Publications and Preprints

Asymptotics of extensions of simple \(\mathbb Q\)algebras, with BĂ©ranger Seguin. (arXiv)
Abstract: We answer various questions concerning the distribution of extensions of a given central simple algebra \(K\) over a number field. Specifically, we give asymptotics for the count of inner Galois extensions \(LK\) of fixed degree and center with bounded discriminant. We also relate the distribution of outer extensions of \(K\) to the distribution of field extensions of its center \(Z(K)\). This paper generalizes the study of asymptotics of field extensions to the noncommutative case in an analogous manner to the program initiated by Deschamps and Legrand to extend inverse Galois theory to skew fields.

Sampling cubic rings. (arXiv, code)
Abstract: We explain how to construct a uniformly random cubic integral domain \(S\) of given signature with \(\textrm{disc}(S)\leq T\) in expected time \(\widetilde{\mathcal O}(\log T)\).

Malle's conjecture with multiple invariants. (arXiv)
Abstract: We define invariants \(\operatorname{inv}_1,\dots,\operatorname{inv}_m\) of Galois extensions of number fields with a fixed Galois group. Then, we propose a heuristic in the spirit of Malle's conjecture which asymptotically predicts the number of extensions that satisfy \(\operatorname{inv}_i\leq X_i\) for all \(X_i\). The resulting conjecture is proved for abelian Galois groups. We also describe refined Artin conductors that carry essentially the same information as the invariants \(\operatorname{inv}_1,\dots,\operatorname{inv}_m\).

Polynomials vanishing at lattice points in a convex set. (arXiv)
Abstract: Let \(P\) be a bounded convex subset of \(\mathbb R^n\) of positive volume. Denote the smallest degree of a polynomial \(p(X_1,\dots,X_n)\) vanishing on \(P\cap\mathbb Z^n\) by \(r_P\) and denote the smallest number \(u\geq0\) such that every function on \(P\cap\mathbb Z^n\) can be interpolated by a polynomial of degree at most \(u\) by \(s_P\). We show that the values \((r_{d\cdot P}1)/d\) and \(s_{d\cdot P}/d\) for dilates \(d\cdot P\) converge from below to some numbers \(v_P,w_P>0\) as \(d\) goes to infinity. The limits satisfy \(v_P^{n1}w_P \leq n!\cdot\operatorname{vol}(P)\). When \(P\) is a triangle in the plane, we show equality: \(v_Pw_P = 2\operatorname{vol}(P)\). These results are obtained by looking at the set of standard monomials of the vanishing ideal of \(d\cdot P\cap\mathbb Z^n\) and by applying the BernsteinKushnirenko theorem. Finally, we study irreducible Laurent polynomials that vanish with large multiplicity at a point. This work is inspired by questions about Seshadri constants.

Integral BrauerManin obstructions for sums of two squares and a power, J. London Math. Soc., 2(88):599618, 2013. (arXiv, DOI)
Abstract: We use BrauerManin obstructions to explain failures of the integral Hasse principle and strong approximation away from \(\infty\) for the equation \(x^2+y^2+z^k=m\) with fixed integers \(k\geq3\) and \(m\). Under Schinzel's hypothesis (H), we prove that BrauerManin obstructions corresponding to specific Azumaya algebras explain all failures of strong approximation away from \(\infty\) at the variable \(z\). Finally, we present an algorithm that, again under Schinzel's hypothesis (H), finds out whether the equation has any integral solutions.