This survey presents recent Helly-type geometric theorems published since the appearance of the last comprehensive survey, more than ten years ago. We discuss how such theorems continue to be influential in computational geometry and in optimization.

Helly's Theorem: New Variations and Applications

We study separating algebras for rings of invariants of finite groups. We describe a separating subalgebra for invariants of p-groups in characteristic p using only transfers and norms. Also we give an explicit construction of a separating set for invariants of groups acting diagonally.

https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/2009/0016/0006/MRL-2009-0016-0006-a011.pdf

Separating invariants for modular p-groups and groups acting diagonally

We consider an arbitrary representation of the additive group Ga over a field of characteristic zero and give an explicit description of a finite separating set in the corresponding ring of invariants.

/pdf/separating-invariants-for-arbitrary-linear-actions-of-the-2zkjtw4nqt.pdf

Separating invariants for arbitrary linear actions of the additive group

It is proved that the universal degree bound for separating polynomial invariants of a finite abelian group (in non-modular characteristic) is typically strictly smaller than the universal degree bound for generators of polynomial invariants. More precisely, these degree bounds can be equal only if the group is cyclic or is the direct sum of r even order cyclic groups where the number of two-element direct summands is not less than the integer part of the half of r. A characterization of separating sets of monomials is given in terms of zero-sum sequences over abelian groups. © 2017 American Mathematical Society.

/pdf/degree-bound-for-separating-invariants-of-abelian-groups-3ibtncyy22.pdf

Degree bound for separating invariants of abelian groups

It is proven that for any representation over a field of characteristic $$0$$
 of the non-abelian semidirect product of a cyclic group of prime order $$p$$
 and the group of order $$3$$
 the corresponding algebra of polynomial invariants is generated by elements of degree at most $$p+2$$
 . We also determine the exact universal degree bound for separating systems of polynomial invariants of this group in characteristic not dividing $$3p$$
 .

The Noether number of the non-abelian group of order 3p

It is shown that for a linear algebraic group G over a field of characteristic zero, there is a natural number \kappa(G) such that if a system of Zariski closed cosets in G has empty intersection, then there is a subsystem consisting of at most \kappa(G) cosets with empty intersection. This is applied to the study of algebraic group actions on product varieties.

Mátyás Domokos

Papers

Helly dimension of algebraic groups

Helly dimension of algebraic groups