arXiv subjects

Dragomir Grozev

Publications and source records attributed to Dragomir Grozev.

3 recordsLinked to original sources

arXiv2026

Weighted Generalizations of Zagier's Phenomenon

We study sums associated with the action of $PGL_2(\mathbb Z)$ on continuous piecewise polynomial functions with exactly two real roots, both irrational. We form weighted sums of the positive parts of their normalized transforms, using nonnegative weights compatible with translation, reflection, and inversion. Under suitable continuity, finiteness, and convergence assumptions, we prove that these sums are well defined, bounded, $1$-periodic, and continuous on $\mathbb R$, and satisfy a reciprocal functional equation. We also extend the construction to finite families of distinct function orbits. Our framework recovers Zagier's constancy result and includes the full family of quadratic sums for which Bengoechea proved convergence.

math.NT↗

arXiv2026

A Note on the Converse Sendov Problem

For a polynomial of degree $n$ whose zeros lie in the closed unit disk, we determine the largest possible distance from a prescribed critical point of modulus $r$ to the nearest zero. If $n$ is even, the sharp radius is $\sqrt{1-r^2}$; if $n$ is odd, the sharp radius is strictly smaller for $r\in(0,1)$ and depends on $n$. Equality cases are also determined. The proof is based on the logarithmic-derivative identity and elementary geometric considerations.

math.CV↗

arXiv2026

On Ulam's Segment Motion Problem

We study extremal rigid motions of a unit segment in $\mathbb{R}^d$, $d\ge 2$. Given two prescribed positions of a unit segment, we consider continuous motions transforming the initial position into the final one and investigate the total length of the trajectories traced by its endpoints. This minimization problem was posed by Ulam~\cite{Ulam1960} and solved by Gurevich~\cite{Gurevich1977} and Dubovitskii~\cite{Dubovitskii1976}. Two natural lower bounds are given by the sum of the endpoint displacements and by the angle between the initial and final directions of the segment. We characterize all pairs of segment positions for which either of these lower bounds is attained. In arbitrary dimension, we obtain complete characterizations of the equality cases for both the endpoint-displacement bound and the angular bound.

math.MG↗