Torben Grabbel | Searcharxiv

arXiv2026

The Ehrlich--Aberth root finder is not generally convergent

We show that the well-known Ehrlich--Aberth root finding method is not generally convergent. More precisely, we show that there is an open subset of polynomials of any degree $d\ge 7$, and an open subset of initial conditions, such that the iteration of the Ehrlich--Aberth method converges to an attracting periodic cycle of period $4$, rather than to a vector of roots. The proof is constructive in the sense that we present explicit values for the polynomial and the $4$-cycle. The search is highly non-trivial: we found no attracting cycles for lower degrees/periods, and the size of the attracting domain relative to a ``unit cube'' in the space of polynomials and starting vectors is on the order of $10^{-127}$.

math.NA↗

arXiv2026

On symmetries of hyperbolic lattices of large rank

For an even, integral hyperbolic lattice $L$, the symmetry group of $L$ is the quotient of the group of isometries of $L$ by the Weyl subgroup of $(-2)$-reflections. Following Nikulin, the exceptional lattice of $L$ is defined as the sublattice generated by elements that have finite orbit under the symmetry group of $L$. We prove that every hyperbolic lattice of rank at least $46$ has trivial exceptional lattice. In particular, every such lattice admits a symmetry of maximal Salem degree.

math.NT↗