Related papers: On equations defining coincident root loci

We give three algebraic equations which allow a geometric classification of all spectral types of equilibria of a given $m$-dimensional dynamical system, and we analyse them thoroughly in dimension 3 and 4. The loci defined by these…

We introduce some natural families of distributions on rooted binary ranked plane trees with a view toward unifying ideas from various fields, including macroevolution, epidemiology, computational group theory, search algorithms and other…

We consider extremal problems related to decks and multidecks of rooted binary trees (a.k.a. rooted phylogenetic tree shapes). Here, the deck (resp. multideck) of a tree $T$ refers to the set (resp. multiset) of leaf induced binary subtrees…

In this paper, we prove a number of results providing either necessary or sufficient conditions guaranteeing that the number of real roots of real polynomials of a given degree is either less or greater than a given number. We also provide…

We introduce augmented and restricted base loci of cycles and we study the positivity properties naturally defined by these base loci.

For the general monic cubic and quartic with real coefficients, polynomial conditions on the coefficients are derived as directly and as simply as possible from the Sturm sequence that will determine the real and complex root multiplicities…

We determine the multiplicities of a class of roots for Nichols algebras of diagonal type of rank two, and identify the corresponding root vectors. Our analysis is based on a precise description of the relations of the Nichols algebra in…

Conditionals play a key role in different areas of logic and probabilistic reasoning, and they have been studied and formalized from different angles. In this paper we focus on the de Finetti's notion of conditional as a three-valued…

Working over the split octonions over an algebraically closed field, we solve all polynomial equations in which all the coefficients but the constant term are scalar. As a consequence, we calculate the n-th roots of an octonion.

We show that every affine or projective algebraic variety defined over the field of real or complex numbers is homeomorphic to a variety defined over the field of algebraic numbers. We construct such a homeomorphism by choosing a small…

We investigate the notion of associated graded coalgebra (algebra) of a bialgebra with respect to a subbialgebra (quotient bialgebra) and characterize those which are bialgebras of type one in the framework of abelian braided monoidal…

We study the combinatorics of hyperplane arrangements over arbitrary fields. Specifically, we determine in which situation an arrangement and its reduction modulo a prime number have isomorphic lattices via the use of minimal strong…

We study approximations of compact linear multivariate operators defined over Hilbert spaces. We provide necessary and sufficient conditions on various notions of tractability. These conditions are mainly given in terms of sums of certain…

The analysis of solutions to algebraic equations is further simplified. A couple of functions and their analytic continuation or root findings are required.

Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…

We study quasilinear Beltrami equations, the complex coefficients of which depend on the unknown function. In terms of the so-called tangential dilatation, we have found conditions under which these equations have homeomorphic…

The Hilbert functions and the regularity of the graded components of local cohomology of a bigraded algebra are considered. Explicit bounds for these invariants are obtained for bigraded hypersurface rings.

We formulate several polynomial identities. One side of these identities has a nice simple form. Whereas the other has a form of a polynomial whose coefficients contain binomial coefficients double factorials or (and) rising factorials. The…

The conditions for cubic equations, to have 3 real roots and 2 of the roots lie in the closed interval $[-1, 1]$ are given. These conditions are visualized. This question arises in physics in e.g. the theory of tops.

We study a generalization of conditional probability for arbitrary ordered vector spaces. A related problem is that of assigning a numerical value to one vector relative to another. We characterize the groups for which these generalized…