Related papers: Bound the difference of two singular moduli

The aim of this paper is to obtain a uniform bound for a certain class of submodules from the following theorem: Let $(R,\frak m)$ be a local ring, let $M$ be a finite $R$--module of dimension $d\ge 1$ and let $\frak q$ be an ideal of $R$…

Pila and Tsimerman proved in 2017 that for every $k$ there exists at most finitely many $k$-tuples $(x_1,\ldots, x_k)$ of distinct non-zero singular moduli with the property "$x_1, \ldots,x_k$ are multiplicatively dependent, but any proper…

We prove a lower bound for the modulus of the amplitude for a two-body process at large scattering angle. This is based on the interplay of the analyticity of the amplitude and the positivity properties of its absorptive part. The…

The determinantal complexity of a polynomial $P \in \mathbb{F}[x_1, \ldots, x_n]$ over a field $\mathbb{F}$ is the dimension of the smallest matrix $M$ whose entries are affine functions in $\mathbb{F}[x_1, \ldots, x_n]$ such that $P =…

The modulus of a polynomial-like (PL) map is an important invariant that controls distortion of the straightening map and, hence, geometry of the corresponding PL Julia set. Lower bounds on the modulus, called complex a priori bounds, are…

We study lower bounds for the norm of the product of polynomials and their applications to the so called \emph{plank problem.} We are particularly interested in polynomials on finite dimensional Banach spaces, in which case our results…

By focusing on the X-matrix part of a density matrix of two qubits we provide an algebraic lower bound for the concurrence. The lower bound is generalized for cases beyond two qubits and can serve as a sufficient condition for…

This work concerns the distance in 2-norm from a matrix polynomial to a nearest polynomial with a specified number of its eigenvalues at specified locations in the complex plane. Perturbations are allowed only on the constant coefficient…

We define two new families of polynomials that generalize permanents and prove upper and lower bounds on their determinantal complexities comparable to the known bounds for permanents. One of these families is obtained by replacing…

Let Lambda be a set of three integers and let C_Lambda be the space of 2pi-periodic functions with spectrum in Lambda endowed with the maximum modulus norm. We isolate the maximum modulus points x of trigonometric trinomials T in C_Lambda…

This paper contributes to the solution of the Poincare problem, which is to bound the degree of a (generalized algebraic) leaf of a (singular algebraic) foliation of the complex projective plane. The first theorem gives a new sort of bound,…

We establish bounds for the Castelnuovo-Mumford regularity of a finitely generated graded module and its symmetric powers in terms of the degrees of the generators of the module and the degrees of their relations. We extend to modules (and…

We obtain conditions guaranteeing that weak solutions of the differential inequality $$ \sum_{|\alpha| = m} \partial^\alpha a_\alpha (x, u) \ge f (x) g (|u|) \quad \mbox{in } \Omega \setminus S, $$ has a removable singular set $S \subset…

In this paper we shall use the boundary Schwarz lemma of Osserman to obtain some generalizations and refinements of some well known results concerning the maximum modulus of the polynomials with restricted zeros due to Turan, Dubinin and…

This paper concerns the bounds for spectral norm distance from a normal matrix polynomial $P(\lambda)$ to the set of matrix polynomials that have $\mu$ as a multiple eigenvalue. Also construction of associated perturbations of $P(\lambda)$…

We initiate a classification of complex polynomials f of degree d having the top Betti number of the general fibre close to the maximum. We find a range in which the polynomial must have isolated singularities and another range where it may…

We study an analogue of a classical arithmetic problem over the ring of polynomials. We prove that $m = 5$ is the minimal number such that the sums of any two distinct polynomials in a set of $m$ polynomials over $\F_2[x]$ cannot all be of…

We improve upon the upper bounds for the cardinality of the value set of a multivariable polynomial map over a finite field using the polytope of the polynomial. This generalizes earlier bounds only dependent on the degree of a polynomial.

We present sharp bounds for $\sum_{i=1}^n \alpha_i x_i - \prod_{i=1}^n x_i^{\alpha_i}$ in terms of the variance of the vector $(x_1^{1/2},...,x_n^{1/2})$.

Let S be a finite subset of a field. For multivariate polynomials the generalized Schwartz-Zippel bound [2], [4] estimates the number of zeros over Sx...xS counted with multiplicity. It does this in terms of the total degree, the number of…