Related papers: On rational points on classifying stacks and Malle…
The log-rank conjecture is a longstanding open problem with multiple equivalent formulations in complexity theory and mathematics. In its linear-algebraic form, it asserts that the rank and partitioning number of a Boolean matrix are…
Using class field theory, we prove a restriction on the intersection of the maximal abelian extensions associated with different number fields. This restriction is then used to improve a result of Rosen and Silverman about the linear…
We consider the set of $n\times n$ matrices with rational entries having numerator and denominator of size at most $H$ and obtain upper and lower bounds on the number of such matrices of a given rank and then apply them to count such…
We put a new conjecture on primes from the point of view of its binary expansions and make a step towards justification.
Banderier, Marchal, and Wallner considered Young tableaux with walls, which are similar to standard Young tableaux, except that local decreases are allowed at some walls. In this work, we prove a conjecture of Fuchs and Yu concerning the…
We show that Zilber's conjecture that complex exponentiation is isomorphic to his pseudo-exponentiation follows from the a priori simpler conjecture that they are elementarily equivalent. An analysis of the first-order types in…
The principle of Batyrev and Manin and its variants gives a precise conjectural interpretation for the dominant term for the number of points of bounded height on an algebraic variety for which the opposite of the canonical line bundle is…
We answer a question by Niederreiter concerning the enumeration of a class of subspaces of finite dimensional vector spaces over finite fields by proving a conjecture by Ghorpade and Ram.
We study the asymptotic growth of the number of rational points of bounded height on smooth projective split toric varieties with Picard rank 2 over number fields, with respect to Arakelov height functions associated with big metrized line…
This expository paper discusses some conjectures related to visibility and blockers for sets of points in the plane.
We investigate infinitary wellfounded systems for linear logic with fixed points, with transfinite branching rules indexed by some closure ordinal $\alpha$ for fixed points. Our main result is that provability in the system for some…
In this paper we make a series of numerical experiments to support Greenberg's $p$-rationality conjecture, we present a family of $p$-rational biquadratic fields and we find new examples of $p$-rational multiquadratic fields. In the case of…
We explain how to deduce from the multi-height analysis of rational points on a toric stack (respectively on a toric variety) the asymptotic study of the number of rational points of bounded orbifold anticanonical height (respectively…
Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence in terms of relatively simple invariants. Where…
We construct an algorithm for solving the following problem: given a number field $K$, a positive integer $N$, and a positive real number $B$, determine all points in $\mathbb P^N(K)$ having relative height at most $B$. A theoretical…
Martin's Conjecture states that every definable function on the Turing degrees is either constant or increasing, and that every increasing function is an iterate of the Turing jump. This classification has already been corroborated for the…
An asymptotic formula is established for the number of rational points of bounded height on a non-singular quartic del Pezzo surface with a conic bundle structure.
In this exposition-type note we present detailed proofs of certain assertions concerning several algebraic properties of the cone and cylinder algebras. These include a determination of the maximal ideals, the solution of the B\'ezout…
In this article we prove lower and upper bounds for class numbers of algebraic curves defined over finite fields. These bounds turn out to be better than most of the previously known bounds obtained using combinatorics. The methods used in…
Closely following recent ideas of J. Borcea, we discuss various modifications and relaxations of Sendov's conjecture about the location of critical points of a polynomial with complex coefficients. The resulting open problems are formulated…