Related papers: The Alpha Problem & Line Count Configurations

In this paper we are concerned with a long-standing conjecture of Huneke and Wiegand. We introduce a new class of ideals and prove thateach ideal from such class satisfies the conclusion of the conjecture in question. We also study the…

Milnor formulated a conjecture about rational linear independence of some special Hurwitz zeta values. The second and third authors along with Ram Murty studied this conjecture and suggested an extension of Milnor's conjecture. In this…

We prove quantitative versions of Borel and Harish-Chandra's theorems on reduction theory for arithmetic groups. Firstly, we obtain polynomial bounds on the lengths of reduced integral vectors in any rational representation of a reductive…

Let $A/\mathbb{Q}$ be an elliptic curve with split multiplicative reduction at a prime $p$. We prove (an analogue of) a conjecture of Perrin-Riou, relating $p$-adic Beilinson$-$Kato elements to Heegner points in $A(\mathbb{Q})$, and a large…

For an irrational $\alpha\in \mathbb{R}$, we consider additive problems with the set of primes satisfying $\lVert\alpha p\rVert\leq \frac{1}{p^\tau}$ for some fixed $\tau>0$. In particular, we show that there exist infinitely many…

Given a linear space L in affine space A^n, we study its closure L' in the product of projective lines (P^1)^n. We show that the degree, multigraded Betti numbers, defining equations, and universal Grobner basis of its defining ideal I(L')…

We consider the isoperimetric inequality on the class of high-dimensional isotropic convex bodies. We establish quantitative connections between two well-known open problems related to this inequality, namely, the thin shell conjecture, and…

Let $L/K$ be a Galois extension of number fields. The problem of counting the number of prime ideals $\mathfrak p$ of $K$ with fixed Frobenius class in $\mathrm{Gal}(L/K)$ and norm satisfying a congruence condition is considered. We show…

A classic theorem of Euclidean geometry asserts that any noncollinear set of $n$ points in the plane determines at least $n$ distinct lines. Chen and Chv\'atal conjectured that this holds for an arbitrary finite metric space, with a certain…

In these notes, we refine Mitsui's Prime Number Theorem from 1957, which for a number field $K$ predicts how many prime elements there are in bounded convex sets in $K \otimes_{\mathbf Q} \mathbf R$, by incorporating potential Siegel zeros…

We study several consequences of the packing problem, a conjecture from combinatorial optimization, using algebraic invariants of square-free monomial ideals. While the packing problem is currently unresolved, we successfully settle the…

We apply ideas from the theory of limits of dense combinatorial structures to study order types, which are combinatorial encodings of finite point sets. Using flag algebras we obtain new numerical results on the Erd\H{o}s problem of finding…

We study the growth of the local $L^2$-norms of the unitary Eisenstein series for reductive groups over number fields, in terms of their parameters. We derive a \emph{poly-logarithmic} bound on an average, for a large class of reductive…

In this work we deal with the problem of support estimation under shape restrictions. The shape restriction we deal with is an extension of the notion of convexity named alpha-convexity. Instead of assuming, as in the convex case, the…

In this note we generalize the main result in [DIV: R. Di Gennaro, G. Ilardi, J. Valles, Singular hypersurfaces characterizing the Lefschetz properties J. Lond. Math. Soc. (2) 89 (2014), no. 1, 194-212] on artinian ideals failing Lefschetz…

This paper is a natural sequel to [22] in that it tackles problems of the same nature. Here one aims at the ideal theoretic and homological properties of a class of ideals of general plane fat points whose second symbolic powers hold…

This article discusses the modular representation theory of finite groups of Lie type from the viewpoint of Broue's abelian defect group conjecture. We discuss both the defining characteristic case, the inspiration for Alperin's weight…

In this paper, we present an alternative proof of a finiteness theorem due to Bombieri, Masser and Zannier concerning intersections of a curve in the multiplicative group of dimension n with algebraic subgroups of dimension n-2. The proof…

In previous work, the first author, Ghioca, and the third author introduced a broad dynamical framework giving rise to many classical sequences from number theory and algebraic combinatorics. Specifically, these are sequences of the form…

In the present paper, we study a purely inseparable counterpart of Abhyankar's conjecture for the affine line in positive characteristic, and prove its validity for all the finite local non-abelian simple group schemes in characteristic…