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Rolle Models in the Real and Complex World
Numerous problems of analysis (real and complex) and geometry (analytic, algebraic, Diophantine e.a.) can be reduced to calculation of the “number of...
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Upper bounds for the number of isolated critical points via the Thom–Milnor theorem
We apply the Thom–Milnor theorem to obtain the upper bounds on the amount of isolated (1) critical points of a potential generated by several fixed...
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Recent Developments
In this short chapter, a number of important developments and advances are summarized that mostly occurred after the 1989 publication of Einführung...
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Beyond this book
In this final chapter, we outline some of the natural directions of further study for a reader of this book, and point out a few interesting recent...
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On the Index of the Gradient of a Real Invertible Polynomial
AbstractWe present a number of observations concerning the so-called invertible polynomials introduced and studied in a series of papers on...
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A Survey on Computational Aspects of Polynomial Amoebas
This article is a survey on the topic of polynomial amoebas. We review results of papers written on the topic with an emphasis on its computational...
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Some open problems in low dimensional dynamical systems
The aim of this paper is to share with the mathematical community a list of 33 problems that I have found along the years in my research. I believe...
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Continuity of the Asymptotics of Expected Zeros of Fewnomials
In “Random complex fewnomials, I,” the limiting formula of the (normalized) expected distribution of complex zeros of a system of k random m -variate...
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Composite factors of binomials and linear systems in roots of unity
In this paper we completely classify binomials in one variable which have a nontrivial factor which is composite, i.e., of the shape g ( h ( x )) for...
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Upper Bounds on Betti Numbers of Tropical Prevarieties
We prove upper bounds on the sum of Betti numbers of tropical prevarieties in dense and sparse settings. In the dense setting the bound is in terms...
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Zero distribution of random polynomials
We study global distribution of zeros for a wide range of ensembles of random polynomials. Two main directions are related to almost sure limits of...
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Complexity of Sparse Polynomial Solving: Homotopy on Toric Varieties and the Condition Metric
This paper investigates the cost of solving systems of sparse polynomial equations by homotopy continuation. First, a space of systems of n -variate...
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30 years of collaboration
We highlight some of the most important cornerstones of the long standing and very fruitful collaboration of the Austrian Diophantine Number Theory...
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Hölder-Type Global Error Bounds for Non-degenerate Polynomial Systems
Let F := ( f 1 , …, f p ): ℝ n → ℝ p be a polynomial map, and suppose that S := { x ∈ ℝ n : f i ( x ) ≤ 0, i = 1, …, p }≠ ∅ . Let d := max i =1, …, p deg f ...
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Multiplicity operators
For functions of a single complex variable, zeros of multiplicity greater than k are characterized by the vanishing of the first k derivatives. There...
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Multiplicities of Noetherian Deformations
The Noetherian class is a wide class of functions defined in terms of polynomial partial differential equations. It includes functions appearing...
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Pfaffian intersections and multiplicity cycles
We consider the problem of estimating the intersection multiplicity between an algebraic variety and a Pfaffian foliation, at every point of the...