Related papers: Explicit arithmetic intersection theory and comput…
The Kauffman bracket skein algebra of a surface is a generalization of the Jones polynomial invariant for links and plays a principal role in the Witten-Reshetikhin- Turaev topological quantum field theory. However, the multiplicative…
A survey on algorithms for computing discrete logarithms in Jacobians of curves over finite fields.
The height of a toric variety and that of its hypersurfaces can be expressed in convex-analytic terms as an adelic sum of mixed integrals of their roof functions and duals of their Ronkin functions. Here we extend these results to the…
We derive various inequalities involving the intersection number of the curves contained in geodesics and tight geodesics in the curve graph. While there already exist such inequalities on tight geodesics, our method applies in the setting…
An important problem in computational arithmetic geometry is to find changes of coordinates to simplify a system of polynomial equations with rational coefficients. This is tackled by a combination of two techniques, called minimisation and…
In this paper, we continue the study of intersections of closed curves on translation surfaces, initiated in by S. Cheboui, A. Kessi and D. Massart for a family of arithmetic Veech surfaces and the author, E. Lanneau and D. Massart for a…
In this paper we study duality for evaluation codes on intersections of d hypersurfaces with given d-dimensional Newton polytopes, so called toric complete intersection codes. In particular, we give a condition for such a code to be…
This research monograph focuses on the arithmetic, over number fields, of surfaces fibred into curves of genus 1 over the projective line, and of intersections of two quadrics in projective space. The first half takes up and develops…
Based on Nielsen fixed point theory and Gr\"{o}bner-Shirshov basis, we obtain a simple method to compute geometric intersection numbers and self-intersection geometric numbers of loops on surfaces.
In this short note we prove a formula for local heights on elliptic curves over number fields in terms of intersection theory on a regular model over the ring of integers.
Let $C$ be a genus $2$ curve with Jacobian isomorphic to the square of an elliptic curve with complex multiplication by a maximal order in an imaginary quadratic field of discriminant $-d<0$. We show that if the stable model of $C$ has bad…
In this paper, we prove an explicit arithmetic intersection formula between arithmetic Hirzebruch-Zagier divisors and arithmetic CM cycles in a Hilbert modular surface over $\mathbb Z$. As applications, we obtain the first `non-abelian'…
We present an algorithm for computing curves and families of curves of prescribed degree and geometric genus on real rational surfaces.
We estimate from below the number of lines meeting each of given 4 disjoint smooth closed curves in a given cyclic order in the real projective 3-space and in a given linear order in the Euclidean 3-space. Similarly, we estimate the number…
We approximately compute the correspondence degree (as defined by Lazarsfeld and Martin) between two unbalanced complete intersections. This is accomplished by showing that the procedure of taking a subvariety of a product $Y \times Y'$ and…
A long standing conjecture of Richter and Thomassen states that the total number of intersection points between any $n$ simple closed Jordan curves in the plane, so that any pair of them intersect and no three curves pass through the same…
We give necessary and sufficient topological conditions for a simple closed curve on a real rational surface to be approximable by smooth rational curves. We also study approximation by smooth rational curves with given complex…
A high-order quadrature algorithm is presented for computing integrals over curved surfaces and volumes whose geometry is implicitly defined by the level sets of (one or more) multivariate polynomials. The algorithm recasts the implicitly…
First constructed by Beilinson for curves defined over an algebraically closed field; R\"ossler and Szamuely generalized Beilinson's height pairing to the higher dimensional setting. In this paper we study this pairing and relate it to the…
Let K be a field of positive characteristic. When V is a linear variety in K^n and G is a finitely generated subgroup of K^*, we show how to compute the intersection of V and G^n effectively using heights. We calculate all the estimates…