Related papers: Signed counts of real simple rational functions
A compactly supported distribution is called invertible in the sense of Ehrenpreis-H\"ormander if the convolution with it induces a surjection from $\mathcal{C}^{\infty}(\mathbb{R}^{n})$ to itself. We give sufficient conditions for radial…
We discuss two elementary constructions for covers with fixed ramification in positive characteristic. As an application, we compute the number of certain classes of covers between projective lines branched at 4 points and obtain…
We study rational functions admitting a continuous extension to the real affine space. First of all, we focus on the regularity of such functions exhibiting some nice properties of their partial derivatives. Afterwards, since these…
We study the asymptotic behaviour of random factorizations of the $n$-cycle into transpositions of fixed genus $g>0$. They have a geometric interpretation as branched covers of the sphere and their enumeration as Hurwitz numbers was…
Double Hurwitz numbers count covers of the projective line by genus g curves with assigned ramification profiles over 0 and infinity, and simple ramification over a fixed branch divisor. Goulden, Jackson and Vakil have shown double Hurwitz…
This article shows that counting or computing the small eigenvalues of the Witten Laplacian in the semi-classical limit can be done without assuming that the potential is a Morse function as the authors did in [LNV]. In connection with…
In this article, to each generic real meromorphic function (i.e., having only simple branch points in the appropriate sense) we associate a certain combinatorial gadget which we call the park of a function. We show that the park determines…
We use generating functions over group rings to count polynomials over finite fields with the first few coefficients prescribed and a factorization pattern prescribed. In particular, we obtain different exact formulas for the number of…
Various methods can obtain certified estimates for roots of polynomials. Many applications in science and engineering additionally utilize the value of functions evaluated at roots. For example, critical values are obtained by evaluating an…
Double Hurwitz numbers have at least four equivalent definitions. Most naturally, they count covers of the Riemann sphere by genus g curves with certain specified ramification data. This is classically equivalent to counting certain…
In this paper, we study a problem that is in a sense a reversal of the Hurwitz counting problem. The Hurwitz problem asks: for a generic target -- $\mathbb P^1$ with a list of $n$ points $q_1,\dots,q_n\in \mathbb P^1$ -- and partitions…
HMC sets are hereditarily at most countable sets. We rework a substantial part of univariate real analysis in a form in which only HMC real functions are used. In such countable real analysis we carry out Hilbert's proof of transcendence of…
This article explores the model-dependent nature of set cardinality, emphasizing that cardinality is not absolute but varies across different axiomatic frameworks. Although Cantor's diagonal argument shows the real numbers are…
Entrywise functions preserving Loewner positivity have been studied by many authors, most notably Schoenberg and Rudin. Following their work, it is known that functions preserving positivity when applied entrywise to positive semidefinite…
Euler's totient function, $\varphi(n)$, which counts how many of $0,1,\dots,n-1$ are coprime to $n$, has an explicit asymptotic lower bound of $n/\log \log n$, modulo some constant. In this note, we generalise $\varphi$; given an…
In this note we generalize and prove a recent conjecture of Varchenko concerning the number of critical points of a (multivalued) meromorphic function $\phi$ on an algebraic manifold. Under certain conditions, this number turns out to…
We develop several notions of multiplicity for linear factors of multivariable polynomials over different arithmetics (hyperfields). The key example is multiplicities over the hyperfield of signs, which encapsulates the arithmetic of…
To answer a question by Rettich and Serafin, we define a model of set theory in which there exists a locally countable $\varPi^1_2$ graph on a subset of the real line, which is not generated by a countable family of projective (or even…
We consider the problem of finding the set of classical polylogarithmic functions $\text{Li}_n$ with branching locus determined by the solution of $p_1\cdot p_2\cdot \ldots \cdot p_n=0$, where $p_1,\ldots, p_n$ are irreducible polynomials…
We resolve a conjecture of Albert and Bousquet-Melou enumerating quarter-plane walks with fixed horizontal and vertical projections according to their upper-right-corner count modulo 2. In doing this, we introduce a signed…