Related papers: Integral points of bounded height on toric varieti…
We give a formula for the component at p of the p-adic height pairing of a divisor of degree 0 on a hyperelliptic curve. We use this to give a Chabauty-like method for finding p-adic approximations to p-integral points on such curves when…
We extend work of Heath-Brown and Salberger, based on the determinant method, to provide a uniform upper bound for the number of integral points of bounded height on an affine surface, which are subject to a polynomial congruence condition.…
We prove matching upper and lower bounds for the average of the 6-torsion of class groups of quadratic fields. Furthermore, we count the number of integer solutions on an affine quartic threefold.
We consider certain systems of three linked simultaneous diagonal equations in ten variables with total degree exceeding five. By means of a complification argument, we obtain an asymptotic formula for the number of integral solutions of…
We derive a new bound for some bilinear sums over points of an elliptic curve over a finite field. We use this bound to improve a series of previous results on various exponential sums and some arithmetic problems involving points on…
Assuming Lang's conjectured lower bound on the heights of non-torsion points on an elliptic curve, we show that there exists an absolute constant C such that for any elliptic curve E/Q and non-torsion point P in E(Q), there is at most one…
We generalize results about local heights previously proved in the case of discrete absolute values to arbitrary non-archimedean absolute values of rank 1. First, this is done for the induction formula of Chambert-Loir and Thuillier. Then…
We characterise and investigate co-Higgs sheaves and associated algebraic and combinatorial invariants on toric varieties. In particular, we compute explicit examples.
We consider an integral operator $\mathcal{I}$, special instances of which was studied in various contexts. Using an appropriate transformation we write this operator in terms of weighted composition operators. Then, we provide a…
Given an extension of number fields $E \subset F$ and a projective variety $X$ over $F$, we compare the problem of counting the number of rational points of bounded height on $X$ with that of its Weil restriction over $E$. In particular, we…
We propose some problems on the classification of toric manifolds from the viewpoint of topology and survey related results.
In this note, we establish an asymptotic formula for the number of rational points of bounded height on the singular cubic surface $$ x_0(x_1^2 + x_2^2)=x_3^3 $$ with a power-saving error term, which verifies the Manin-Peyre conjectures for…
In this paper, we obtain an asymptotic formula for the number of integral solutions to a system of diagonal equations. We obtain an asymptotic formula for the number of solutions with variables restricted to smooth numbers as well. We…
We have developed in the past several algorithms with intrinsic complexity bounds for the problem of point finding in real algebraic varieties. Our aim here is to give a comprehensive presentation of the geometrical tools which are…
We prove a theorem giving the asymptotic number of binary quartic forms having bounded invariants; this extends, to the quartic case, the classical results of Gauss and Davenport in the quadratic and cubic cases, respectively. Our…
We present and expand some existing results on the Zariski closure of cyclic groups and semigroups of matrices. We show that, with the exclusion of isolated points, their irreducible components are toric varieties. Additionally, we…
We study the distribution of $S$-integral points on $\mathrm{SL}_2$-orbit closures of binary forms and prove an asymptotic formula for the number of $S$-integral points of bounded height on $\mathrm{SL}_2$-orbit closures of binary forms.…
We establish upper bounds for the complexity of Seifert fibered manifolds with nonempty boundary. In particular, we obtain potentially sharp bounds on the complexity of torus knot complements.
We prove new bounds on the number of incidences between points and higher degree algebraic curves. The key ingredient is an improved initial bound, which is valid for all fields. Then we apply the polynomial method to obtain global bounds…
We give explicit formulas for the dimensions and the degrees of $A$-discriminant varieties introduced by Gelfand-Kapranov-Zelevinsky. Our formulas can be applied also to the case where the $A$-discriminant varieties are higher-codimensional…