Related papers: On continuous selections of polynomial functions
We prove that every vertically nearly separately continuous function defined on a product of a strong PP-space and a topological space and with values in a strongly $\sigma$-metrizable space with a special stratification, is a pointwise…
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…
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…
Let $K$ be an algebraically closed field with an absolute value. This note gives an elementary proof of the classical result that the roots of a polynomial with coefficients in $K$ are continuous functions of the coefficients of the…
A deep approximation is an approximating function defined by composing more than one layer of simple functions. We study deep approximations of functions of one variable using layers consisting of low-degree polynomials or simple conformal…
The second author proved that the set of post-critically finite polynomials of given degree is a set of bounded height, up to change of variables. Motivated by an observation about unicritical polynomials, we complement this by proving that…
We consider several coding discretizations of continuous functions which reflect their variation at some given precision. We study certain statistical and combinatorial properties of the sequence of finite words obtained by coding a typical…
We give a combinatorial classification for the class of postcritically fixed Newton maps of polynomials as dynamical systems. This lays the foundation for classification results of more general classes of Newton maps. A fundamental…
A transcendental entire function is called criniferous if every point in its escaping set can eventually be connected to infinity by a curve of escaping points. Many transcendental entire functions with bounded singular set have this…
The paper studies complementary choice functions, i.e. monotonic and consistent choice functions. Such choice functions were introduced and used in the work \cite{RY} for investigation of matchings with complementary contracts. Three…
Polynomial ensembles are determinantal point processes associated with (non necessarily orthogonal) projections onto polynomial subspaces. The aim of this survey article is to put forward the use of recurrence coefficients to obtain the…
In this paper some classes of local polynomial functions on abelian groups are characterized by the properties of their variety. For this characterization we introduce a numerical quantity depending on the variety of the local polynomial…
Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts.…
We study an infinite class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. This generalizes a sequence of sparse polynomials which arises in a natural way as graph theoretic…
We propose a general method for optimization with semi-infinite constraints that involve a linear combination of functions, focusing on the case of the exponential function. Each function is lower and upper bounded on sub-intervals by…
Assume that X is a metrizable separable space, and each clopen-valued lower semicontinuous multivalued map Phi from X to Q has a continuous selection. Our main result is that in this case, X is a sigma-space. We also derive a partial…
Given a univariate polynomial, its abscissa is the maximum real part of its roots. The abscissa arises naturally when controlling linear differential equations. As a function of the polynomial coefficients, the abscissa is H{\"o}lder…
The problem "A general characterization of uniqueness polynomial for non-critically injective polynomials" has been remained open since the last two decades. In this paper, we explore this open problem. To this end, we initiate a new…
We prove that if nonlinear complex polynomials of the same degree have orbits with infinite intersection, then the polynomials have a common iterate. We also prove a special case of a conjectured dynamical analogue of the Mordell-Lang…
We present counting methods for some special classes of multivariate polynomials over a finite field, namely the reducible ones, the s-powerful ones (divisible by the s-th power of a nonconstant polynomial), and the relatively irreducible…