Related papers: Multiple discriminants and critical values of a mu…
The multiplier $\lambda_n$ of a periodic orbit of period $n$ can be viewed as a (multiple-valued) algebraic function on the space of all complex quadratic polynomials $p_c(z)=z^2+c$. We provide a numerical algorithm for computing critical…
The method of self-similar root approximants has earlier been shown to provide accurate interpolating formulas for functions for which small-variable expansions are given and the behaviour of the functions at large variables is known. Now…
We consider a scalar-valued implicit function of many variables, and provide two closed formulae for all of its partial derivatives. One formula is based on products of partial derivatives of the defining function, the other one involves…
We give a first-order definition of key polynomials, we show the links with previous definitions, that it is relevant to study key degrees, and to use a kind of valuations that we call partially multiplicative. We also prove or reprove…
In this paper, we obtain new results on the critical points of a polynomial. We discuss the Sendov conjecture for polynomials of degree nine.
In this paper we develop with considerable details a theory of multivector functions of a $p$-vector variable. The concepts of limit, continuity and differentiability are rigorously studied. Several important types of derivatives for these…
Positive and negative quadratic forms are well known and widely used. They are multivariate homogeneous polynomials of degree two taking positive or negative values respectively for any values of their arguments not all zero. In the present…
We consider various definitions of degrees of discrete functions and establish relations between the number of relevant (essential) variables and degrees of two- and three-valued functions. Keywords: relevant variable, sensitivity, degree…
In this paper, we show that for a unicritical polynomial having a priori bounds, the unique conformal measure of minimal exponent has no atom at the critical point. For a conformal measure of higher exponent, we give a necessary and…
In this paper, we discuss the joint value distribution of $L$-functions in a suitable class. We obtain joint large deviations results in the central limit theorem for these $L$-functions and some mean value theorems, which give evidence…
We study structured optimization problems with polynomial objective function and polynomial equality constraints. The structure comes from a multi-grading on the polynomial ring in several variables. For fixed multi-degrees we determine the…
We consider critical points of a class of functionals on compact four-dimensional manifolds arising from Regularized Determinants for conformally covariant operators, whose explicit form was derived in [10], extending Polyakov's formula.…
In this note a critical point result for differentiable functionals is exploited in order to prove that a suitable class of one-dimensional fractional problems admits at least one non-trivial solution under an asymptotical behaviour of the…
In the present paper we extend the multiplicative integral to complex-valued functions of complex variable. The main difficulty in this way, that is the multi-valued nature of the complex logarithm, is avoided by division of the interval of…
It is shown that a polynomial with a Cremer periodic point has a non-accessible critical point in its Julia set provided that the Cremer periodic point is approximated by small cycles.
We study rather general multiple zeta-functions whose denominators are given by polynomials. The main aim is to prove explicit formulas for the values of those multiple zeta-functions at non-positive integer points. We first treat the case…
We describe a method to evaluate multivariate polynomials over a finite field and discuss its multiplicative complexity.
A classification is provided of functors, in particular polynomial ones, from a category with a zero object in which every object is a finite sum of copies of a generating object, into an abelian category. This classification is extended to…
We extend to several variables an earlier result of ours, according to which an entire function of one variable of sufficiently small exponential type, having all derivatives of even order taking integer values at two points, is a…
The absolute separation of a polynomial is the minimum nonzero difference between the absolute values of its roots. In the case of polynomials with integer coefficients, it can be bounded from below in terms of the degree and the height…