Related papers: Generalized Hurwitz matrices, generalized Euclidea…

Let $F := (f_1, \ldots, f_p) \colon {\Bbb R}^n \to {\Bbb R}^p$ be a polynomial map, and suppose that $S := \{x \in {\Bbb R}^n \ : \ f_i(x) \le 0, i = 1, \ldots, p\} \ne \emptyset.$ Let $d := \max_{i = 1, \ldots, p} \deg f_i$ and…

Let $\mathbb{F}$ be a field. We show that given any $n$th degree monic polynomial $q(x)\in \mathbb{F}[x]$ and any matrix $A\in\mathbb{M}_n(\mathbb{F})$ whose trace coincides with the trace of $q(x)$ and consisting in its main diagonal of…

We present a polynomial partitioning theorem for finite sets of points in the real locus of an irreducible complex algebraic variety of codimension at most two. This result generalizes the polynomial partitioning theorem on the Euclidean…

We prove expressions for the inequalities in Hermite's theorem which are conditions for a real polynomial to have real zeros. These expressions generalize the discriminant of a quadratic polynomial and the expression of J. Mar\'ik for a…

A key property of an algebraic variety is whether it is absolutely irreducible, meaning that it remains irreducible over the algebraic closure of its defining field, and determining absolute irreducibility is important in algebraic geometry…

Let $H$ be a Hardy field (a field consisting of germs of real-valued functions at infinity that is closed under differentiation) and let $f \in H$ be a subpolynomial function. Let $\mathcal{P} = \{2, 3, 5, 7, \dots \}$ be the (naturally…

We classify certain $\mathbb{Z}_2 $-graded extensions of generalized Haagerup categories in terms of numerical invariants satisfying polynomial equations. In particular, we construct a number of new examples of fusion categories, including:…

We study "pure-cycle" Hurwitz spaces, parametrizing covers of the projective line having only one ramified point over each branch point. We start with the case of genus-0 covers, using a combination of limit linear series theory and group…

Let $A$ be a matrix with nonnegative real entries. A nonnegative factorization of size $k$ is a representation of $A$ as a sum of $k$ nonnegative rank-one matrices. The space of all such factorizations is a bounded semialgebraic set, and we…

We introduce new logarithmic Hurwitz spaces $\mathcal{LH}^{\mathbb{Z}_{(p)}}_A$ and $\mathcal{LH}^{\mathbb{F}_{p}}_{A,\Xi}$ over $\mathbb{Z}_{(p)}$ and $\mathbb{F}_p$ respectively that in the mixed characteristic case can be considered as a…

In this article, for positive integers $n\geq m\geq 1$, the parameter spaces for the isomorphism classes of the generic point arrangements of cardinality $n$, and the antipodal point arrangements of cardinality $2n$ in the Eulidean space…

Let $M^n\subset\mathbb R^{n+1}$ be the graph of a $C^2$-real valued function defined in a closed ball of $\mathbb R^n$. In this work, we obtain upper bounds for $\inf_M|H|$ and $\inf_M|R|$, where $H$ and $R$ are, respectively, the mean…

Let $\mu$ be a probability measure on $\mathbb{Z}$ that is not a Dirac mass and that has finite support. We prove that if the coefficients of a monic polynomial $f(x)\in\mathbb{Z}[x]$ of degree $n$ are chosen independently at random…

We prove an effective form of Hilbert's irreducibility theorem for polynomials over a global field $K$. More precisely, we give effective bounds for the number of specializations $t\in \mathcal{O}_K$ that do not preserve the irreducibility…

In the paper [E. Jim\'enez-Fern\'andez, J. Rodr\'{\i}guez-L\'opez, E. A. S\'anchez-P\'erez, Fuzzy Sets and Systems 406 (2021),66-81], a McShane-Whitney extension theorem is presented for real-valued fuzzy Lipschitz maps between fuzzy metric…

We consider two applications of the strata of differentials of the second kind (all residues equal to zero) with fixed multiplicities of zeros and poles: Positivity: In genus $g=0$ we show any associated divisorial projection to…

We show that the signed counts of normalized real polynomials, as defined by Itenberg and Zvonkine, provide the signed counts of genus zero real ramified coverings of the Riemann sphere with a point of total ramification and several other…

We give conditions for the monodromy group of a Hurwitz space over the configuration space of branch points to be the full alternating or symmetric group on the degree. Specializing the resulting coverings suggests the existence of many…

We study the irreducibility of Wronskian Hermite polynomials labelled by partitions. It is known that these polynomials factor as a power of x times a remainder polynomial. We show that the remainder polynomial is irreducible for the…

In 1891, Hurwitz introduced the enumeration of genus $g$, degree $d$, branched covers of the Riemann sphere with simple ramification over prescribed points and no branching elsewhere. He showed that for fixed degree $d$, the enumeration…