Related papers: Zeros of Dirichlet Polynomials via a Density Crite…

We obtain asymptotic bounds on the number of natural numbers less than $X$ satisfying $\gcd{\left(n,\lfloor P(n) \rfloor\right)}=1$, under some diophantine conditions on the coefficient of $x$ in $P$, and show that the density of such…

We discuss the existence of zero coefficients in the powers of the determinant polynomial of order $n$. D. G. Glynn proved that the coefficients of the $m$th power of the determinant polynomial are all nonzero, if $m = p-1$ with a prime…

We consider a Dirichlet series $\sum_{n=1}^{\infty}a_n^{-s}$, where $a_n$ satisfies a linear recurrence of arbitrary degree with integer coefficients. Under suitable hypotheses, we prove that it has a meromorphic continuation to the complex…

We derive a boundary monotonicity formula for a class of biharmonic maps with Dirichlet boundary conditions. A monotonicity formula is crucial in the theory of partial regularity in super-critical dimensions. As a consequence of such a…

We show the existence of a weak solution of a semilinear elliptic Dirichlet problem on an arbitrary open set. We make no assumptions about the open set, very mild regularity assumptions on the semilinearity, plus a coerciveness assumption…

This paper investigates lower bounds on the number of zeros and poles of a general Dirichlet series in a disk of radius $r$ and gives, as a consequence, an affirmative answer to an open problem of Bombieri and Perelli on the bound.…

We prove a quantitative version of the Polynomial Szemeredi Theorem for difference sets. This result is achieved by first establishing a higher dimensional analogue of a theorem of Sarkozy (the simplest non-trivial case of the Polynomial…

We introduce certain special polynomials in an arbitrary number of indeterminates over a finite field. These polynomials generalize the special polynomials associated to the Goss zeta function and Goss-Dirichlet $L$-functions over the ring…

We study the existence of positive solutions on the half-line $[0,\infty)$ for the nonlinear second order differential equation \[ \bigl(a(t)x^{\prime}\bigr)^{\prime}+b(t)F(x)=0,\quad t\geq0, \] satisfying Dirichlet type conditions, say…

In this paper we provide a complete answer to a question by Heyman and Shparlinski concerning the natural density of polynomials which are irreducible by Eisenstein's criterion after applying some shift. The main tool we use is a local to…

We show that by working over the absolute base $\mathbb S$ (the categorical version of the sphere spectrum) instead of $\mathbb S[\pm 1]$ improves our previous Riemann-Roch formula for $\overline{{\rm Spec\,}\mathbb Z}$. The formula equates…

We consider multivariate polynomials and investigate how many zeros of multiplicity at least $r$ they can have over a Cartesian product of finite subsets of a field. Here r is any prescribed positive integer and the definition of…

In this paper, we consider the problem of determining the density of monic polynomials over $\mathbb{Z}_p$ with squarefree discriminant over various subsets of the set of monic polynomials over $\mathbb{Z}_p$ of fixed degree. We compute the…

It was observed by Bump et al. that Ehrhart polynomials in a special family exhibit properties similar to the Riemann {\zeta} function. The construction was generalized by Matsui et al. to a larger family of reflexive polytopes coming from…

If the coefficients of polynomials are selected by some random process, the zeros of the resulting polynomials are in some sense random. In this paper the author rephrases the above in more precise language, and calculates the joint…

We refine an idea of Deodhar, whose goal is a counting formula for Kazhdan-Lusztig polynomials. This is a consequence of a simple observation that one can use the solution of Soergel's conjecture to make ambiguities involved in defining…

We prove the discrete analogue of Kakeya conjecture over $\mathbb{R}^n$. This result suggests that a (hypothetically) low dimensional Kakeya set cannot be constructed directly from discrete configurations. We also prove a generalization…

A polynomial is a direct sum if it can be written as a sum of two non-zero polynomials in some distinct sets of variables, up to a linear change of variables. We analyze criteria for a homogeneous polynomial to be decomposable as a direct…

We formulate and discuss a necessary and sufficient condition for polynomials to be dense in a space of continuous functions on the real line, with respect to Bernstein's weighted uniform norm. Equivalently, for a positive finite measure…

A novel method with two variations is proposed with which the number of positive and negative zeros of a polynomial with real coefficients and degree $n$ can be restricted with significantly better determinacy than that provided by the…