We apply and extend some well known and some recent tech niques from algebraic residue theory in order to relate to each other two major subjects of algebraic and numerical computing that is computa tions with structured matrices and solving a system of polynomial equa tions In the rst part of our paper we extend the Toeplitz and Hankel structures of matrices and some of their known properties ...

We report on a case study on combining proof planning with computer algebra systems. We construct proofs for basic algebraic properties of residue classes as well as for isomorphisms between residue classes using di erent proving techniques, which are implemented as strategies in a multi-strategy proof planner. We show how these techniques help to successfully derive proofs in our domain and ex...

For any prime $p$, let $y(p)$ denote the smallest integer $y$ such that every reduced residue class $\pmod p$ is represented by product of some subset $\{1,\dots,y\}$. It easy to see at least as large quadratic nonresidue p$; we prove $y(p) \ll_\varepsilon p^{1/(4 \sqrt e)+\varepsilon}$, thus strengthening Burgess's classical result. This result intermediate strength between two other results, ...