are computed, with one representative per orbit being…
It was first pointed out by Weil that we can use classical invariant theory to compute the Jacobian of a genus one curve. The invariants required for curves of degree n = 2,3,4 were already known to the nineteenth centuary invariant…
Provided a special function of one variable and some of its derivatives can be accurately computed over a finite range, a method is presented to build a series of polynomial approximations of the function with a defined relative error over…
This expository paper reviews some of the recent uses of computational algebraic geometry in classical and quantum optimization. The paper assumes an elementary background in algebraic geometry and adiabatic quantum computing (AQC), and…
The modular curves serve as excellent objects for testing conjectures in arithmetic geometry. They possess a natural geometric definition in contrast with rather nontrivial structure. On the other hand, they are well-studied from the…
These lecture notes consist of an introduction to moduli spaces in algebraic geometry, with a strong emphasis placed on examples related to the theory of quiver representations. The goal is to provide the background necessary to understand…
In a previous joint article with F. Abu Salem, we gave efficient algorithms for Jacobian group arithmetic of "typical" divisor classes on C_{3,4} curves, improving on similar results by other authors. At that time, we could only state that…
These informal notes are an expanded version of lectures on the moduli space of elliptic curves given at Zhejiang University in July, 2008. Their goal is to introduce and motivate basic concepts and constructions (such as orbifolds and…
We discuss a selection of recent developments in arithmetic combinatorics having to do with ``approximate algebraic structure'' together with some of their applications.
In this paper we propose two proximal gradient algorithms for fractional programming problems in real Hilbert spaces, where the numerator is a proper, convex and lower semicontinuous function and the denominator is a smooth function, either…
Much of arithmetic geometry is concerned with the study of principal bundles. They occur prominently in the arithmetic of elliptic curves and, more recently, in the study of the Diophantine geometry of curves of higher genus. In particular,…
Studies in thermodynamics often require the reduction of some first or second order partial derivatives in terms of a smaller basic set. A simple algorithm to perform such a reduction is presented here, together with a review of earlier…
The Chord algorithm is a popular, simple method for the succinct approximation of curves, which is widely used, under different names, in a variety of areas, such as, multiobjective and parametric optimization, computational geometry, and…
We show how our p-adic method to compute Galois representations occurring in the torsion of Jacobians of algebraic curves can be adapted to modular curves. The main ingredient is the use of "moduli-friendly" Eisenstein series introduced by…
We obtain a coarse relationship between geometric intersection numbers of curves and the sum of their subsurface projection distances with explicit quasi-constants. By using this relationship, we give applications in the studies of the…
A simply laced Dynkin diagram gives rise to a family of curves over $\mathbb{Q}$ and a coregular representation, using deformations of simple singularities and Vinberg theory respectively. Thorne has conjectured and partially proven a…
We describe an algorithm for computing a $\Q$-rational model for the quotient of a modular curve by an automorphism group, under mild assumptions on the curve and the automorphisms, by determining $q$-expansions for a basis of the…