Related papers: Arithmetic Nullstellensatz and Nonstandard Methods
We give a conditional proof of the Uniform Boundedness Conjecture of Morton and Silverman in the case of polynomials over number fields, assuming a standard conjecture in arithmetic geometry. Our technique simultaneously yields a dynamical…
Understanding bounds for the effective differential Nullstellensatz is a central problem in differential algebraic geometry. Recently, several bounds have been obtained using Dicksonian and antichains sequences (with a given growth rate).…
Factoring ideals in integral domains is a central topic in multiplicative ideal theory. In the present paper we study monoids of ideals and consider factorizations of ideals into multiplicatively irreducible ideals. The focus is on the…
We give bounds for the number and the size of the primes $p$ such that a reduction modulo $p$ of a system of multivariate polynomials over the integers with a finite number $T$ of complex zeros, does not have exactly $T$ zeros over the…
Moduli of rings and quadrilaterals are frequently applied in geometric function theory, see e.g. the Handbook by K\"uhnau. Yet their exact values are known only in a few special cases. Previously, the class of planar domains with polygonal…
The complexity of computing the solutions of a system of multivariate polynomial equations by means of Groebner bases computations is upper bounded by a function of the solving degree. In this paper, we discuss how to rigorously estimate…
We investigate two invariants of Noetherian semiperfect rings, namely the depth and a new invariant we call the "delooping level". These give lower and upper bounds for the finitistic dimension, respectively. As first theorems, we give a…
In this article, we explore the problem of constructing high-dimensional expanders through the study of relations between expansion constants over different rings. We investigate expansion constants of integer matrices regarded as morphisms…
This paper studies the notions of star and semistar operations over a polynomial ring. It aims at characterizing when every upper to zero in $R[X]$ is a $*$-maximal ideal and when a $*$-maximal ideal $Q$ of $R[X]$ is extended from $R$, that…
In this article, we obtain upper bounds on the number of irreducible factors of some classes of polynomials having integer coefficients, which in particular yield some of the well known irreducibility criteria. For devising our results, we…
This article investigates multiset dimensions in zero divisor graphs (ZD-graphs) associated with rings. Through rigorous analysis, we establish general bounds for the multiset dimension (Mdim) in ZD-graphs, exploring various commutative…
We present a new method for obtaining norm bounds for random matrices, where each entry is a low-degree polynomial in an underlying set of independent real-valued random variables. Such matrices arise in a variety of settings in the…
A sharp upper bound for the maximum integer not belonging to an ideal of a numerical semigroup is given and the ideals attaining this bound are characterized. Then the result is used, through the so-called Feng-Rao numbers, to bound the…
This paper establishes new upper bounds for the right eigenvalues of monic matrix polynomials over the quaternion division algebra. The noncommutative nature of quaternion multiplication presents fundamental challenges in eigenvalue…
We present sharp lower bounds for the A-numerical radius of semi-Hilbertian space operators. We also present an upper bound. Further we compute new upper bounds for the $B$-numerical radius of $2 \times 2$ operator matrices where $B =…
The aim of this article is to introduce standard bases of ideals in polynomial rings with respect to a class of orderings which are not necessarily semigroup orderings. Our approach generalises the concept of standard bases with respect to…
In this paper, we present a generic parametrization of generically zero-dimensional parametric polynomial systems. More specifically, we study the specialization properties of the Rational Univariate Representation and derive bounds on the…
In this paper we exhibit an example of a three-dimensional regular local domain (A, n) having a height-two prime ideal P with the property that the extension PA^ of P to the n-adic completion A^ of A is not integrally closed. We use a…
We prove an upper bound for the number of rational points of bounded height on irreducible affine hypersurfaces. More precisely, given an irreducible polynomial $f \in \mathbb{Z}[X_1, \dots, X_n]$, we prove an upper bound on the number of…
Polynomials whose coefficients, roots, and critical points lie in the ring of rational integers are called nice polynomials. In this paper, we present a general method for investigating such polynomials. We extend our results from the ring…