Polypols are natural generalizations of polytopes, with boundaries given by non-linear algebraic hypersurfaces.We describe polypols in the plane and in 3-space that admit a unique adjoint hypersurface and study them from an algebro-geometric perspective. We relate planar polypols to positive geometries introduced originally in particle physics, and identify the adjoint curve of a planar polypol with the numerator of the canonical differential form associated with the positive geometry.We settle several cases of a conjecture by Wachspress claiming that the adjoint curve of a regular planar polypol does not intersect its interior. In particular, we provide a complete characterization of the real topology of the adjoint curve for arbitrary convex polygons. Finally, we determine all types of planar polypols such that the rational map sending a polypol to its adjoint is finite, and explore connections of our topic with algebraic statistics.

Given a linear ordinary differential operator T with polynomial coefficients, we study the class of closed subsets of the complex plane such that T sends any polynomial (respectively, any polynomial of degree exceeding a given positive integer) with all roots in a given subset to a polynomial with all roots in the same subset or to 0. Below we discuss some general properties of such invariant subsets, as well as the problem of existence of the minimal under inclusion invariant subset.

We initiate the study of a natural generalisation of the classical Bochner-Krall problem asking which linear ordinary differential operators possess sequences of eigenpolynomials satisfying linear recurrence relations of finite length; the classical case corresponds to the 3-term recurrence relations with real coefficients subject to some extra restrictions. We formulate a general conjecture and prove it in the first non-trivial case of operators of order 3. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

In the late 80s, V. Arnold and V. Vassiliev initiated the topological study of the space of real univariate polynomials of a given degree d and with no real roots of multiplicity exceeding a given positive integer. Expanding their studies, we consider the spaces of real monic univariate polynomials of degree d whose real divisors avoid sequences of root multiplicities, taken from a given poset Θ of compositions which is closed under certain natural combinatorial operations. In this paper, we concentrate on the fundamental group of and of some related topological spaces. We find explicit presentations for the groups π₁() in terms of generators and relations and show that in a number of cases they are free with rank bounded from above by a quadratic function in d. We also show that π₁(PcΘd) stabilizes for d large. The mechanism that generates π₁() has similarities with the presentation of the braid group as the fundamental group of the space of complex monic degree d polynomials with no multiple roots and with the presentation of the fundamental group of certain ordered configuration spaces over the reals which appear in the work of Khovanov. We further show that the groups π₁() admit an interpretation as special bordisms of immersions of one-manifolds into the cylinder , whose images avoid the tangency patterns from Θ with respect to the generators of the cylinder.

We consider the evolutes of plane real-algebraic curves and discuss some of their complex and real-algebraic properties. In particular, for a given degree d ⩾ 2, we provide lower bounds for the following four numerical invariants:

• (1) the maximal number of times a real line can intersect the evolute of a real-algebraic curve of degree d;
• (2) the maximal number of real cusps which can occur on the evolute of a real-algebraic curve of degree d;
• (3) the maximal number of crunodes which can occur on the dual curve to the evolute of a real-algebraic curve of degree d;
• (4) the maximal number of crunodes which can occur on the evolute of a real-algebraic curve of degree d.

Nous considérons les développées des courbes algébriques réelles planes et discutons certaines de leurs propriétés dans le cas complexe et réel. En particulier, pour un degré donné d ⩾ 2, nous fournissons des bornes inférieures pour les quatre invariants numériques suivants :

En particulier, pour un degré donné d ⩾ 2, nous fournissons des bornes inférieures pour les quatre invariants numériques suivants :

• (1) le nombre maximal de fois qu’une droite réelle peut intersecter la développée d’une courbe algébrique réelle de degré d ;
• (2) le nombre maximal de points de rebroussement réels pouvant survenir sur la développée d’une courbe algébrique réelle de degré d ;
• (3) le nombre maximal de points double ordinaires pouvant survenir sur la courbe duale à la développée d’une courbe algébrique réelle de degré d ;
• (4) le nombre maximal de points double ordinaires pouvant survenir sur la développée d’une courbe algébrique réelle de degré d.

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 counterexamples to two earlier conjectures refining Descartes rule of signs.

In this paper, we initiate the study of a new interrelation between linear ordinary differential operators and complex dynamics which we discuss in detail in the simplest case of operators of order 1. Namely, assuming that such an operator T has polynomial coefficients, we interpret it as a continuous family of Hutchinson operators acting on the space of positive powers of linear forms. Using this interpretation of T, we introduce its continuously Hutchinson invariant subsets of the complex plane and investigate a variety of their properties. In particular, we prove that for any T with non-constant coefficients, there exists a unique minimal under inclusion invariant set M^T_CH and find explicitly what operators T have the property that M^T_CH=C.

Motivated by the classical Rodrigues' formula, we study below the root asymptotic of the polynomial sequence R-[alpha n],R-n,R-P(z) = d([alpha n]) P-n(z)/dz([alpha n]), n = 0, 1, ... where P(z) is a fixed univariate polynomial, alpha is a fixed positive number smaller than deg P, and [alpha n] stands for the integer part of alpha n. Our description of this asymptotic is expressed in terms of an explicit harmonic function uniquely determined by the plane rational curve emerging from the application of the saddle point method to the integral representation of the latter polynomials using Cauchy's formula for higher derivatives. As a consequence of our method, we conclude that this curve is birationally equivalent to the zero locus of the bivariate algebraic equation satisfied by the Cauchy transform of the asymptotic root-counting measure for the latter polynomial sequence. We show that this harmonic function is also associated with an abelian differential having only purely imaginary periods and the latter plane curve belongs to the class of Boutroux curves initially introduced in Bertola (Anal Math Phys 1: 167-211, 2011), Bertola and Mo(Adv Math 220(1): 154-218, 2009). As an additional relevant piece of information, we derive a linear ordinary differential equation satisfied by {R-[alpha n],R-n,R- P(z)} as well as higher derivatives of powers of more general functions.

The Grassmann convexity conjecture, formulated in [8], gives a conjectural formula for the maximal total number of real zeroes of the consecutiveWronskians of an arbitrary fundamental solution to a disconjugate linear ordinary differential equation with real time. The conjecture can be reformulated in terms of convex curves in the nilpotent lower triangular group. The formula has already been shown to be a correct lower bound and to give a correct upper bound in several small dimensional cases. In this paper we obtain a general explicit upper bound.

The isodynamic points of a plane triangle are known to be the only pair of its centers invariant under the action of the Möbius group M on the set of triangles, Kimberling (Encyclopedia of Triangle Centers, http://faculty.evansville.edu/ck6/encyclopedia). Generalizing this classical result, we introduce below the isodynamic map associating to a univariate polynomial of degree d≥3 with at most double roots a polynomial of degree (at most) 2d−4 such that this map commutes with the action of the Möbius group M on the zero loci of the initial polynomial and its image. The roots of the image polynomial will be called the isodynamic points of the preimage polynomial. Our construction naturally extends from univariate polynomials to binary forms and further to their ratios.